Area is the number of square units needed to cover a surface or figure and relates to a 2D object. The surface area is the area covering a 3D object.

Surface area has lots of applications, for example:

In manufacturing we may need to calculate the cost of making boxes or sheet metal parts.
In construction, surface area affects planning (how much to buy) and costs (how much to charge) in connection with items like wallboard, shingles, and paint.

Many objects have complex shapes to increase their surface area: the inside of your lungs, intestines, and brain; air purifiers, or radiators.

We will start by looking at how to find the surface area of a rectangular prism.

Ideas

Surface area of rectangular prisms

Surface area of rectangular prisms

Rectangular prisms have three pairs of congruent faces. We can see below how we could break the rectangular prism above into three pairs of congruent rectangles. To find the total surface area, we must add up the area of all of the faces.

The image shows 4 rectangular prisms with opposite pairs of faces shaded the same color.

The image shows a rectangular prism with length,l, width,w, and height,h.

\text{Surface area of a prism} = \text{Sum of areas of faces}

If we are just looking at a rectangular prism, we can use a formula instead of adding up all 6 faces separately.

This is a rectangular prism with length l, width w, and height h.

As we saw with the figure above, there are three pairs of congruent rectangles.

The top and bottom which are both l \times w
The left and right which are l \times h
The front and back which are w \times h

Since there are two of each of these rectangles we get the formula below.

SA=2lw+2lh+2wh

Examples

Example 1

Consider the following cube with a side length equal to 6 \text{ cm}.

The image shows a cube with a side length of 6 centimeters.

Find the total surface area.

Worked Solution

Create a strategy

We can use the surface area of cube formula: \text{Surface area}=6\times \text{side}^{2}, where 6=\text{no. of faces}.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 6\times (6)^{2}	Substitute the values
	\displaystyle =	\displaystyle 6 \times 36	Evaluate the square
	\displaystyle =	\displaystyle 216\text{ cm}^{2}	Evaluate the multiplication

Example 2

Consider the following rectangular prism with length, width and height equal to 12 \text{ m}, 6 \text{ m} and 4 \text{ m} respectively.

The image shows a rectangular prism with a height of 4 meters, width of 6 meters, and length of 12 meters.

Find the surface area of the prism.

Worked Solution

Create a strategy

We can use the surface area of rectangular prism formula: SA=2lw + 2lh + 2wh, where \\l=\text{length},\, w=\text{width}, and h=\text{height}.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2(12\times 6) + 2(12 \times 4) + 2(6 \times 4)	Substitute the values of l, w, \text{and} \,h
	\displaystyle =	\displaystyle 2(72) + 2(48) +2(24)	Evaluate the multiplication inside the brackets
	\displaystyle =	\displaystyle 144 + 96 + 48	Evaluate the multiplication
	\displaystyle =	\displaystyle 288\text{ m}^{2}	Evaluate the addition

Idea summary

\displaystyle \text{Surface area of a prism} = \text{Sum of areas of faces}

\bm{\text{Surface area of a prism}}

is the sum of the areas of faces.

\displaystyle SA=2lw+2lh+2wh

\bm{SA}

is the surface area of the prism.

\bm{l}

is the length of the prism.

\bm{w}

is the width of the prism.

\bm{h}

is the height of the prism.

Outcomes

7.G.A.3

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

7.G.B.6

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

7.06 Surface area of right prisms

Introduction