Pyramid in Maths: Definition, Shape, Types and Formula

Pyramid in Maths: The ancient pyramids are some of the oldest man-made structures in the world, which still stir a sense of wonder. Pyramids are the most recognizable structures, with a square as a base and triangles as sides. The Pyramid shape or design has been used in different forms of architecture and is observed in industrial scenarios.

In this article, let us learn the concept of the pyramid, the formula of a pyramid, types of pyramids, how to measure the area of a pyramid, and its shape in detail. Moreover, students can also refer to solved examples related to the structure of a pyramid for a better understanding of the concept.

Pyramid Shape Definition

A Pyramid is a polyhedron formed by joining a polygonal base and a point known as the apex.

Apex: Apex is the vertex at the top of the Pyramid.

Base: Base is the bottom of the Pyramid.

Pyramid Shape

A pyramid is a three-dimensional shape. A pyramid has a polygonal base and flat triangular faces that meet at a point known as the apex. A pyramid is formed by connecting the bases to an apex. Each edge of the base is connected to the apex and forms the triangular face, called the lateral face. If the base of the pyramid is a triangle, then it is a triangular pyramid. Similarly, if the base of the pyramid is a square, then it is a square pyramid.

Lateral face: If each edge of the base is connected to the apex, and forms the triangular face, then it is known as the lateral face.

Types of Pyramid

A pyramid can be classified into different models based on the shape of the base. Some of them are discussed below.

Triangular Pyramid

If the base of the pyramid has three sides, and it is in a triangular shape, then the pyramid is called a triangular pyramid. The properties of triangular pyramid are:

• Number of Faces: \((3+1) = 4\)
• Number of Vertices: \((3+1) = 4\)
• Number of Edges: \(2(3) = 6\)

Square Pyramid

The base of the pyramid with four sides and has the shape of a square, then it is called a square pyramid. The properties of a pyramid, which is square in share are:

• Number of Faces: \((4+1) = 5\)
• Number of Vertices: \((4+1)= 5\)
• Number of Edges: \(2(4) = 8\)

Pentagonal Pyramid

The base of the pyramid with five sides and is in the shape of a pentagon, then it is called a pentagonal pyramid. The properties of the pentagonal pyramid are:

• Number of Faces: \((5+1)=6\)
• Number of Vertices: \((5+1)=6\)
• Number of Edges: \(2(5)= 10\)

Hexagonal Pyramid

The base of the pyramid with six sides and is in the shape of a hexagon, then it is called a hexagonal pyramid. The properties of hexagonal pyramid are:

• Number of Faces: \((6+1)=7\)
• Number of Vertices: \((6+1)=7\)
• Number of Edges: \(2(6)= 12\)

Right Pyramid vs Oblique Pyramid

Now let’s understand the concept of the right pyramid and oblique pyramid concepts in the following section:

Right Pyramid

If the apex of a pyramid is exactly over the middle of the base, then it is called a right pyramid.

Oblique Pyramid

If the apex of a pyramid is not exactly over the middle of its base, then this pyramid is called an oblique pyramid. Oblique pyramids are also called non-right pyramids.

Regular vs Irregular Pyramid

Regular pyramids have its faces as regular polygons, since all sides and all angles are same.

Irregular pyramids have irregular polygons as its base.

Pyramid: Formula

The total surface area of a pyramid:

The total surface area of pyramid \( = \,\frac{1}{2}\, \times \,Pl\, + \,B\,\;{\rm{square}}\,\,{\rm{units}}\)

where,

\(“P”\) is the perimeter of the base.

\(“l”\) is the slant height. 

\(“B”\) is the base area.

The volume of a pyramid

The volume of the pyramid \( = \frac{1}{3}{\rm{(Base}}\,{\rm{area) \times (Height)\;cubic}}\,{\rm{units}}\).

Pyramid Shape Examples

Some of the Pyramid examples are:

1. The roof of the house.
2. Giza Pyramids in Egypt.
3. Pyramid shaped paperweights
4. Pyramid shaped bottles
5. Pyramid shaped tents

Applications of Pyramid

1. Great Giza Pyramid was built in Egypt. Pharaoh’s burial chamber lays deep inside the Pyramids.
2. Egypt’s largest pyramid can trap electric and magnetic energy into its chambers to spark higher levels of energy.
3. A Pyramid helps to regenerate the flow of energy circulation in the human body.
4. A Pyramid strengthens the power of intention and harmonizes the environment.
5. Pyramids are used for the fast healing process.
6. Pyramids provide effective high-energy environments for meditation.
7. Pyramids help to reduce the stress level and tension in the physical body.
8. To reduce the number of accidents on the road, traffic officers are now using Pyramid structures cones.

Solved Examples on Pyramid

Q-1: Find the length of the side base if the volume of a \(6\)-foot-tall square pyramid is \(8\) cubic feet?
A: The volume of the pyramid  \(V = \frac{1}{3} \times ({\rm{Base}}{\mkern 1mu} {\rm{area}}) \times ({\rm{Height}})\;{\rm{cubic}}{\mkern 1mu} {\rm{units}}\)

\(8 = \frac{1}{3} \times \) Area of the base\(×6\)

\(\therefore \) Area of the base \({\rm{ = }}\,{\rm{side}} \times {\rm{side}}\)

 Area of the base \(= \frac{{\rm{8}}}{{\rm{2}}}{\rm{ = 4}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Then, the side of the base is  \({\rm{ = }}\sqrt {\rm{4}} {\rm{ = 2}}\,{\rm{cm}}\)

Therefore, \({\rm{2}}\,{\rm{cm}}\) is the side of the square pyramid.

Q-2: Find the volume of the square pyramid if its base area is  \({\rm{81 c}}{{\rm{m}}^2}\) and its height is \({\rm{ = 9}}\,{\rm{cm}}\).
A: Given: The base area of the square pyramid \({\rm{81 c}}{{\rm{m}}^2}\)

Height  \({\rm{ = 9}}\,{\rm{cm}}\)

Thus, the volume of the square pyramid \( = \frac{1}{3}{\rm{(Base}}\,{\rm{area) \times (Height)\;cubic}}\,{\rm{units}}\)

The volume of a square pyramid \( = \left( {\frac{1}{3}} \right)\left( {81} \right)\left( 9 \right)\)

\( = (81)(3)\)

\( \Rightarrow V = 243\,\;{\rm{c}}{{\rm{m}}^3}\)

Hence, the volume of the square pyramid is \({\rm{243\;c}}{{\rm{m}}^{\rm{3}}}\).

 Q-3: Find the total surface area of the square pyramid if each side of the base measures \({\rm{16\;cm}}\), and the slant height is \({\rm{18\;cm}}\), and the altitude is \({\rm{15\;cm}}\).
A: Given: As the base is a square, the perimeter of the base \({\rm{ = 4 \times side}}\)

\(P = 4(16)\)

\( \Rightarrow P\, = \,64\,{\rm{cm}}\)

The area of the base \({\rm{ = sid}}{{\rm{e}}^{\rm{2}}}\)

\(B = {16^2} = 256\;\,{\rm{c}}{{\rm{m}}^2}\)

We know, the total surface area of the square pyramid \( = \frac{1}{2} \times P{\rm{l}} + B\) square units

The total surface area of the square pyramid \({\rm{ = }}\left[ {\left( {\frac{{\rm{1}}}{{\rm{2}}}} \right){\rm{(64)(18)}}} \right]{\rm{ + (256)}}\)

\(TSA = 576 + 256\)

\(TSA = 832\;\,{\rm{c}}{{\rm{m}}^2}\)

Hence, the total surface area of the square pyramid is \({\rm{832\;c}}{{\rm{m}}^{\rm{2}}}\).

Q-4: Find the volume of a regular square pyramid with altitude \({\rm{18}}\,{\rm{cm}}\) and base sides \({\rm{10}}\,{\rm{cm}}\)
A: The volume of the pyramid  \(V = \frac{1}{3}({\rm{Base}}{\mkern 1mu} {\rm{area}}) \times ({\rm{Height}})\,{\rm{cubic}}{\mkern 1mu} {\rm{units}}\)

Base area \({\rm{ = 1}}{{\rm{0}}^{\rm{2}}}{\rm{ = 100\;c}}{{\rm{m}}^{\rm{2}}}\)

\(V = \frac{1}{3} \times (100) \times (18) = 600\,\;{\rm{c}}{{\rm{m}}^3}\)

Therefore, the volume is \({\rm{600\;c}}{{\rm{m}}^{\rm{3}}}\).

Q-5: What is the volume of a pyramid with a height of \({\rm{15}}\,{\rm{cm}}\) and a square base with a side length of \({\rm{8}}\,{\rm{cm}}\)?
A: First we need to calculate the area of the base.

Area of the base \(A = {\mathop{\rm side}\nolimits}  \times {\rm{side}}\)

\(A = 8 \times 8 = 64\;\,{\rm{c}}{{\rm{m}}^2}\)

The volume of the pyramid  \(V = \frac{1}{3}({\rm{Base}}\,{\rm{area}}) \times ({\rm{Height}})\,{\rm{cubic}}\,{\rm{units}}\)

\(V = \frac{1}{3} \times \) Area of the base\({\rm{ \times 15}}\)

\(V = \frac{1}{3} \times 64 \times 15\)

\(V = 320\;\,{\rm{c}}{{\rm{m}}^3}\)

Therefore, the volume of a pyramid is \({\rm{320\;c}}{{\rm{m}}^{\rm{3}}}\).

Summary

A pyramid is a three-dimensional structure whose outer surfaces are triangular and coverage to a single step at the top. The pyramid is classified into different types based on the shape of the base. The Great Pyramid of Giza, is structured on the pyramid concept. This article helps to learn in detail about pyramids, types of pyramids, formulae, some solved examples, and FAQs.

Frequently Asked Question About Pyramid

Following are the FAQs on pyramid:

Q-1: What is inside a pyramid in Egypt?

A: Pharaoh’s burial chamber lays deep inside the pyramids. 

Q-2: What is the definition of pyramid?

A: A pyramid is a polyhedron formed by joining a polygonal base and a point known as the apex.

Q-3: What is the history of pyramids?

A: The pyramids were built by the Egyptians during when Egypt was one of the richest and powerful civilizations in the world. The pyramids – like the great Pyramids of Giza – are the most significant man-made structures in the history of the world.

Q-4: What did the first pyramid look like?

A: The first pyramid was built as a structure-like tomb with ascending layers of huge limestone blocks.

Q-5: How do you know if it is a pyramid scheme?

A: A pyramid scheme is a business model of making money that requires members by promising of giving the payments or service for enrolling other members into the scheme, instead of supplying investments or selling the products.

Q-6: How do you measure the pyramids?

A: Pyramids are measured by using the formulae

The Total Surface Area of Pyramid  \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times (Perimeter \times slant}}\,{\rm{height) + (Base}}\,{\rm{area)\;square}}\,{\rm{units}}\)

The volume of the pyramid \( = \frac{1}{3}{\rm{(Base}}\,{\rm{area) \times (Height)cubic}}\,{\rm{units}}\)

Q-7: What is a right pyramid?

A: If the apex of a pyramid is exactly over the middle of the base, then it is called a right pyramid.

Q-8: What do you call the vertex of a pyramid?

A: Apex is the vertex at the top of the Pyramid.

Q-9: What is a 6 sided pyramid called?

A: \(6\) sided pyramid is called a hexagonal pyramid.

Q-10: What is a \(5\) sided pyramid called?

A: \(5\) sided pyramid is called a pentagonal pyramid.

Q-11: How do you find the total surface area of a pyramid?

A: Total surface area of a pyramid is finding by using the formula

The Total Surface Area of Pyramid \({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times (Perimeter \times slant}}\,{\rm{height) + (Base}}\,{\rm{area)\;square}}\,{\rm{units}}\)

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