Let’s learn how to find the surface area and volume of a rectangular prism.
What is a Rectangular Prism?
Prisms are three-dimensional objects with two equal bases or ends, flat surfaces or sides, and the same cross-section along its length. A cube is a prism, but unlike a cube that has 6 equal square faces, a rectangular prism has six rectangular faces and 12 edges. It has a length, width, and height that make up 3 pairs of equal rectangular faces: top-bottom, left-right, and front-back. Examples of objects shaped like a rectangular prism are shoe boxes, books, buildings, and cabinets.
How to Find the Surface Area of a Rectangular Prism:
The total surface area of a rectangular prism is the sum of all the areas of its six rectangular sides. Recall that the area of a rectangle is the product of its length and width: A = l • w.
Related Reading: Area of a Rectangle – Formula & Examples
Use one of these formulas to find the surface area of a rectangular prism:
A = 2lw + 2lh + 2hw
or
SA = 2 (lw + lh + hw)
where: l = length of the prism; w = width of the prism; and h = height of the prism.
Note: Surface areas are expressed in cubic units such as in2, cm2, km2, m2. Make sure all units are the same before you compute the surface area.
Solve for Surface Area:
Find the surface area of a rectangular prism with the following measurements:
l = 8cm, w = 4cm, h = 2cm
Plug the figures into the formula for surface area and solve.
SA = 2 (lw + lh + hw)
SA = 2[(8cm • 4cm)+(8cm • 2cm )+(2cm • 4cm)]
SA = 2 [(32cm²) + (16cm²) +(8cm²)]
SA = 2 [56cm²]
SA = 112cm²
Therefore, the surface area of the rectangular prism is 112cm².
How to Find the Volume of a Rectangular Prism:
The volume of a rectangular prism is the total amount of space it takes up, and can be defined as the product of its length, width, and height.
Use this formula to find the volume of a rectangular prism:
V = l • w • h
where l = length of the prism; w = width of the prism; and h = height of the prism.
The volume of a rectangular prism also tells its capacity – or the amount of space inside an object that can be filled. Volumes are expressed in cubic units such as m3, km3, and cm3. If the volume refers to the prism’s capacity, it can also be expressed in liters (L) or milliliters (mL).
Note: Remember that all measurement units should be the same before you compute the volume.
Solve for Volume:
Find the volume of a rectangular prism with the following measurements:
l = 7cm, w = 3.5cm, h = 2.5cm
Plug the figures into the formula for volume and solve.
V = l • w • h
V = 7cm • 3.5cm • 2.5cm
V = 61.25cm3
Therefore, the volume of the rectangular prism is 61.25cm3.
Example #1: Finding SA & V of a Rectangular Prism when given Length, Width, and Height
Find the surface area and volume of the rectangular prism with the following measurements:
l = 11cm, w = 5cm, h = 3cm
Solution for Example #1:
We’ll start by finding the surface area first.
Step 1. Write the given figures:
l = 11cm, w = 5cm, h = 3cm.
Step 2. Plug the figures into the surface area formula and perform the needed operations.
SA = 2 (lw + lh + hw)
= 2 [(11cm • 5cm) + (11cm • 3cm) + (3cm • 5cm)]
=2 [(55cm²) +(33cm²) + (15cm²)]
= 2 [103cm²]
SA = 206cm²
Therefore, the surface area of the rectangular prism is 206cm².
Lastly, let’s find the volume by substituting the given figures into the formula for volume and multiplying them.
V = l • w • h
V = 11cm • 5cm • 3cm
V = 165cm3
Therefore, the volume of the rectangular prism is 165cm3.
Example #2: Finding the Width & SA of a Rectangular Prism when given Length, Height, and Volume
Find the width and surface area of the rectangular prism with the following measurements: l = 9m, h = 5m, and V = 270m3
Solution for Example #2:
First, find the width using the given volume, length, and height.
Step 1. Write the given figures:
l = 9m, h = 5m, and V = 270m3
Step 2. Multiply the volume formula V = l • w • h by (1 / lh) to find the formula for width.
(1 / lh) V = (l • w • h) (1 / lh)
V / lh = w
Step 3. Compute the width using the formula found in Step 2.
w = V / lh
w = 270m3 / (9m • 5m)
= 270m3 / 45m²
w = 6m
Therefore, the width of the rectangular prism is 6m.
Lastly, find the surface area.
Step 1. Write the given figures:
l = 9m, h = 5m, and w = 6m
Step 2. Substitute the figures into the surface area formula and solve.
SA = 2 (lw + lh + hw)
= 2 [(9m • 6m) + (9m • 5m) + (5m • 6m)]
= 2 [(54m²) + (45m²) + (30m²)]
= 2 [129m²]
SA = 258m²
Therefore, the surface area of the rectangular prism is 258m².
Example #3: Comparing Volumes of Rectangular Prisms
How many rectangular barrels can fit in a rectangular container with a length of 5m, width, of 3m, and height of 4m if each barrel has a length of 0.5m, width of 15.m, and height of 2.5m?
Solution for Example #3:
Determine how many barrels can fit in the container by dividing the container’s volume by the barrel’s volume.
Firstly, find the volume of the container.
Step 1. Write the measurements for the container:
l = 5m, w = 3m, h = 4m
Step 2. Plug the figures into the volume formula.
Vcontainer = l • w • h
= 5m • 3m • 4m
Vcontainer = 60m3
Therefore, the volume of the container is 60m3.
Next, find the volume for the barrels by following the same steps.
Step 1. Write the measurements for the barrel:
l = 0.5m, w = 1.5m, h = 0.25m
Step 2. Plug the figures into the volume formula.
Vbarrel = l • w • h
= 0.5m • 1.5m • 0.25m
Vbarrel = 0.1875m3
Therefore, the volume of each barrel is 0.1875m3.
Lastly, determine how many barrels the container can hold by dividing the container’s volume by the barrel’s volume.
Vcontainer / Vbarrel = 60m3 / 0.1875 m3 = 320
Therefore, the rectangular container can hold 320 barrels.
Thank you for reading. We hope it’s effective! Always feel free to revisit this page if you ever have any questions about the surface area and volume of a rectangular prism.
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