triangular pyramid problem solving

Triangular Pyramid

A triangular pyramid is a geometric solid with a triangular base, and all three lateral faces are also triangles with a common vertex. The tetrahedron is a triangular pyramid with equilateral triangles on each face. Four triangles form a triangular pyramid. Triangular pyramids are regular, irregular, and right-angled.

A three-dimensional shape with all its four faces as triangles is known as a triangular pyramid.

What is a Triangular Pyramid?

A triangular pyramid is a 3D shape in which all the faces are triangles . It is a pyramid with a triangular base connected by four triangular faces where 3 faces meet at one vertex. If it is a right triangular pyramid, the base is a right-angled triangle while the other faces are isosceles triangles .

Triangular Pyramid Nets

The net pattern is different for different types of solids. Nets are useful to find the surface area of ​​solids. A triangular pyramid net is a pattern that forms when its surface is laid flat, showing each triangular face in the two-dimensional (2D) form. The triangular pyramid net consists of four triangles.

Let us do a small activity to learn more about the net of a triangular pyramid. Take a sheet of paper. You can observe two different nets of a triangular pyramid shown below. Copy this on the sheet of paper. Cut it along the edge and fold it as shown in the picture below. The folded paper forms a triangular pyramid.

Types of Triangular Pyramid

Like any other geometrical figure, triangular pyramids can also be classified into regular and irregular pyramids. The different types of triangular pyramids are explained below.

Regular Triangular Pyramid

A regular triangular pyramid has equilateral triangles as its faces. Since it is made of equilateral triangles, all its internal angles measure 60°.

Irregular Triangular Pyramid

An irregular triangular pyramid also has triangular faces, but they are not equilateral . The internal angles in each plane add up to 180° as they are triangular. Unless a triangular pyramid is specifically mentioned as irregular, all triangular pyramids are assumed to be regular triangular pyramids.

Right Triangular Pyramid

A right triangular pyramid (a three-dimensional figure) has the base as a right-angled triangle and the apex is aligned above the center of the base. It has 1 right-angled base, 6 edges, 3 triangular faces, and 4 vertices.

Properties of a Triangular Pyramid

The properties of a triangular pyramid help us to identify a pyramid from a given set of figures quickly and easily. The different properties of a triangular pyramid are:

• A triangular pyramid has 4 triangular faces, 6 edges, and 4 vertices.
• 3 edges meet at each of its vertex.
• A triangular pyramid has no parallel faces.
• A regular triangular pyramid has equilateral triangles for all its faces. It has 6 planes of symmetry .
• Triangular pyramids can be regular, irregular, and right-angled.

Triangular Pyramid Formulas

There are various formulas that are used to calculate the volume and surface area of triangular pyramids. Observe the following figure to relate to the formulas given below:

• The volume of a triangular pyramid is calculated with the formula, Volume of Triangular Pyramid = 1/3 × Base Area × Height; where we multiply the area of ​​the triangular base by the height of the pyramid (measured from base to top) and then divide that product by 3 as shown in the formula.
• The total surface area of a triangular pyramid is calculated with the formula, Total Surface area of a Triangular Pyramid = Base Area + 1/2 (Perimeter of the base × Slant Height) ; where 'slant height' is the distance from its triangular face along the center of the face to the apex.
• Now consider a regular triangular pyramid made of equilateral triangles of side 'a' . The two main formulas of a regular triangular pyramid are: Volume of a Regular Triangular Pyramid = a 3 /6√2 and Total Surface Area of a Regular Triangular Pyramid = √3a 2

Tips on Triangular Pyramid

• A triangular pyramid has 4 faces, 6 edges, and 4 vertices. All four faces are triangular in shape.
• The tetrahedron is a triangular pyramid having congruent equilateral triangles for each of its faces.

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Triangular Pyramid Examples

Example 1: If two congruent triangular pyramids are stuck together along their base they form a triangular bipyramid. How many faces, edges, and vertices does this bipyramid have?

Solution: This triangular bipyramid has 6 triangular faces, 9 edges, and 5 vertices.

Example 2: Find the volume of a regular triangular pyramid with a side length measuring 5 units. (Round off the answer to 2 decimal places)

Solution: We know that for a triangular pyramid whose side is a volume is: a 3 /6√2. Substituting a = 5, we get

Volume = 5 3 /6√2

= 125/8.485

∴ The volume of the triangular pyramid is 14.73 units 3

Example 3: Each edge of a regular triangular pyramid is of length 6 units. Find its total surface area.

Solution: The total surface area of a regular triangular pyramid of side a is: √3a 2 . Substituting a = 6, we get,

TSA = √3 × 6 2 = √3 × 6 × 6

∴ Total Surface Area = 62.35 units 2

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Practice Questions on Triangular Pyramid

Faqs on triangular pyramid, what is a triangular pyramid in math.

A triangular pyramid is a 3D shape in which all the faces are triangles . It is a geometric solid with a triangular base, and all three lateral faces are also triangles with a common vertex.

What is the Volume of a Triangular Pyramid?

The volume of a triangular pyramid is the space inside the pyramid in a three-dimensional (3D) plane. The formula for the volume of a triangular pyramid is expressed as, Volume of a Triangular Pyramid = 1/3 × Base Area × Height, where, base area is the area of ​​the triangular base and height is the height of the pyramid measured from the base to the top vertex.

What is the Surface Area of a Triangular Pyramid?

The surface area of a triangular pyramid is the sum of the areas of all of the faces of the pyramid. The formula that is used to find the surface area of a triangular pyramid is, Total Surface area of Triangular Pyramid = Base Area + 1/2 (Perimeter of the base × Slant Height); where the slant height is the distance from the triangular face along the center of the face to the apex.

What is the Base of a Triangular Pyramid?

The base of a triangular pyramid is also a triangle which means there are 4 triangular faces in a triangular pyramid.

Give an Example of a Triangular Pyramid.

A common example of a triangular pyramid is the Pyramix or the Rubik's triangle which has 1 triangular base and 3 other triangular faces.

How many Faces, Edges and Vertices does a Triangular Pyramid have?

A triangular pyramid has 4 triangular faces, 6 edges (sides) and 4 vertices (corners). The 4 triangular faces include the base of the pyramid which is also a triangle.

How Many Triangles Form a Triangular Pyramid?

Four triangles form a triangular pyramid out of those, one is the base and the other three triangles are the three lateral faces of the pyramid.

How Many Vertices Does a Triangular Pyramid have?

A triangular pyramid has 4 vertices. The three edges of the pyramid intersect at each vertex. Out of the 4 vertices, 3 vertices are at the base of the triangular pyramid and one vertex is at the top.

What are the Different Types of Triangular Pyramids?

There are three types of triangular pyramids and those are - regular, irregular, and right-angled triangular pyramids.

Is a Triangular Pyramid made of Equilateral Triangles?

All triangular pyramids are not made of equilateral triangles. When a triangular pyramid is made up of equilateral triangles, it is called a regular triangular pyramid. It has four faces, including the base.

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Last modified on August 3rd, 2023

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Triangular pyramid.

A triangular pyramid is a polyhedron with a triangular base bounded by three lateral faces meeting at a common point, known as the apex.

The lateral faces are triangular.

Tents and combination puzzles are some real-life example of a triangular pyramid shape.

How many faces, vertices, and edges does a triangular pyramid have?

A triangular pyramid has 4 faces, 4 vertices, and 6 edges. Since all the 4 faces are triangular, a triangular pyramid is also called a tetrahedron .

A net for a triangular pyramid can illustrate its shape from a 2-D view. This net can be folded along the dotted lines to form a triangular pyramid as shown in the diagram below.

Based on the regularity of the base, a triangular pyramid can be – (1) Regular triangular pyramid , (2) Irregular triangular pyramid .

A regular triangular pyramid is a pyramid that has its base in shape of an equilateral triangle. So it is also called an equilateral triangular pyramid.

In contrast, when the base of a triangular pyramid is an irregular, it is an irregular triangular pyramid .

Based on the position of its apex, a a triangular pyramid can be (1) Right triangular pyramid , (2) Oblique triangular pyramid .

A right triangular pyramid is a pyramid that has its apex right above its base center. So, an imaginary line drawn from the apex intersects the base at its center at a right angle. This line is its height.

In contrast, when the apex is away from the base center, the pyramid is an oblique triangular pyramid .

Like all other polyhedrons, we can calculate the surface area and the volume of a triangular pyramid.

The formula is:

Volume ( V ) = ${\dfrac{1}{3}Bh}$, here B = base area, h = height

Let us solve some examples involving the above formula.

Find the volume of a regular triangular pyramid with a base area of 97 cm 2 and a height of 26 cm.

As we know, Volume ( V ) = ${\dfrac{1}{3}Bh}$, here B = 97 cm 2 , h = 26 cm ∴ V = ${\dfrac{1}{3}\times 97\times 26}$ = 840.6 cm 3

Find the volume of a right triangular pyramid with a base of 21 cm, a base height of 8 cm, and a height of 17 cm.

As we know, Volume ( V ) = ${\dfrac{1}{3}Bh}$ B = ${\dfrac {1}{2}bH}$, here b = base, H = base height plugging the value of B, we get, Volume ( V ) = ${\dfrac{1}{6}bHh}$ , here b = 21 cm, H = 8 cm, h = 17 cm ∴ V = ${\dfrac{1}{6}\times 21\times 8\times 17}$ = 476 cm 3

Finding the volume of a regular triangular pyramid when the BASE and HEIGHT are known

Find the volume of a regular triangular pyramid with a base of 7 cm, and a height of 16 cm.

We will use an alternative formula here. Volume ( V ) =  ${ \dfrac{\sqrt{3}}{12}b^{2}h }$, here b = 7 cm, h = 16 cm ∴ V = ${\dfrac{\sqrt{3}}{12}\times 7^{2}\times 16}$ = 21.21762 × 1/3 × 16 = 113.16 cm 3

Surface Area

Surface Area (SA) = ${B+\dfrac{1}{2}Ps}$, here B = base area, P = base perimeter, s = slant height,

Also ${\dfrac{1}{2}Ps}$ = lateral surface area ( LSA )  

∴  SA  = B +  LSA

Let us solve some examples to understand the above concept better.

Find the surface area of a regular triangular pyramid with a base area of 62.35 cm 2 , a base perimeter of 36 cm, and a slant height of 7 cm.

As we know, Total Surface Area ( TSA ) = ${B+\dfrac{1}{2}Ps}$, here B = 62.35 cm 2 , P = 36 cm, s = 7 cm ∴ TSA = ${62.35+\dfrac{1}{2}\times 36\times 7}$ = 188.35 cm 2

Find the lateral and total surface area of a triangular pyramid with a base of 8 cm, a base height of 4.6 cm, and a slant height of 9 cm.

As we know, Lateral Surface Area ( LSA ) = ${\dfrac{1}{2}Ps}$ P = 3 × b, here b = 8 cm ∴ LSA = ${\dfrac{3}{2}bs}$, here b = 8 cm, s = 9 cm ∴ LSA = ${\dfrac{3}{2}\times 8\times 9}$ = 108 cm 2 Total Surface Area ( TSA ) = B + LSA Now, B = ${\dfrac{1}{2}bH}$, here b = 8 cm, H = 4.6 cm ∴ TSA =  ${\dfrac{1}{2}\times bH+LSA}$, here b = 8 cm, H = 4.6 cm, LSA = 108 cm 2 ∴ TSA = ${\dfrac{1}{2}\times 8\times 4.6 +108}$ = 126.4 cm 2

Finding the surface area of an regular triangular pyramid when the BASE and SLANT HEIGHT are known

Find the surface area of a regular triangular pyramid with a base of 11 mm, and a slant height of 7 mm.

Here we will use an alternative formula for a regular (equilateral) triangular pyramid. Total Surface Area ( TSA ) = ${\dfrac{\sqrt{3}}{4}b^{2}+\dfrac{1}{2}\times 3bs}$, here b = 11 mm,s = 7 mm ∴ TSA = ${\dfrac{\sqrt{3}}{4}\times 11^{2}+\dfrac{1}{2}\times 3\times 11\times 7}$ = 167.89 mm 2

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