Surface area of a rectangular prism

Here you will learn about the surface area of a rectangular prism and how to calculate it.

Students will first learn about the surface area of a rectangular prism as part of geometry in 6th grade.

What is the surface area of a rectangular prism?

The surface area of a rectangular prism is the total area of all of the faces of a rectangular prism.

Rectangular prisms have three pairs of identical faces – top and bottom, front and back, and left and right. Notice that it is always the opposite faces that are identical. Rectangular prisms are also known as cuboids.

To calculate the total surface area of a rectangular prism, calculate the area of each rectangular face and add them all together.

For example,

You can also use the net of a rectangular prism to find the surface area. The rectangular prism above unfolded into its net would be:

Surface area is measured in square units (e.g. mm^2, cm^2, m^2 etc).

What is the surface area of a rectangular prism?

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – Geometry (6.G.A.4)
  Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

How to calculate the surface area of a rectangular prism

In order to work out the surface area of a rectangular prism:

1. Calculate the area of each face.
2. Add the six areas together.
3. Include the units.

Surface area of a rectangular prism examples

Example 1: surface area of a rectangular prism

Calculate the surface area of the rectangular prism.

1. Calculate the area of each face.

The area of the bottom is 9 \times 3=27 \, cm^2.

The top face is the same as the bottom face, so the area of the top is also 27 \, cm^2.

The area of the front is 9 \times 4=36 \, cm^2.

The back face is the same as the front face, so the area of the back is also 36 \, cm^2.

The area of the right hand side is 3 \times 4=12 \, cm^2.

The left side face is the same as the right side face, so the area of the left side is also 12 \, cm^2.

It will make our work clearer if you use a table:

2Add the six areas together.

Total surface area: 27+27+36+36+12+12=150

3Include the units.

The measurements on the rectangular prism are in cm, so the total surface area of the rectangular prism is 150 \mathrm{~cm}^2.

Example 2: surface area of a rectangular prism

Calculate the surface area of the rectangular prism.

Calculate the area of each face.

Example 3: surface area of a cube

Calculate the surface area of this cube.

Example 4: surface area with different units

Given that the base of this rectangular prism is the bottom rectangle, calculate the lateral surface area.

Calculate the area of each face.

Notice that one of the measurements is in meters and the rest are in centimeters. Before you can calculate any areas, ensure all units are the same.

In this case, 0.1 \, m = 10 \, cm, so you can use 10 \, cm.

For the lateral surface area of a rectangular prism, do not include the base area (bottom and top).

Example 5: surface area with a missing length

The surface area of this rectangular prism is 328 \mathrm{~cm}^2. Find the missing length.

Calculate the area of each face.

Use the variable x to represent the missing side.

Example 6: surface area word problem – missing height

Helena was making rectangular boxes out of wood. If she used 200 \mathrm{~cm}^2 of wood for the rectangular box shown above, what is the missing height?

Calculate the area of each face.

You can unfold the rectangular prism, and use the net to find the area of each face:

The base is a rectangle that measures 2 \, cm by 6 \, cm. The area of each base is 12 \, cm^2.

Add the six areas together.

Total area of the bases: 12+12=24

Subtract the area of the bases from the total amount of wood that Helena used, to see how much was used on the lateral faces: 200-24=176

The total area of the faces left is 176 \mathrm{~cm}^2.

Remember that the edges in a prism are always equal, so if you were to fold up the net, the 2 \, cm and 6 \, cm sides of the rectangle would combine to form an edge with each corresponding rectangle – making their lengths equal.

Labeling the missing length as x, this leaves the area of 4 faces:

• 6 \times x or 6 x
• 2 \times x or 2 x
• 6 \times x or 6 x
• 2 \times x or 2 x

Together they equal 176 \mathrm{~cm}^2, so the total of the missing areas can be written as:

6 x+2 x+6 x+2 x=176

or

16 x=176

Since 16 \times 11=176, the missing side length is 11.

Teaching tips for the surface area of a rectangular prism

• Provide opportunities that connect the surface area of rectangular prisms to real world applications. The connection to wrapping gifts or painting the sides of a prism are popular examples. Also allow students to think critically about each real life connection.
  
  Asking questions like “When wrapping a gift, is it possible to use more paper than the surface area? Why or why not?” or “When painting a rectangular prism in the real world, what is a situation where you might not paint the entire surface area?”

• Worksheets can be a useful practice tool for this topic, but make sure that students also have a sufficient amount of practice with real world 3D shapes and nets. Solving problems on worksheets require students to be able to work with 2D representations of 3D shapes.
  
  This is much easier for students if they have worked extensively with these shapes in the real world and can easily picture and rotate them in their minds.

Easy mistakes to make

• Calculating volume instead of surface area
  Volume and surface area are different things – volume is the space within the shape and is measured in cubic units, whereas surface area is the total area of the faces and is measured in square units. To find surface area, you need to work out the area of each face and add them together.

• Thinking all lateral faces are equal
  Only opposite sides are equal in a rectangular prism. This means that many rectangular prisms have 3 different sets of congruent rectangles.
  For example,
  

  

  
  It is possible for all lateral faces to be equal, but only if the base is a square.
  For example,
  

  

Related surface area lessons

• Surface area
• Surface area of a prism
• Surface area of a cylinder
• Surface area of a triangular prism
• Surface area of a pyramid
• Surface area of a cone
• Surface area of a sphere
• Surface area of a cube
• Surface area of a hemisphere

The next lessons are

• Pythagorean theorem
• Trigonometry
• Circle math

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