The volume of a square pyramid is obtained by multiplying the area of the base of the pyramid by its height and dividing the product by three. On the other hand, the surface area of a square pyramid is defined as the sum of the areas of all the faces of the pyramid.
Here, we will look at the formulas we can use to calculate the volume and surface area of square pyramids. Then, we will use these formulas to solve some practice problems.
How to find the volume of a square prism
The volume of any pyramid is found by multiplying the area of its base times the length of its height and dividing by three. This means that we have the following formula:
$latex \text{Volume}=\frac{1}{3}\text{Area base}\times \text{Height}$
We know that the base of a square pyramid is a square. Furthermore, we also know that the area of a square is found by squaring one of its side lengths. Therefore, we have the following formula:
| $latex V=\frac{1}{3}{{l}^2}\times h$ |
where, l is the length of one side of the square base, and h is the height of the pyramid.
How to find the surface area of a square pyramid
The surface area of a square pyramid is equivalent to the sum of the areas of all the faces of the pyramid. These pyramids have a square face at the base and four lateral triangular faces. Since the square base has sides of the same length, its area is equal to the length of one of the sides squared.
On the other hand, we know that the area of any triangle is equal to one-half the length of the base times the height of the triangle. In square pyramids, the bases of the triangular faces are the sides of the square base.
This means that the bases of the four triangular faces are equal, therefore their areas are equal. Thus, we have the following formula:
| $latex A_{S}={{l}^2}+2lh$ |
where l represents the length of one of the sides of the square base and h represents the slant height of the triangular faces.
Volume and area of a square pyramid – Examples with answers
EXAMPLE 1
What is the volume of a square pyramid that has a height of 5 m and sides of length 4 m?
Solution
We have the following lengths
- Sides of the square, $latex l=4$
- Height of pyramid, $latex h=5$
Therefore, we use these values in the volume formula:
$latex V=\frac{1}{3}{{l}^2}\times h$
$latex V=\frac{1}{3}{{(4)}^2}\times (5)$
$latex V=\frac{1}{3}(16)\times (5)$
$latex V=26.67$
The volume is equal to 26.67 m³.
EXAMPLE 2
What is the surface area of a pyramid that has a square base with sides of 3 m and triangular faces with a height of 4 m?
Solution
We have the following values:
- Base sides, $latex l=3$
- Height of triangles, $latex h=4$
We use this information in the formula for surface area:
$latex A_{S}={{l}^2}+2lh$
$latex A_{S}={{3}^2}+2(3)(4)$
$latex A_{S}=9+24$
$latex A_{S}=33$
The surface area is 33 m².
EXAMPLE 3
If we have a pyramid with a height of 6 m and a square base with sides of 5 m, what is its volume?
Solution
From the question, we get the following information:
- Sides of the square, $latex l=5$
- Height of pyramid, $latex h=6$
By substituting these values into the volume formula, we have:
$latex V=\frac{1}{3}{{l}^2}\times h$
$latex V=\frac{1}{3}{{(5)}^2}\times (6)$
$latex V=\frac{1}{3}(25)\times (6)$
$latex V=50$
The volume is equal to 50 m³.
EXAMPLE 4
If a square pyramid has sides 5 m long and triangular faces 6 m high, what is its surface area?
Solution
We have the following information:
- Base sides, $latex l=5$
- Height of triangles, $latex h=6$
We use the formula for the surface area with the given information:
$latex A_{S}={{l}^2}+2lh$
$latex A_{S}={{5}^2}+2(5)(6)$
$latex A_{S}=25+60$
$latex A_{S}=85$
The surface area is 85 m².
EXAMPLE 5
What is the volume of a pyramid that has a height of 9 m and a square base with sides of length 8 m?
Solution
We observe the following information:
- Sides of the square, $latex l=8$
- Height of pyramid, $latex h=9$
Using these values in the volume formula, we have:
$latex V=\frac{1}{3}{{l}^2}\times h$
$latex V=\frac{1}{3}{{(8)}^2}\times (9)$
$latex V=\frac{1}{3}(64)\times (9)$
$latex V=192$
The volume is equal to 192 m³.
EXAMPLE 6
A pyramid has a square base with sides 10 m long and triangular faces 7 m high. What is its surface area?
Solution
From the question, we have the following:
- Base sides, $latex l=10$
- Height of triangles, $latex h=7$
We solve using the formula for surface area:
$latex A_{S}={{l}^2}+2lh$
$latex A_{S}={{10}^2}+2(10)(7)$
$latex A_{S}=100+140$
$latex A_{S}=240$
The surface area is 240 m².
EXAMPLE 7
A square pyramid has a volume of 96 m³. If its sides are 6 m long, what is the length of its height?
Solution
We have the following values:
- Sides of the square, $latex l=6$
- Volume, $latex V=96$
In this case, we have the volume and we want to find the length of the height of the pyramid. Therefore, we use the volume formula and solve for h:
$latex V=\frac{1}{3}{{l}^2}\times h$
$latex 96=\frac{1}{3}{{(6)}^2}\times h$
$latex 96=\frac{1}{3}(36)\times h$
$latex 96=12 h$
$latex h=8$
The length of the height is equal to 8 m.
EXAMPLE 8
What is the surface area of a square pyramid that has sides 11 m long and triangular faces 12 m high?
Solution
We have the following information:
- Base sides, $latex l=11$
- Height of triangles, $latex h=12$
We use the formula for surface area with these values:
$latex A_{S}={{l}^2}+2lh$
$latex A_{S}={{11}^2}+2(11)(12)$
$latex A_{S}=121+264$
$latex A_{S}=385$
The surface area is 385 m².
EXAMPLE 9
If a square pyramid has a volume of 75 m³ and a height of 9 m, what is the length of its height?
Solution
We have the following information:
- Height, $latex h=9$
- Volume, $latex V=75$
We use the volume formula and solve for it:
$latex V=\frac{1}{3}{{l}^2}\times h$
$latex 75=\frac{1}{3}{{l}^2}\times (9)$
$latex 75=3{{l}^2}$
$latex {{l}^2}=25$
$latex l=5$
The length of the sides is equal to 5 m.
Volume and area of a square pyramid – Practice problems
If a square pyramid has a volume of 12 m3 and a height of 4 m, what are the lengths of its sides?
Write the answer in the input box.
See also
Interested in learning more about geometric pyramids? Take a look at these pages:
