Explainer: Surface Areas of Rectangular Prism and Cubes

In this explainer, we will learn how to calculate surface areas of rectangular prisms and cubes and use this to solve real-life problems.

Let us first recall the definition of a rectangular prism.

Definition: Rectangular Prism

A rectangular prism is a box-shaped solid object. It has six faces that are rectangles. It is described by its length, width, and height.

In this explainer, we are interested in the surface area of rectangular prisms. We usually distinguish between lateral surface area and total surface area.

Definitions

The lateral surface area of a prism is the total surface area of only the lateral sides of the prism (that is, of the faces that are not the bases).

The total surface area of a prism is the sum of the areas of its lateral sides plus those of the bases.

How to Find the Surface Area of a Rectangular Prism

A rectangular prism has three pairs of congruent rectangular faces whose areas are ๐‘™โ‹…๐‘ค, ๐‘คโ‹…โ„Ž and โ„Žโ‹…๐‘™, where ๐‘™ is the length, ๐‘ค is the width, and โ„Ž is the height of the prism, as shown in the diagram. To help us visualize all the faces of a rectangular prism, it is helpful to draw its net, also as shown in the diagram.

The surface area of a rectangular prism is, therefore, SArectangularprism=2โ‹…(๐‘™โ‹…๐‘ค+๐‘คโ‹…โ„Ž+โ„Žโ‹…๐‘™).

Let us now look at some questions to see how the surface area of a rectangular prism is calculated.

Example 1: Finding the Surface Area of a Rectangular Prism

Find the surface area of the rectangular prism shown.

Answer

A rectangular prism has three pairs of congruent rectangular faces whose areas are ๐‘™โ‹…๐‘ค, ๐‘คโ‹…โ„Ž and โ„Žโ‹…๐‘™, where ๐‘™ is the length, ๐‘ค is the width, and โ„Ž is the height of the prism. Therefore, the surface area of a rectangular prism is SArectangularprism=2โ‹…(๐‘™โ‹…๐‘ค+๐‘คโ‹…โ„Ž+โ„Žโ‹…๐‘™).

Here, the length is 39 m, the width is 29 m, and the height is 30 m. After having checked that all the lengths are expressed in the same unit, we can plug them into our equation: SArectangularprism=2โ‹…(39โ‹…29+29โ‹…30+30โ‹…39)=6,342m2.

The surface area of the rectangular prism is 6,342 m2.

Example 2: Finding the Surface Area of a Cube

Find the surface area of a cube of length 11 centimeters.

Answer

A cube is a particular rectangular prism with 6 congruent faces that are squares. Its surface area is then six times the area of one of its faces. In this question, the side of the cube is 11 centimeters. The area of one of its faces in square centimeters is then 112. That is, 121 cm2. The surface area of the cube is six times this area; that is, 6ร—121=726cm2.

Now that we have seen how to find the surface area of a rectangular prism when we know its three dimensions, let us look at a couple of questions where we need to derive some missing dimensions of the rectangular prism from the information given in the question.

Example 3: Finding the Surface Area of Half of a Cube

Suppose the length of each edge of an ice cube is 19 centimeters. The cube is then cut horizontally in half into two smaller rectangular prisms. Determine the surface area of one of the two prisms.

Answer

We need to visualize how the cube is cut into two equal rectangular prisms: only one of the three dimensions is halved, while the two others remain 19 cm.

A rectangular prism has three pairs of congruent rectangular faces whose areas are ๐‘™โ‹…๐‘ค, ๐‘คโ‹…โ„Ž and โ„Žโ‹…๐‘™, where ๐‘™ is the length, ๐‘ค is the width, and โ„Ž is the height of the prism. Therefore, the surface area of a rectangular prism is SArectangularprism=2โ‹…(๐‘™โ‹…๐‘ค+๐‘คโ‹…โ„Ž+โ„Žโ‹…๐‘™).

Here, the length is 19 cm, the width is 19 cm, and the height is 19รท2=9.5cm. After having checked that all the lengths are expressed in the same unit, we can plug them into our equation: SArectangularprism=2โ‹…(19โ‹…19+19โ‹…9.5+9.5โ‹…19)=1,444cm2.

The surface area of the rectangular prism is 1,444 cm2.

Example 4: Finding the Surface Area of a Cube Knowing the Perimeter of Its Base

If the perimeter of the base of a cube is 54.4 cm, find its total surface area.

Answer

A cube is a particular rectangular prism with 6 congruent faces that are squares. Its surface area is then six times the area of one of its faces. To find the area of one face, we need to know the side of the cube. However, in this question, we do not know the side of the cube; we only know the perimeter of its base. Its base is one of its faces; since they are all congruent, there is no need to specify which one is considered the base. A square has four sides of equal length, so the perimeter of a square is four times the length of the side. Hence, we have 54.4=4โ‹…๐‘ , where ๐‘  is the side of the cube. The side is, therefore, one quarter of the perimeter. That is,๐‘ =54.44=13.6cm.

The area of one face is Aface=๐‘ 2=13.62=184.96cm2.

The surface area of the cube is six times the area of one of its faces:SAcube=6โ‹…Aface=6โ‹…184.96=1,109.76cm2.

The surface area of the cube is 1,109.76 cm2.

We are going to look at a question where we know the surface area of a cube but want to find the area of one face of the cube.

Example 5: Finding the Area of a Face of a Cube Knowing Its Surface Area

The surface area of a cube is 1,020 cm2. What is the area of one face of the cube?

  1. 170 cm2
  2. 255 cm2
  3. 85 cm2
  4. 340 cm2

Answer

A cube is a particular rectangular prism with 6 congruent faces that are squares. Its surface area is then six times the area of one of its faces. Therefore, we can writeSAcube=6โ‹…Aface.

It means that the area of one face is one-sixth of the surface area of the cube:Aface=SAcube6.

We are given the surface area of the cube in the question. It is 1,020 cm2. Hence, we have Aface=1,0206=170cm2.

The area of one face of the cube is 170 cm2.

Key Points

  1. A rectangular prism is a box-shaped solid object. It has six faces that are rectangles. It is described by its length, width, and height.
  2. A rectangular prism has three pairs of congruent rectangular faces whose areas are ๐‘™โ‹…๐‘ค, ๐‘คโ‹…โ„Ž and โ„Žโ‹…๐‘™, where ๐‘™ is the length, ๐‘ค is the width, and โ„Ž is the height of the prism. Therefore, the surface area of a rectangular prism is SArectangularprism=2โ‹…(๐‘™โ‹…๐‘ค+๐‘คโ‹…โ„Ž+โ„Žโ‹…๐‘™).
  3. A cube is a particular rectangular prism with 6 congruent faces that are squares. Its surface area is then six times the area of one of its faces.
  4. Remember to make sure that all dimensions are given in the same unit before performing any calculation.

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