The surface area of a rectangular prism is equal to the total area occupied by the prism. The surface area is a two-dimensional measurement, so it can be measured in m², cm², among others. To calculate the surface area, we have to add the individual areas of all the faces of the rectangular prism. In total, we have 6 faces in a rectangular prism, so we add the areas of the six faces. The area of each face depends on two of the three dimensions of the prism, length, base, and height.

Here, we will learn about the formula that we can use to find the surface area of rectangular prisms. In addition, we will solve some problems, in which we will apply this formula.

GEOMETRY
formula for the surface area of a rectangular prism

Relevant for

Learning about the surface area of a rectangular prism.

See examples

GEOMETRY
formula for the surface area of a rectangular prism

Relevant for

Learning about the surface area of a rectangular prism.

See examples

Formula to find the surface area of a rectangular prism

A rectangular prism has two parallel rectangular bases and four rectangular faces. The surface area of the rectangular prism is the sum of the area of the lateral faces and the rectangular bases. The unit of measurement of the surface area of the rectangular prism is square units.

Let’s consider the following rectangular prism with its dimensions:

diagram of a rectangular prism with dimensions

The formula for the surface area of this rectangular prism is:

A_{S}=2(bl+lh+hb)

where,

  • b is the length of the base
  • l is the length of the width of the prism
  • h is the length of the height of the prism

This formula is derived by considering that the parallel faces in a rectangular prism have the same area.


Surface area of a rectangular prism – Examples with answers

The formula for the surface area of rectangular prisms is applied to solve the following examples. Each example has its respective solution, but it is recommended that you try to solve the problems yourself.

EXAMPLE 1

A rectangular prism has a base of 5 m, a width of 4 m, and a height of 4 m. What is its surface area?

We can obtain the following information:

  • Base, b=5
  • Width, l=4
  • Height, h=4

We use the formula for surface area with these values. Therefore, we have:

A_{S}=2(bl+lh+hb)

A_{S}=2((5)(4)+(4)(4)+(4)(5))

A_{S}=2(20+16+20)

A_{S}=2(56)

A_{S}=112

The surface area is 112 m².

EXAMPLE 2

What is the surface area of a rectangular prism that has a base of 7 m, a width of 6 m, and a height of 8 m?

From the question, we have the following values:

  • Base, b=7
  • Width, l=6
  • Height, h=8

We substitute  these values in the formula for surface area:

A_{S}=2(bl+lh+hb)

A_{S}=2((7)(6)+(6)(8)+(8)(7))

A_{S}=2(42+48+56)

A_{S}=2(146)

A_{S}=292

The surface area is 292 m².

EXAMPLE 3

If a rectangular prism has a base of 8 m, a height of 12 m, and a width of 11 m. What is its surface area?

We have the following information:

  • Base, b=8
  • Width, l=11
  • Height, h=12

Using these values in the formula for surface area, we have:

A_{S}=2(bl+lh+hb)

A_{S}=2((8)(11)+(11)(12)+(12)(8))

A_{S}=2(88+132+96)

A_{S}=2(316)

A_{S}=632

The surface area is 632 m².

EXAMPLE 4

What is the length of the height of a rectangular prism that has a surface area of 148 m² if its base is 6 m and its width is 4 m?

We have the following information:

  • Base, b=6
  • Width, l=4
  • Surface area, A=148

Here, we have the surface area and we want to find the length of the height. Therefore, we use the formula and solve for h:

A_{S}=2(bl+lh+hb)

148=2((6)(4)+(4)h+(6)(h))

148=2(24+10h)

74=24+10h)

10h=74-24

10h=50

h=5

The length of the height is 5 m.

EXAMPLE 5

What is the length of the height of a rectangular prism that has a surface area of 340 m², a width of 5 m, and a base of 8 m?

We have the following values:

  • Base, b=8
  • Width, l=5
  • Surface area, A=340

We use the formula for surface area with these values and solve for h:

A_{S}=2(bl+lh+hb)

340=2((8)(5)+(5)h+(8)(h))

340=2(40+13h)

170=40+13h)

13h=170-40

13h=130

h=10

The length of the height is 10 m.


Surface area of a rectangular prism – Practice problems

The following problems can be used to practice using the formula for the surface area of regular prisms. If you need help with these problems, you can look at the solved examples above.

If a rectangular prism has a base of 7m, a width of 11m, and a height of 5m, what is its surface area?

Choose an answer






What is the surface area of a rectangular prism that has a base of 11m, a width of 9m, and a height of 13m?

Choose an answer






A rectangular prism has a surface area of 188 {{m}^2}, a base of 6m, and a width of 4m. What is the length of its height?

Choose an answer






If a rectangular prism has a surface area of 382 {{m}^2}, a base of 7m, a width of 8m, what is the length of its height?

Choose an answer







See also

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