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Pythagorean Theorem and Problems with Solutions

Explore some simple proofs of the Pythagorean theorem and its converse and use them to solve problems. Detailed solutions to the problems are also presented.

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Pythagorean Theorem – Definition, Formula, Problems

In mathematic, the Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its other two sides . Another way of stating the theorem is that the sum of the areas of the squares formed by the sides of a right triangle equals the area of the square whose side is the hypotenuse. The theorem is a key relation in Euclidean geometry. It is named for the Greek philosopher Pythagorus.

Remember: The Pythagorean theorem only applies to right triangles!

Pythagorean Theorem Formula

The formula for the Pythagorean theorem describes the relationship between the sides a and b of a right triangle to its hypotenuse, c . A right triangle is one containing a 90° or right angle. The hypotenuse is the side of the triangle opposite from the right angle (which is the largest angle in a right triangle).

a 2 + b 2 = c 2

Solving for a, b, and c

Rearranging the equation gives the formulas solving for a, b, and c:

• a = (c 2 – b 2 ) ½
• b = (c 2 – a 2 ) ½
• c = (a 2 + b 2 ) ½

How to Solve the Pythagorean Theorem – Example Problems

For example, find the hypotenuse of a right triangle with side that have lengths of 5 and 12.

Start with the formula for the Pythagorean theorem and plug in the numbers for the sides a and b to solve for c .

a 2 + b 2 = c 2 5 2 + 12 2 = c 2 c 2 = 5 2 + 12 2 = 25 + 144 = 169 c2 = 169 c = √169 or 169 ½ = 13

For example, solve for side b of a triangle where a is 9 and the hypotenuse c is 15.

a 2 + b 2 = c 2 9 2 + b 2 = 15 2 b 2 = 15 2 – 9 2 = 225 – 81 = 144 b = √144 = 12

Now, let’s combine a bit of algebra with the geometry. Solve for x where the sides of a right triangle are 5x and 4x +5 and the hypotenuse has a length of 8x -3.

a 2 + b 2 = c 2 (5x) 2 + (4x +5) 2 = (8x-3) 2

The (4x + 5) 2 and (8x -3)2 terms are the squares of binomial expressions. So, expanding the equation gives the following:

25x 2 + (4x +5)(4x +5) = (8x -3)(8x -3) 25x 2 = 16×2 + 20x +20x + 25 = 64x – 24x – 24x + 9

Combine like terms:

41x 2 + 40x + 25 = 64x 2 – 48x + 9

Rewrite the equation and solve for x.

0 = 23x 2 – 88x – 16

Apply the quadratic equation and solve for x:

x = [-b ± √(b 2 -4ac)]/2a x = [-(-88) ± √[-88 2 – 4(23)(-16)] / 2(23) = [88 ± √(7744 + 1472)] / 46 = (88 ± 96) / 46

So, there are two answers:

x = (88 + 96)/46 = 4 and (88 – 96).46 = -4/23

A triangle does not have a negative length for its side, so x is 4.

Plugging in”4″ in place of x, the sides of the right triangle are 20, 21, and 29.

Pythagorean Triples

Pythagorean triples are integers a, b, and c, that represent the sides of a right triangle and satisfy the Pythagorean theorem. Here is the list of Pythagorean triples for integers with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)

Proof of the Pythagorean Theorem

There are more proofs for the Pythagorean theorem than for any other theorem in geometry! At least 370 proofs are known. Some of these proofs use the parallel postulate. Some rely on the complementarity of acute angles in a right triangle. Proofs using shearing use the properties of parallelograms.

History – Did Pythagoras Discover the Pythagorean Theorem?

While the Pythagorean theorem takes its name from Pythagorus, he did not discover it. Exactly who gets the credit or whether many different places made the discovery independently is a matter of debate. The Mesopotamians made calculations using the formula as early as 2000 BC, which was over a thousand years before Pythagorus. A papyrus from the Egyptian Middle Kingdom, dating between 2000 and 1786 BC, references a math problem describing Pythagorean triples. The Baudhayana Shulba Sutra from India (dating between the 8th and 5th century BC) lists both Pythagorean triples and the Pythagorean theorem. The “Gougu theorem” from China offers a proof for the Pythagorean theorem, which came into use long before its oldest surviving description from the 1st century BC.

Pythagorus of Samos lived between 570 and 495 BC. While he was not the original person who formulated the Pythagorean theorem, he (or his students) may have introduced its proof to ancient Greece. In any case, his philosophical treatment of math left a lasting impression on the world.

• Bell, John L. (1999). The Art of the Intelligible: An Elementary Survey of Mathematics in its Conceptual Development . Kluwer. ISBN 0-7923-5972-0.
• Heath, Sir Thomas (1921). “ The ‘Theorem of Pythagoras ‘”. A History of Greek Mathematics (2 Vols.) (Dover Publications, Inc. (1981) ed.). Oxford: Clarendon Press. ISBN 0-486-24073-8.
• Maor, Eli (2007). The Pythagorean Theorem: A 4,000-Year History . Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-12526-8.
• Swetz, Frank; Kao, T. I. (1977). Was Pythagoras Chinese?: An Examination of Right Triangle Theory in Ancient China . Pennsylvania State University Press. ISBN 0-271-01238-2.

Related Posts

Module 11: Geometry

Using the pythagorean theorem to solve problems, learning outcomes.

• Use the pythagorean theorem to find the unknown length of a right triangle given the two other lengths

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[/latex] BCE.

Remember that a right triangle has a [latex]90^\circ [/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\circ [/latex] angle is called the hypotenuse, and the other two sides are called the legs. See the triangles below.

In a right triangle, the side opposite the [latex]90^\circ [/latex] angle is called the hypotenuse and each of the other sides is called a leg.

The Pythagorean Theorem

In any right triangle [latex]\Delta ABC[/latex],

[latex]{a}^{2}+{b}^{2}={c}^{2}[/latex]

where [latex]c[/latex] is the length of the hypotenuse [latex]a[/latex] and [latex]b[/latex] are the lengths of the legs.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation [latex]\sqrt{m}[/latex] and defined it in this way:

[latex]\text{If }m={n}^{2},\text{ then }\sqrt{m}=n\text{ for }n\ge 0[/latex]

For example, we found that [latex]\sqrt{25}[/latex] is [latex]5[/latex] because [latex]{5}^{2}=25[/latex].

We will use this definition of square roots to solve for the length of a side in a right triangle.

Use the Pythagorean Theorem to find the length of the hypotenuse.

Use the Pythagorean Theorem to find the length of the longer leg.

Kelvin is building a gazebo and wants to brace each corner by placing a [latex]\text{10-inch}[/latex] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

In the following video we show two more examples of how to use the Pythagorean Theorem to solve application problems.

• Question ID 146918, 146916, 146914, 146913. Authored by : Lumen Learning. License : CC BY: Attribution
• Solve Applications Using the Pythagorean Theorem (c only). Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/2P0dJxpwFMY . License : CC BY: Attribution
• Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Pythagorean Theorem

How to Use The Pythagorean Theorem

The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$ \overline{c}$$ is always the hypotenuse . Remember that this formula only applies to right triangles .

Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above. Look at the following examples to see pictures of the formula.

Conceptual Animation of Pythagorean Theorem

Demonstration #1.

More on the Pythagorean theorem

Demonstration #2

Video tutorial on how to use the pythagorean theorem.

Step By Step Examples of Using the Pythagorean Theorem

Example 1 (solving for the hypotenuse).

Use the Pythagorean theorem to determine the length of X.

Identify the legs and the hypotenuse of the right triangle .

The legs have length 6 and 8 . $$X $$ is the hypotenuse because it is opposite the right angle.

Substitute values into the formula (remember 'C' is the hypotenuse).

$ A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2 $

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X $

Example 2 (solving for a Leg)

The legs have length 24 and $$X$$ are the legs. The hypotenuse is 26.

$ \red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2 $

$ \red x^2 + 24^2= 26^2 \\ \red x^2 + 576= 676 \\ \red x^2 = 676 - 576 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10 $

Practice Problems

Find the length of X.

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse .

The legs have length 14 and 48 . The hypotenuse is X.

$ A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2 $

Solve for the unknown.

$ 14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x} $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest tenth.

Remember our steps for how to use this theorem. This problems is like example 2 because we are solving for one of the legs .

The legs have length 9 and X . The hypotenuse is 10.

$ A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2 $

$ 9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4 $

Use the Pythagorean theorem to calculate the value of X. Round your answer to the nearest hundredth.

The legs have length '10' and 'X'. The hypotenuse is 20.

$ A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2 $

$ 10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32 $

• Pythagorean Theorem Shell Problem Cool challenging problem from involving pythagoren theorem, isosceles triangles and pattern recognition
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• Pythagorean Theorem Calculator

Study Guides > Prealgebra

Using the pythagorean theorem to solve problems, learning outcomes.

• Use the pythagorean theorem to find the unknown length of a right triangle given the two other lengths

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[/latex] BCE. Remember that a right triangle has a [latex]90^\circ [/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\circ [/latex] angle is called the hypotenuse, and the other two sides are called the legs. See the triangles below.

The Pythagorean Theorem

Licenses & Attributions

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• Question ID 146918, 146916, 146914, 146913. Authored by: Lumen Learning. License: CC BY: Attribution .

CC licensed content, Specific attribution

• Prealgebra. Provided by: OpenStax License: CC BY: Attribution . License terms: Download for free at http://cnx.org/contents/ [email protected] .

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MathBootCamps

The pythagorean theorem with examples.

The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle. Even though it is written in these terms, it can be used to find any of the side as long as you know the lengths of the other two sides. In this lesson, we will look at several different types of examples of applying this theorem.

Table of Contents

• Examples of using the Pythagorean theorem
• Solving applied problems (word problems)
• Solving algebraic problems

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Applying the Pythagorean theorem (examples)

In the examples below, we will see how to apply this rule to find any side of a right triangle triangle. As in the formula below, we will let a and b be the lengths of the legs and c be the length of the hypotenuse. Remember though, that you could use any variables to represent these lengths.

In each example, pay close attention to the information given and what we are trying to find. This helps you determine the correct values to use in the different parts of the formula.

Find the value of \(x\).

The side opposite the right angle is the side labelled \(x\). This is the hypotenuse. When applying the Pythagorean theorem, this squared is equal to the sum of the other two sides squared. Mathematically, this means:

\(6^2 + 8^2 = x^2\)

Which is the same as:

\(100 = x^2\)

Therefore, we can write:

\(\begin{align}x &= \sqrt{100}\\ &= \bbox[border: 1px solid black; padding: 2px]{10}\end{align}\)

Maybe you remember that in an equation like this, \(x\) could also be –10, since –10 squared is also 100. But, the length of any side of a triangle can never be negative and therefore we only consider the positive square root.

In other situations, you will be trying to find the length of one of the legs of a right triangle. You can still use the Pythagorean theorem in these types of problems, but you will need to be careful about the order you use the values in the formula.

Find the value of \(y\).

The side opposite the right angle has a length of 12. Therefore, we will write:

\(8^2 + y^2 = 12^2\)

This is the same as:

\(64 + y^2 = 144\)

Subtracting 64 from both sides:

\(y^2 = 80\)

\(\begin{align}y &= \sqrt{80} \\ &= \sqrt{16 \times 5} \\ &= \bbox[border: 1px solid black; padding: 2px]{4\sqrt{5}}\end{align}\)

In this last example, we left the answer in exact form instead of finding a decimal approximation. This is common unless you are working on an applied problem.

Applications (word problems) with the Pythagorean theorem

There are many different kinds of real-life problems that can be solved using the Pythagorean theorem. The easiest way to see that you should be applying this theorem is by drawing a picture of whatever situation is described.

Two hikers leave a cabin at the same time, one heading due south and the other headed due west. After one hour, the hiker walking south has covered 2.8 miles and the hiker walking west has covered 3.1 miles. At that moment, what is the shortest distance between the two hikers?

First, sketch a picture of the information given. Label any unknown value with a variable name, like x.

Due south and due west form a right angle, and the shortest distance between any two points is a straight line. Therefore, we can apply the Pythagorean theorem and write:

\(3.1^2 + 2.8^2 = x^2\)

Here, you will need to use a calculator to simplify the left-hand side:

\(17.45 = x^2\)

Now use your calculator to take the square root. You will likely need to round your answer.

\(\begin{align}x &= \sqrt{17.45} \\ &\approx 4.18 \text{ miles}\end{align}\)

As you can see, it will be up to you to determine that a right angle is part of the situation given in the word problem. If it isn’t, then you can’t use the Pythagorean theorem.

Algebra style problems with the Pythagorean theorem

There is one last type of problem you might run into where you use the Pythagorean theorem to write some type of algebraic expression. This is something that you will not need to do in every course, but it does come up.

A right triangle has a hypotenuse of length \(2x\), a leg of length \(x\), and a leg of length y. Write an expression that shows the value of \(y\) in terms of \(x\).

Since no figure was given, your first step should be to draw one. The order of the legs isn’t important, but remember that the hypotenuse is opposite the right angle.

Now you can apply the Pythagorean theorem to write:

\(x^2 + y^2 = (2x)^2\)

Squaring the right-hand side:

\(x^2 + y^2 = 4x^2\)

When the problem says “the value of \(y\)”, it means you must solve for \(y\). Therefore, we will write:

\(y^2 = 4x^2 – x^2\)

Combining like terms:

\(y^2 = 3x^2\)

Now, use the square root to write:

\(y = \sqrt{3x^2}\)

Finally, this simplifies to give us the expression we are looking for:

\(y = \bbox[border: 1px solid black; padding: 2px]{x\sqrt{3x}}\)

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The Pythagorean theorem allows you to find the length of any of the three sides of a right triangle. It is one of those things that you should memorize, as it comes up in all areas of math, and therefore in many different math courses you will probably take. Remember to avoid the common mistake of mixing up where the legs go in the formula vs. the hypotenuse and to always draw a picture when one isn’t given.

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15 Pythagorean Theorem Practice Problems For 8th Grade

Beki christian.

Pythagorean Theorem practice problems involve using the relationship between the sides of a right triangle to calculate missing side lengths in triangles. The Pythagorean Theorem is introduced in 8th grade and is used to solve a variety of problems across high school.

Here, you’ll find a selection of Pythagorean Theorem questions that demonstrate the different types of questions students are likely to encounter in 8th grade.

What is the Pythagorean Theorem?

How to answer pythagorean theorem questions, pythagorean theorem in real life, pythagorean theorem in 8th grade, pythagorean theorem practice problems.

The Pythagorean Theorem is the geometric theorem that states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the two shorter sides of the triangle.

This can be written as a^2+b^2=c^2 for a triangle labeled like this:

15 Pythagoras Theorem Practice Problems

Wish you could have the 15 Pythagoras Theorem questions from this blog in a ready-to-go worksheet? We've done just that!

1 – Label the sides of the triangle a , b , and c . Note that the hypotenuse, the longest side of a right triangle, is opposite the right angle and will always be labeled .

2 – Write down the formula and substitute the values>

3 – Calculate the answer. You may be asked to give your answer in an exact form or round to a given degree of accuracy, such as a certain number of decimal places or significant figures.

Pythagorean Theorem has many real-life uses, including in architecture and construction, navigation and surveying.

Pythagorean Theorem is usually introduced in middle school, as it is a part of the 8th grade Common Core Math Standards.

The emphasis in middle school is on students being able to:

• Explain the Pythagorean Theorem;
• Use the theorem to solve mathematical and real-world problems – with both 2D and 3D figures;
• Use the theorem to calculate the distance between two points on a coordinate grid.

The process for solving any Pythagoras Theorem problem always begins by identifying the relevant right-angled triangle and labeling the sides a , b , c. If there is not a diagram in the question, it can be helpful to draw one.

Where necessary, round your answers to 3 significant figures.

1. A ship sails 6 \, km East and then 8 \, km North. Find the ship’s distance from its starting point.

The ship is 10 kilometers from its starting point.

2. A ladder is 5 \, m long. The base of the ladder is 3 \, m from the base of a vertical wall. How far up the wall does the ladder reach?

The ladder reaches 4 meters up the wall.

3. Alex and Sam start from the same point. Alex walks 400 meters west. Sam walks x meters south, until they are 600 \, m apart from each other. How far does Sam walk?

4. A television’s size is the measurement from the upper left hand corner of the television to the bottom right hand corner. Find the size of this television.

39.7 inches

55.1 inches

5. The pole of a sailing boat is supported by a rope from the top of the pole to an anchor point on the deck. The pole is 4 \, m long and the rope is 4.5 \, m long. Calculate the distance from the base of the pole to the anchor point of the rope on the deck.

6. Work out the length of the diagonal of a square with 8 \, cm sides.

The diagonal of the square has a length of 11.3 centimeters.

7. ABC is an isosceles triangle.

Work out the height of the triangle.

8. ABCD is an isosceles trapezoid.

Work out the length of AD.

9. Here is a cm square grid. Calculate the distance between the points A and B.

10. Which is a right angled triangle?

Not a right angled triangle because Pythagorean Theorem doesn’t work.

Right angled triangle because Pythagorean Theorem works.

11. PQRS is made from two right angled triangles.

Work out the length of QR.

Triangle \text{PQS:}

Triangle \text{QRS}

12. Here is a pattern made from right angled triangles. Work out the length x.

Triangle \text{ABC:}

Triangle \text{ACD:}

13. Here is a pyramid.

Work out the height of the pyramid.

14. Here is a cuboid.

Work out the length AG.

Give your answer in its exact form.

Length of \text{BG:}

Length of \text{AG:}

15. Here is a right angled triangle.

Form an equation and use it to work out the value of x.

x=4 \, or \, x=12

x cannot be 4 as you can’t have a negative side length so x=12

In middle school, students…

• prove the Pythagorean Theorem;
• use the Pythagorean Theorem with trigonometric ratios to solve problems;
• use the Pythagorean Theorem in proofs.

Pythagoras Theorem may feature in questions alongside other topics, such as trigonometry, circle theorems or algebra.

The Pythagorean Theorem is used to calculate a missing length in a right triangle . If you have a right angled triangle and you know two of the lengths, label the sides of the triangle a,b and c (c must be the hypotenuse – the longest side). Pythagorean Theorem is a^2+b^2=c^2. Substitute the values you know into Pythagorean Theorem and solve to find the missing side.

The hypotenuse of a right triangle is the longest side. If you know the lengths of the other two sides, you can find the length of the hypotenuse by squaring the two shorter sides, adding those values together and then taking the square root. By doing this you are finding c in a^2+b^2=c^2

If your triangle is a right triangle and you know two of the sides, you can use Pythagorean Theorem to find the length of the third side. To do this, label the sides a , b and c (with c being the hypotenuse – the longest side). Substitute the values you know into a^2+b^2=c^2 and solve to find the missing side.

Looking for more Pythagorean theorem math questions?

• Ratio questions
• Algebra questions
• Probability questions
• Trigonometry questions
• Venn diagram questions
• Long division questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by former UK Secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary and middle school teacher Kathleen Epperson.

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Pythagorean Theorem Calculator

How to use the pythagorean theorem calculator, what is the pythagorean theorem, how do i use pythagorean theorem, what is the hypotenuse formula, other considerations when dealing with triangles, the pythagorean theorem calculator in the making.

This Pythagorean theorem calculator will calculate the length of any of the missing sides of a right triangle, provided you know the lengths of its other two sides. This includes calculating the hypotenuse. The hypotenuse of the right triangle is the side opposite the right angle, and is the longest side. This side can be found using the hypotenuse formula, another term for the Pythagorean theorem when it's solving for the hypotenuse.

Recall that a right triangle is a triangle with an angle measuring 90 degrees. The other two angles must also total 90 degrees, as the sum of the measures of the angles of any triangle is 180. Read on to answer "what is the Pythagorean theorem and how is it used?"

The Pythagorean theorem calculator is one of the most accessible tools you will come across, despite the name being scary. All you need is any two of the three sides of a right triangle, and you are all set.

Let's take a look at the steps to use our Pythagorean theorem calculator.

Input leg a of the right triangle.

Next, input leg b of the right triangle.

And that is it. The tool will display the following results:

a. The hypotenuse (c) of the right triangle.

b. The area of the triangle.

c. And the perimeter of the triangle.

You don't necessarily have to input legs a and b. If you know the hypotenuse value and need to know any other legs, input the information accordingly.

The default unit is centimeter (cm) for length and cm² for area. You can change these to one of the listed of options according to your requirements. Remember to change the unit before you input the values.

How about we try to understand an example ? For instance, you are climbing a ladder to your roof, and you get curious about the length of the ladder . You notice that the distance from the roof to the ground is leg a , and the distance from the wall to the ladder's base is leg b . And the ladder itself is the hypotenuse. You input the measurements in the tool as leg a and b, 4 m and 3 m , respectively. (Remember to change the units from centimeters to meters first). The calculator instantly determines the hypotenuse as 5 m , the area as 6 m² , and the perimeter as 12 m .

Next time you get curious about any setup that looks like a right triangle, you can use our Pythagorean theorem calculator to determine the lengths of the legs.

The Pythagorean theorem describes how the three sides of a right triangle are related in Euclidean geometry. It states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse . You can also think of this theorem as the hypotenuse formula. If the legs of a right triangle are a and b and the hypotenuse is c , the formula is:

a² + b² = c²

The theorem was credited to the ancient Greek philosopher and mathematician Pythagoras, who lived in the sixth century BC. Although it was previously used by the Indians and Babylonians, Pythagoras (or his students) were credited with being the first to prove the theorem. It should be noted that there is no concrete evidence that Pythagoras himself worked on or proved this theorem.

Here's how to use Pythagorean theorem:

• Input the two lengths that you have into the formula. For example, suppose you know one leg a = 4 and the hypotenuse c = 8.94 . We want to find the length of the other leg b .
• After the values are put into the formula, we have 4² + b² = 8.94² .
• Square each term to get 16 + b² = 80 .
• Combine like terms to get b² = 64 .
• Take the square root of both sides of the equation to get b = 8 . Go ahead and check it with an online Pythagorean theorem calculator!

Note that if you are solving for a or b , rearrange the equation to isolate the desired variable before combining like terms and taking the square root

The Pythagorean theorem calculator will solve for the sides in the same manner that we displayed above. We have included the method to show you how you can solve your problem if you prefer to do it by hand.

The hypotenuse formula simply takes the Pythagorean theorem and solves for the hypotenuse, c . To solve for the hypotenuse, we simply take the square root of both sides of the equation a² + b² = c² and solve for c . When doing so, we get c = √(a² + b²) . This is just a reformulation of the Pythagorean theorem and is often associated with the name hypotenuse formula .

Notice the sides of a triangle have a certain degree of gradient or slope. We can use the slope calculator to determine the slope of each side. In a right triangle, the sides that form the right angle will have slopes whose product is -1. The formula for slope, if you wish to calculate by hand, is:

(y₂ − y₁)/(x₂ − x₁)

You can also figure out the missing side lengths and angles of a right triangle using the right triangle calculator . If the angles given in the problem are in degrees and you want to convert to radians or radians to degrees, check out our angle converter . There is an easy way to convert degrees to radians and radians to degrees.

If the angle is in radians:

• Multiply by 180/π .

If the angle is in degrees:

• Multiply by π/180 .

Sometimes you may encounter a problem where two lengths are missing. In such cases, the Pythagorean theorem calculator won't help – you will use trigonometric functions to solve for these missing pieces. Don't worry! We have an excellent trigonometric functions calculator available for you.

Indeed, all maths enthusiasts would be happy to have access to a Pythagorean theorem calculator. Even the students who have to complete their assignments would be thrilled. Now imagine how happy Mateusz and Piotr were when they decided to make a tool for one of the most sought-after mathematics concepts and successfully did so.

Mateusz Mucha is the brain behind Omni Calculator. His deep love for numbers with strategic vision and operational expertise is a testament to his career. He believes in a hands-on approach in all aspects of life, whether it is leadership, building some innovative calculator or digital product, or participating in a cycling marathon. He is a well-balanced blend of exemplary leadership and vision, with strategic thinking, innovation, and attention to detail being a few of the skills in his arsenal.

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What is the hypotenuse given legs 7 and 9?

The hypotenuse is 11.40 .

You need to apply the Pythagorean theorem:

• Recall the formula a² + b² = c² , where a , and b are the legs and c is the hypotenuse.
• Put the length of the legs into the formula: 7² + 9² = c² .
• Squaring gives 49 + 81 = c² . That is, c² = 150 .
• Taking the square root, we obtain c = 11.40 . You can verify the result with an online Pythagorean theorem calculator.

What is the leg in an isosceles triangle with hypotenuse 10?

Each leg has length 10/√2 ≈ 7.07 . To arrive at this answer, we apply the Pythagorean theorem:

• In our case, a = b , so the formula reads 2a² = c² .
• Solving for a , we get a = c/√2 .
• Plugging in c = 10 , we get the final answer: a = 10/√2 ≈ 7.07 .

Absolute value

Christmas tree, regular polygon, significant figures.

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Pythagorean Theorem Word Problems

In these lessons, we will be looking at how to solve different types of word problems using the Pythagorean Theorem.

Related Pages Pythagorean Theorem Converse Of Pythagorean Theorem Applications Of Pythagorean Theorem More Geometry Lessons

How To Solve Word Problems Using The Pythagorean Theorem?

• Determine whether the word problem can be modeled by a right triangle.
• Use the Pythagorean Theorem to find the missing side if you are given two sides.

Example: Shane marched 3 m east and 6 m north. How far is he from his starting point?

Solution: First, sketch the scenario. The path taken by Shane forms a right-angled triangle. The distance from the starting point forms the hypotenuse.

Example: The rectangle PQRS represents the floor of a room.

Ivan stands at point A. Calculate the distance of Ivan from a) the corner R of the room b) the corner S of the room

Example: In the following diagram of a circle, O is the centre and the radius is 12 cm. AB and EF are straight lines.

Find the length of EF if the length of OP is 6 cm.

Examples Of Real Life Pythagorean Theorem Word Problems

Problem 1: A 35-foot ladder is leaning against the side of a building and is positioned such that the base of the ladder is 21 feet from the base of the building. How far above the ground is the point where the ladder touches the building?

Problem 2: The main mast of a fishing boat is supported by a sturdy rope that extends from the top of the mast to the deck. If the mast is 20 feet tall and the rope attached to the deck 15 feet away from the base of the mast, how long is the rope?

Problem 3: If an equilateral triangle has a height of 8, find the length of each side.

Problem 4: Two cyclist start from the same location. One cyclist travels due north and the other due east, at the same speed. Find the speed of each in miles per hour if after two hours they are 17sqrt(2) miles apart.

Problem 5: Two cars start from the same intersection with one traveling southbound while the other travels eastbound going 10 mph faster. If after two hours they are 10sqrt(34) apart, how fast was each car traveling?

Problem 6: A carpet measures 7 feet long and has a diagonal measurement of sqrt(74) feet. Find the width of the carpet.

Problem 7: Jim and Eileen decided to take a short cut through the woods to go to their friend’s house. When they went home they decided to take the long way around the woods to avoid getting muddy shoes. What total distance did they walk to and from their friend’s house? Dimensions are in meters.

Problem 8: Shari went to a level field to fly a kite. She let out all 650 feet of the string and tied it to a stake. Then, she walked out on the field until she was directly under the kite, which was 600 feet from the stake. How high was the kite from the ground?

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Word problems on Pythagorean Theorem

Learn how to solve different types of word problems on Pythagorean Theorem .

Pythagoras Theorem can be used to solve the problems step-by-step when we know the length of two sides of a right angled triangle and we need to get the length of the third side.

Three cases of word problems on Pythagorean Theorem :

Case 1: To find the hypotenuse where perpendicular and base are given.

Case 2: To find the base where perpendicular and hypotenuse are given.

Case 3: To find the perpendicular where base and hypotenuse are given.

Word problems using the Pythagorean Theorem:

1. A person has to walk 100 m to go from position X in the north of east direction to the position B and then to the west of Y to reach finally at position Z. The position Z is situated at the north of X and at a distance of 60 m from X. Find the distance between X and Y.

⇒ 200x = 10000 + 3600

⇒ 200x = 13600

⇒ x = 13600/200

Therefore, distance between X and Y = 68 meters.

Therefore, length of each side is 8 cm.

Using the formula solve more word problems on Pythagorean Theorem.

3. Find the perimeter of a rectangle whose length is 150 m and the diagonal is 170 m.

In a rectangle, each angle measures 90°.

Therefore PSR is right angled at S

Using Pythagoras theorem, we get

⇒ PS = √6400

Therefore perimeter of the rectangle PQRS = 2 (length + width)

                                                          = 2 (150 + 80) m

                                                          = 2 (230) m

                                                          = 460 m

4. A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high. Find the distance of the foot of the ladder from the bottom of the wall.

Let the required distance be x meters. Here, the ladder, the wall and the ground from a right-angled triangle. The ladder is the hypotenuse of that triangle.

According to Pythagorean Theorem,

Therefore, distance of the foot of the ladder from the bottom of the wall = 5 meters.

5. The height of two building is 34 m and 29 m respectively. If the distance between the two building is 12 m, find the distance between their tops.

The vertical buildings AB and CD are 34 m and 29 m respectively.

Draw DE ┴ AB

Then AE = AB – EB but EB = BC

Therefore AE = 34 m - 29 m = 5 m

Now, AED is right angled triangle and right angled at E.

⇒ AD = √169

Therefore the distance between their tops = 13 m.

The examples will help us to solve various types of word problems on Pythagorean Theorem.

Congruent Shapes

Congruent Line-segments

Congruent Angles

Congruent Triangles

Conditions for the Congruence of Triangles

Side Side Side Congruence

Side Angle Side Congruence

Angle Side Angle Congruence

Angle Angle Side Congruence

Right Angle Hypotenuse Side congruence

Pythagorean Theorem

Proof of Pythagorean Theorem

Converse of Pythagorean Theorem

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How to Solve Pythagorean Theorem Word Problems

The Pythagorean Theorem is a cornerstone of mathematics, fundamental in various scientific fields and real-life situations. We will guide you through understanding and solving Pythagorean Theorem word problems, explaining each step meticulously for your convenience.

The Pythagorean Theorem is an equation attributed to the ancient Greek mathematician, Pythagoras. The theorem states that in any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula can be written as \(a^2+b^2=c^2\).

An essential part of understanding the Pythagorean Theorem involves recognizing Pythagorean triplets. These are sets of three integers that satisfy the Pythagorean Theorem. Notable examples are \((3,4,5), (5,12,13)\), and \((8,15,17)\). Recognizing these sets can simplify solving Pythagorean Theorem word problems.

A Step-by-step Guide to Solving Pythagorean Theorem Word Problems

Now that we’ve laid the groundwork, let’s delve into the process of solving word problems involving the Pythagorean Theorem.

Step 1: Analyzing the Problem

The first step in solving any word problem is understanding what the problem is asking. Translate the problem into an understandable format and identify the elements related to the Pythagorean Theorem.

Step 2: Sketching the Problem

Visualizing the problem can be extremely helpful. Sketch the triangle described in the problem and label the sides according to the given information. The right angle is usually denoted by a small square, and the hypotenuse, by the letter c .

Step 3: Applying the Pythagorean Theorem

Apply the Pythagorean Theorem to the problem. Plug in the values of the known sides into the theorem and solve for the unknown side.

Step 4: Verifying the Solution

After you’ve found the value of the unknown side, verify your solution by substituting the values back into the theorem. If the sides satisfy the theorem, you’ve correctly solved the problem.

Example of a Pythagorean Theorem Word Problem

Consider a problem where you’re given a right-angled triangle with one side measuring \(5\) units and the hypotenuse measuring \(13\) units. You’re asked to find the length of the other side.

You can use the Pythagorean Theorem to solve this problem. By substituting \(a=5\) and \(c=13\) into the theorem, you can solve for \(b\).

Calculating this will give you \(b^2=13^2-5^2=144\). Therefore, \(b=\sqrt{144}=12\) units. Your verification will involve substituting \(a=5, b=12\), and \(c=13\) into the theorem. The equation \(5^2+12^2=13^2\) checks out, confirming the solution.

The Pythagorean Theorem is a potent tool for solving geometrical problems involving right-angled triangles. With the steps outlined in this guide, you can confidently tackle any word problem that comes your way. Practice is key to mastery, so take time to solve different problems and apply the theorem in real-world situations.

by: Effortless Math Team about 10 months ago (category: Articles )

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Course: 8th grade   >   Unit 5

• Use Pythagorean theorem to find area of an isosceles triangle
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Pythagorean theorem word problems

• Pythagorean theorem in 3D
• Your answer should be
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Pythagorean Theorem Calculate 4+

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Unlock the power of the Pythagorean Theorem with our intuitive and versatile Pythagorean Theorem Calculator app. Whether you're a student tackling geometry problems, a math enthusiast, or a professional in need of quick triangular solutions, this app is your ultimate companion. Key Features: Effortless Triangle Solving: Solve for any missing side or angle in a right triangle with ease. Our calculator makes complex calculations a breeze, saving you time and effort. Interactive Triangle Visualisation: Visualize triangles dynamically as you input values, helping you understand the theorem better and ensuring accuracy in your calculations. Step-by-Step Solutions: Get a detailed breakdown of how each calculation is performed, perfect for learning and teaching the Pythagorean Theorem. Who Can Benefit: Students: Ace your geometry classes by understanding and applying the Pythagorean Theorem confidently. Teachers: Use this tool to teach the Pythagorean Theorem effectively with interactive visualizations and step-by-step solutions. Professionals: Solve real-world problems that involve right triangles, whether in architecture, engineering, or any field where accurate measurements are crucial. Why Choose Our App: Our Pythagorean Theorem Calculator is designed to simplify your math journey. Whether you're learning, teaching, or applying this fundamental concept, our user-friendly interface and powerful features ensure that you have the right tool at your fingertips. Download the Pythagorean Theorem Calculator app today and start unlocking the mysteries of right triangles effortlessly. Mathematics has never been this accessible and enjoyable!

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