Surface Area and Volume of a Pyramid

Pyramids are among the most recognisable geometric solids. In this article we focus on the regular square pyramid -- the type you will meet most often in school problems.

What is a regular square pyramid?
Key dimensions
Slant height
Net of a pyramid
Base area
Lateral (side) area
Total surface area
Volume
Worked example 1 -- Surface area
Worked example 2 -- Volume and slant height
Summary of formulas
Practice exercises

What is a regular square pyramid?

A regular square pyramid is a 3D solid with:

A square base
Four congruent triangular faces that meet at a single point called the apex
The apex lies directly above the centre of the base

```

╲ ╱

╲ ▲ ╱ ▲ = apex

╲ │ ╱

╲│╱

┌───────┼───────┐

│ │ │

│ base (a×a) │

│ │

└───────────────┘

```

Key dimensions

A regular square pyramid is described by three main measurements:

Symbol	Meaning
$a$	Length of one edge of the square base
$h$	Height -- perpendicular distance from the base to the apex
$h_{s}$	Slant height -- distance from the midpoint of a base edge to the apex, measured along a triangular face

Do not confuse $h$ (the vertical height through the centre) with $h_{s}$ (the slant height along a face). They are not the same.

Slant height

The slant height connects the midpoint of a base edge to the apex. Together with the height $h$ and half the base edge $\frac{a}{2}$ , these three lengths form a right triangle:

```

apex

│╲

h │ ╲ h_s

│ ╲

│_____╲

a/2

```

By the Pythagorean theorem:

h_{s}^{2} = h^{2} + (\frac{a}{2})^{2}

h_{s} = h^{2} + (\frac{a}{2})^{2}

You will need $h_{s}$ to calculate the lateral surface area. Always find it first if it is not given directly.

Net of a pyramid

If you unfold a regular square pyramid you get:

One square (side length $a$ ) -- the base
Four congruent isosceles triangles -- each with base $a$ and height $h_{s}$

Base area

The base is a square with side $a$ :

S_{base} = a^{2}

Lateral (side) area

Each triangular face has base $a$ and height (slant height) $h_{s}$ . The area of one triangle is:

S_{triangle} = \frac{1}{2} \cdot a \cdot h_{s}

There are four such triangles, so the total lateral area is:

S_{lateral} = 4 \cdot \frac{1}{2} \cdot a \cdot h_{s}

S_{lateral} = 2 a \cdot h_{s}

Total surface area

The total surface area is the base plus the lateral area:

S = S_{base} + S_{lateral}

S = a^{2} + 2 a \cdot h_{s}

This can also be factored as $S = a (a + 2 h_{s})$ .

Volume

The volume of any pyramid is one third of the base area times the height:

V = \frac{1}{3} a^{2} h

Why one third? A cube can be divided into three congruent pyramids. Each pyramid therefore has $\frac{1}{3}$ of the cube's volume. This relationship holds for all pyramids, not just those carved from cubes.

Worked example 1 -- Surface area

Problem: A regular square pyramid has base edge

a = 10 cm

and height

h = 12 cm

. Find the total surface area. Step 1 -- Find the slant height:

h_{s} = h^{2} + (\frac{a}{2})^{2} = 1 2^{2} + 5^{2} = 144 + 25 = 169 = 13 cm

Step 2 -- Base area:

S_{base} = a^{2} = 1 0^{2} = 100 cm^{2}

Step 3 -- Lateral area:

S_{lateral} = 2 a \cdot h_{s} = 2 \cdot 10 \cdot 13 = 260 cm^{2}

Step 4 -- Total surface area:

S = 100 + 260

S = 360 cm^{2}

Worked example 2 -- Volume and slant height

Problem: A regular square pyramid has base edge

a = 6 cm

and slant height

h_{s} = 5 cm

. Find the volume. Step 1 -- Find the height from the slant height:

h_{s}^{2} = h^{2} + (\frac{a}{2})^{2}

25 = h^{2} + 9

h^{2} = 16

h = 4 cm

Step 2 -- Calculate the volume:

V = \frac{1}{3} a^{2} h = \frac{1}{3} \cdot 36 \cdot 4 = \frac{144}{3}

V = 48 cm^{3}

Summary of formulas

Quantity	Formula
Slant height	$h_{s} = h^{2} + (a /2)^{2}$
Base area	$S_{base} = a^{2}$
Lateral area	$S_{lateral} = 2 a \cdot h_{s}$
Total surface area	$S = a^{2} + 2 a \cdot h_{s}$
Volume	$V = \frac{1}{3} a^{2} h$

Practice exercises

Put your knowledge to the test:

Surface Area and Volume of a Pyramid – Formulas and Examples

Surface Area and Volume of a Pyramid

Table of contents

What is a regular square pyramid?

Key dimensions

Slant height

Net of a pyramid

Base area

Lateral (side) area

Total surface area

Volume

Worked example 1 -- Surface area

Worked example 2 -- Volume and slant height

Summary of formulas

Practice exercises

Surface Area and Volume of a Pyramid – Formulas and Examples

Surface Area and Volume of a Pyramid

Table of contents

What is a regular square pyramid?

Key dimensions

Slant height

Net of a pyramid

Base area

Lateral (side) area

Total surface area

Volume

Worked example 1 -- Surface area

Worked example 2 -- Volume and slant height

Summary of formulas

Related articles

Practice exercises