# Pyramid

A pyramid is a three-dimensional solid object with a flat base and a pointed end.

The base can be of any shape such as triangle, square, rectangle, or a pentagon. The diagram above shows a pyramid with a square base.

The Pyramids of Giza, in Egypt, is a very common and beautiful example of a pyramid shape.

## Definition

A pyramid is a polyhedron with a polygonal base bounded by triangular faces meeting at a common vertex, known as the apex. The triangular faces are called the lateral faces.

When a pyramid is rested on its base, it points upwards. The square pyramid is the most common type of pyramid.

## Parts

A pyramid has an apex, vertices, a base, lateral faces, and edges.

• Base – The flat face on which a pyramid rests.
• Lateral faces – The triangular faces connecting the base at the apex.
• Vertices – The corners.
• Apex – The common vertex at which the lateral faces of a pyramid meet.
• Edges – Where any 2 faces meet.
• Height – The imaginary straight line drawn right from the apex, perpendicular to the base.
• Slant Height – The imaginary straight line drawn from the mid point of any base edge to the apex. Although the height and the slant height are not parts of a pyramid but those are used for the formulas of pyramids.

## Types

Depending on the base, a pyramid can be of different shapes. Some common types are: triangularrectangularsquarepentagonalhexagonal.

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

A pyramid with n-sided bases, has n + 1 faces, n + 1 vertices, and 2n edges. For example, a triangular pyramid has 4 faces (1 triangular base and 3 triangular lateral faces), 4 vertices, and 6 edges.

A pyramid can be right or oblique, depending on the location of its apex.

## Right vs Oblique Pyramid

1. Right Pyramid – It is a pyramid whose apex is exactly over the middle of its base. The perpendicular line from the apex to the base center is the height.
2. Oblique Pyramid – It is a pyramid whose apex is not exactly over the middle of its base. The perpendicular line from the apex to the base is the height.

A pyramid can also be regular or irregular, depending on the regularity of its base.

## Regular vs Irregular Pyramid

1. Regular Pyramid – A regular pyramid is such a pyramid whose base is a regular polygon (all sides equal). All its lateral faces are same.
2. Irregular Pyramid – an irregular pyramid is such a pyramid whose base is an irregular polygon (sides not equal to each other). Its lateral faces are different.

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

## Formulas

### Surface Area

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

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

SA = B + LSA

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

Find the lateral and total surface area of a square pyramid with a base area of 64 cm2, base perimeter of 32 cm, and slant height of 8.5 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${\dfrac{1}{2}P{s}}$, here P = 32 cm, s = 8.5 cm
LSA = ${\dfrac{1}{2}\times 32\times 8\cdot 5}$
= 136 cm2
Total Surface Area (TSA) = B + LSA, here B = 64 cm2, LSA = 136 cm2
TSA = 64 + 136
= 200 cm2

Find the total surface area of a triangular pyramid with a base area of 7.3 cm2, base perimeter of 12.3 cm, and slant height of 6 cm.

Solution:

As we know,
Total Surface Area (TSA) = ${B+\dfrac{1}{2}Ps}$, here B = 7.3 cm2, P = 12.3 cm, s = 6 cm
TSA = ${7.3+\dfrac{1}{2}\times 12.3\times 6}$
= 44.2 cm2

### Volume

The formula is given below:

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

Let us solve an example to understand the above concept better.

Find the volume of a square pyramid with a base area of 49 cm2, and a height of 12 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}Bh}$, here B = 49 cm2, h = 12 cm
V = ${\dfrac{1}{3}\times 49\times 12}$
= 196 cm3

## Real-life Examples

• Chichen Itza Pyramid, in Yucatán, Mexico.
• Pyramid-shaped tents and temples.
• Football-shoe spikes.
• More Resources:

## 4 thoughts on “Pyramid”

1. Schuyler Martin Ph.D. says:

Well written, easy to follow, a wonderful resource. Thanks.

2. Mr. Ruhnke says:

Hi, there. I was just wondering about horizontal cross-sections of oblique square based pyramids. Will the horizontal cross-section always be a square…as it will always be with a right square based pyramid? Thanks so much.

1. staff says:

Horizontal cross-sections of an oblique square-based pyramid will not always be squares. The shape of the cross-section will depend on the angle of the slant of the pyramid’s sides and the height at which the cross-section is made.

3. Aale Mustafa says:

It was very helpful for me ☺️ and easy to understand, keep it up mathmonks…