The volume of a rectangular pyramid is calculated by multiplying the area of the base by the height of the pyramid and dividing by three. The surface area of a rectangular pyramid is calculated by adding the areas of all the faces of the pyramid.
Here, we will look at the formulas that we can use to calculate the volume and the area of a rectangular pyramid. In addition, we will solve some exercises in which we will apply this formula.
How to find the volume of a rectangular prism
The volume of any pyramid is calculated by multiplying the area of its base by the length of its height and dividing by 3. Therefore, we have the following formula:
$latex \text{Volume} = \frac{1}{3}\text{Area base} \times \text{Height}$
In a rectangular pyramid, its base is a rectangle. Remember that the area of a rectangle is calculated by multiplying the length of its base by the length of its width. Therefore, we have the following formula:
| $latex V=\frac{1}{3}b\times a\times h$ |
where b is the length of the rectangular base, a is the width of the rectangular base, and h es the height of the pyramid.
How to find the surface area of a rectangular pyramid
The surface area of rectangular pyramids is equal to the sum of the areas of all the faces of the pyramid. These pyramids are composed of a rectangular face and four triangular faces. The opposite triangular faces are the same. Therefore, we have to find expressions for the areas of triangles and rectangles.
Remember that the area of a rectangle is equal to the length of its base multiplied by its width. Therefore, for the base, we have the area $latex A=ba$, where b is the length of the base and a is the length of the width.
On the other hand, we know that the area of a triangle is equal to one-half of the triangle’s base multiplied by the height. We have two different bases for the triangles, b and a, which are the lengths of the rectangular base. Also, the slant height of the triangles is the same.
Therefore, we have two different areas for the triangles, $latex \frac{1}{2} bh$ and $latex \frac{1}{2} ah$, where h represents the length of the slant height of the triangular faces. We have two triangles with each of these areas, so the total surface area of the pyramid is:
$latex A_{s}=ba+bh+ah$
Volume and area of a rectangular pyramid – Examples with answers
EXAMPLE 1
What is the volume of a pyramid that has a height of 5 m and a rectangular base with a length of 4 m and a width of 5 m?
Solution
We have the following lengths:
- Pyramid height, $latex h=5$
- Rectangle base, $latex b=4$
- Rectangle width, $latex a=5$
Using these lengths in the volume formula, we have:
$latex V=\frac{1}{3}b\times a\times h$
$latex V=\frac{1}{3}\times4\times 5\times 5$
$latex V=33.3$
The volume is equal to 33.3 m³.
EXAMPLE 2
What is the surface area of a pyramid that has a rectangular base with a width of 5 m and a base of 4 m and where the slant height of the faces is 5 m?
Solution
We have the following values:
- Rectangle width, $latex a=5$
- Rectangle base, $latex b=4$
- Triangles slant height, $latex h=5$
Using these values in the formula for surface area, we have:
$latex A_{s}=ba+bh+ah$
$latex A_{s}=(4)(5)+(4)(5)+(5)(5)$
$latex A_{s}=20+20+25$
$latex A_{s}=65$
The surface area is equal to 65 m².
EXAMPLE 3
If a rectangular pyramid has a base of 6 m, a width of 5 m, and a height of 6 m, what is its volume?
Solution
From the question, we get the following values:
- Pyramid height, $latex h=6$
- Rectangle base, $latex b=6$
- Rectangle width, $latex a=5$
Using these values in the volume formula, we have:
$latex V=\frac{1}{3}b\times a\times h$
$latex V=\frac{1}{3}\times6\times 5\times 6$
$latex V=60$
The volume is equal to 60 m³.
EXAMPLE 4
A pyramid has a rectangular base with a width of 6 m and a base of 7 m. If the slant height of the triangular faces is 8 m, what is the surface area?
Solution
We recognize the following information:
- Rectangle width, $latex a=6$
- Rectangle base, $latex b=7$
- Triangles slant height, $latex h=8$
We substitute these values in the formula for surface area:
$latex A_{s}=ba+bh+ah$
$latex A_{s}=(7)(6)+(7)(8)+(6)(8)$
$latex A_{s}=42+56+48$
$latex A_{s}=146$
The surface area is equal to 146 m².
EXAMPLE 5
A pyramid has a height of 7 m. If its rectangular base has a width of 6 m and a base of 8 m, what is its volume?
Solution
We have the following information:
- Pyramid height, $latex h=7$
- Rectangle base, $latex b=8$
- Rectangle width, $latex a=6$
We use the volume formula with these values:
$latex V=\frac{1}{3}b\times a\times h$
$latex V=\frac{1}{3}\times 8\times 6\times 7$
$latex V=112$
The volume is equal to 112 m³.
EXAMPLE 6
What is the surface area of a rectangular pyramid that has triangular faces with a height of 9 m and a rectangular base with a width of 7 m and a base of 9 m?
Solution
We have the following information:
- Rectangle width, $latex a=7$
- Rectangle base, $latex b=9$
- Triangles slant height, $latex h=9$
By substituting these values in the formula for surface area, we have:
$latex A_{s}=ba+bh+ah$
$latex A_{s}=(9)(7)+(9)(9)+(7)(9)$
$latex A_{s}=63+81+63$
$latex A_{s}=207$
The surface area is equal to 207 m².
EXAMPLE 7
A rectangular pyramid has a volume of 10 m³. If its height is 5 m and its width is 3 m, what is the length of its base?
Solution
We have the following:
- Volume, $latex V=10$
- Pyramid height, $latex h=5$
- Rectangle width, $latex a=3$
In this case, we start with the volume and want to find the length of its base. Therefore, we use the volume formula and solve for b:
$latex V=\frac{1}{3}b\times a\times h$
$latex 10=\frac{1}{3}\times b\times 3\times 5$
$latex 30= b\times 3\times 5$
$latex 30= 15b$
$latex b=2$
The length of the base is 2 m.
EXAMPLE 8
If a rectangular pyramid has a rectangular base with a width of 8 m and a base of 10 m and the height of its triangular faces is 11 m, what is its surface area?
Solution
We have the following information:
- Rectangle width, $latex a=8$
- Rectangle base, $latex b=10$
- Triangles slant height, $latex h=11$
We find the surface area using these values in the formula:
$latex A_{s}=ba+bh+ah$
$latex A_{s}=(10)(8)+(10)(11)+(8)(11)$
$latex A_{s}=80+110+88$
$latex A_{s}=278$
The surface area is equal to 278 m².
EXAMPLE 9
What is the length of the base of a rectangular pyramid that has a volume of 144 m³, a width of 6 m, and a height of 9 m?
Solution
We have the following values:
- Volume, $latex V=144$
- Pyramid height, $latex h=9$
- Rectangle width, $latex a=6$
We use the values given in the volume formula and solve for b:
$latex V=\frac{1}{3}b\times a\times h$
$latex 144=\frac{1}{3}\times b\times 6\times 9$
$latex 432= b\times 6\times 9$
$latex 432= 54b$
$latex b=8$
The length of the base is 8 m.
Volume and area of a rectangular pyramid – Practice problems
If we have a rectangular pyramid with a base of 11 m, a width of 12 m and a height of triangular faces of 10 m, what is its surface area?
Write the answer in the input box.
See also
Interested in learning more about triangular pyramids? Take a look at these pages:
