# Square Pyramid

## Definition

A square pyramid is a pyramid with a square base bounded by four lateral faces meeting at a common point, known as the apex. The lateral faces are triangular.

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

A square pyramid has 5 faces, 8 edges, and 5 vertices.

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

## Types

Based on the position of its apex, a square pyramid can be classified into 2 types â€“ (1) Right square pyramid, and (2) Oblique square pyramid.

### Right and Oblique

A right square pyramid is a pyramid whose apex is aligned right above its base center. So, an imaginary line drawn from the apex intersects the base at its center at a right angle. A square pyramid is usually right.

In contrast, when the apex is away from the base center, the pyramid is an oblique square pyramid. Its height is a perpendicular line from the apex to the base.

A square pyramid has a special type based on its edge lengths.

### Equilateral

An equilateral square pyramid has its 3 lateral faces that are equilateral triangles. It means all the edges of are of equal length.

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

## Formulas

### Volume

The formula is:

Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = base, h = height

Let us solve some examples to understand the concept better.

Find the volume of aÂ squareÂ pyramidÂ with a base of 12 cm, and a height of 6 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 12 cm, h = 6 cm
âˆ´Â VÂ = ${\dfrac{1}{3}\times 12^{2}\times 6}$
= 288 cm3

Find the volume of aÂ squareÂ based pyramidÂ with a base of 4 cm, and a height of 14 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 4 cm, h = 14 cm
âˆ´Â VÂ = ${\dfrac{1}{3}\times 4^{2}\times 14}$,
= 74.66 cm3

### Surface Area

The formula is:

Surface Area (SA) = ${b^{2}+2bs}$, here b = base, s = slant height

Also ${2bs}$ = lateral surface area (LSA)

âˆ´ SA = b2 LSA

Let us solve some examples to understand the concept better.

Find the lateral and total surface area of aÂ squareÂ pyramidÂ with a slant height of 8.5 cm and a base of 8 cm.

Solution:

As we know,
Lateral Surface Area (LSA) = ${2bs}$, here b = 8 cm, s = 8.5 cm
âˆ´ LSAÂ = 2Â Ã— 8 Ã— 8.5
= 136 cm2
Total Surface Area (TSA) = b2 + LSA, here b = 8 cm, LSAÂ = 136 cm2
âˆ´ TSAÂ = Â 82+ 136
= 200 cm2

Find the surface area of aÂ squareÂ based pyramidÂ with a base of 11 cm and a slant height of 15 cm.

Solution:

As we know,
Total Surface Area (TSA) = b2 + 2bs, here b = 11 cm, s = 15 cm
âˆ´ TSAÂ =Â  112+ 2 Ã— 11 Ã— 15
= 451 cm2

Now, let us learn how to find the slant height and height of a square pyramid with some typical examples.

Finding the slant height of a square pyramid when its BASE and SURFACE AREA are known.

Find the slant height of a square pyramid with a base area of 189 cm2 and a base of 9 cm.

Solution:

As we know,
Surface Area (SA) = b2 + 2bs
âˆ´ s = ${\dfrac{SA-b^{2}}{2b}}$, here SA = 189 cm2, b = 9 cm
= ${\dfrac{189-9^{2}}{2\times 9}}$
= 6 cm

Finding the height of a square pyramid when its BASE and VOLUME are known.

Find the height of a square pyramid with a volume of 256 cm3, and a base of 8 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$
âˆ´ h = ${\dfrac{3V}{b^{2}}}$, here V = 256 cm3, b = 8 cm
= ${\dfrac{3\times 256}{8^{2}}}$
= 12 cm