In construction, calculating surface area is a part of planning, for example, calculating the amount of materials to you need to buy, as well as determining costs for a project. Similar calculations are required in manufacturing and design based professions.

Surface area measures the outside of a three-dimensional figure and is measured in \text{units}^2. Some examples of real-life scenarios that require calculating surface area are:

A package shaped like a rectangular prism.

\\\\ \,\\\\\\ \,\\\\\\ \,\\ Determining much wrapping paper will be needed to wrap this package that is shaped like a rectangular prism.

\\\\ \,\\\\\\ \,\\\\\\ \,\\ Determining much paint is needed to paint the inside of a box.

Whether you want to find out the volume of a swimming pool so you know how much water it can hold or find out the available space that can be occupied by a van or a truck in a gargage, the concept of volume is used often in daily life.

The volume of a three dimensional shape is the amount of space that the shape takes up. Volume measures how much space is inside of a three-dimensional figure, or the capacity that the figure can hold and is measured in \text{units}^{3}. Some examples of real-life scenarios that require calculating volume are:

\\\\ \,\\\\\\ \,\\\\\\ \,\\Determining how much water is needed to fill a pool is important for calculating the amount of chemicals needed to keep the water safe for swimming.

A rectangular storage locker with boxes inside.

\\\\ \,\\\\\\ \,\\\\\\ \,\\Determining the capacity of a rectangular storage locker to decide if it is large enough to fit your items or if you need to upgrade to a larger locker.

Examples

Example 1

Ivan is building a storage chest in the shape of a rectangular prism. The chest will be 55\operatorname{cm} long, 41\operatorname{cm} deep, and 39\operatorname{cm} high. Ivan wants to paint the outside of the chest, not including the bottom.

Does this scenario require calculating surface area or volume?

Worked Solution

Apply the idea

Since Ivan is trying to measure how much area will be covered on the outside of the chest, Ivan must calculate the surface area of the rectangular prism.

Reflect and check

If the question had asked how much the chest could hold, we would calculate volume instead.

How much paint will Ivan need to cover the entire chest?

Worked Solution

Create a strategy

A rectangular prism has three pairs of identical faces.

To find the surface area of a rectangular prism, we want to add the areas of all these faces.

Apply the idea

The net of a rectangular prism is made up of three pairs of rectangles, with each pair representing one possible pair of dimensions of the prism.

A rectangular storage chest and its net. height is 39cm, length is 55cm and width is 41cm. The center of the net image is greyed out.

Since this rectangular prism has dimensions of 55\operatorname{cm}, 41\operatorname{cm}, and 39\operatorname{cm}, the pairs of dimensions for the faces of the net will be 55 \operatorname{cm} \cdot \,39 \operatorname{cm} , 41\operatorname{cm} \cdot \,39\operatorname{cm} and 55\operatorname{cm} \cdot \,41\operatorname{cm}.

Because he is not painting the bottom of the chest, the side with the dimension of 55\operatorname{cm} \cdot \, 41\operatorname{cm} will only need to be added one time. The other two faces will be added twice.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2\left( 55\cdot 39 \right) + 2\left( 41\cdot 39\right) + \left(55 \cdot 41\right)
	\displaystyle =	\displaystyle 4290+3198+2255	Evaluate the multiplication
	\displaystyle =	\displaystyle 9743\operatorname{cm} ^{2}	Evaluate the addition

The surface area of the chest is 9743\operatorname{cm} ^{2}.

Example 2

A nesting box in a shape of a rectangular prism with height of 83 centimeters, width of 54 centimeters and length of 29cm centimeters.

This wild animal house is made out of plywood.

The nesting box has a depth of 29 \text{ cm}, a height of 83\text{ cm} and front width of 54\text{ cm}.

If you wanted to fill the box so that it was 50\% full of straw, how much straw would be required?

Worked Solution

Create a strategy

Because we are looking to fill the nesting box, we must calculate the volume. Once we have the volume of the entire nesting box, we will calculate 50\% of the volume to determine how much straw is required to fill the box 50\% full.

Apply the idea

\displaystyle \text{Volume}	\displaystyle =	\displaystyle \text{length } \cdot \text{ width} \cdot \text{height}
	\displaystyle =	\displaystyle 29 \cdot 54 \cdot 83	Substitute the given dimensions
	\displaystyle =	\displaystyle 29\text{ cm}	Evaluate the multiplication

The volume of the nesting box is 129\,978 \text{ cm}^3.

Now, we will find 50\% of the volume. 50\% is half the capacity of the nesting box.

\displaystyle \text{Volume}	\displaystyle =	\displaystyle \dfrac{1}{2}\cdot 129\,978	Substitute the volume
	\displaystyle =	\displaystyle 65\,989	Evaluate the multiplication

The amount of straw needed is 65\,989 \text{ cm}^3

Example 3

Sabrina is making a candle in a cylindrical jar to give as a gift. She must first fill the jar with wax to make the candle and then wrap the candle in wrapping paper. The cylindrical jar has a radius of 3 \text{ in} and a height of 4 \text { in}.

How much wax will Sabrina need to fill the candle jar?

Worked Solution

Create a strategy

Because Sabrina needs to fill the inside of the cylindrical container, she has to calculate the volume. We can use the formula for volume of a cylinder, V=\pi r^2h, to calculate this.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle \pi \cdot 3^2 \cdot 4	Substitute the given values
	\displaystyle =	\displaystyle \pi \cdot 9 \cdot 4	Evaluate the exponent
	\displaystyle =	\displaystyle 113.1	Evaluate the multiplcation

Sabrina will need 113.1 \text{ in}^3 of wax to fill the candle jar.

How much wrapping paper will Sabrina need to wrap the candle?

Worked Solution

Create a strategy

Because Sabrina needs to wrap the outside of the candle, we need to calculate the surface area to determine the amount of wrapping paper needed. We can use the fomula for surface area of a cylinder, SA=2 \pi r^2 + 2 \pi rh, to find this.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2\pi r^2 + 2 \pi r h
	\displaystyle =	\displaystyle 2\cdot \pi \cdot 3^2 + 2\cdot \pi \cdot 3 \cdot 4	Substitute the given values
	\displaystyle =	\displaystyle 2 \cdot \pi \cdot 9 + 2\cdot \pi \cdot 3 \cdot 4	Evaluate the exponents
	\displaystyle =	\displaystyle 56.6+37.7	Evaluate the multiplication
	\displaystyle =	\displaystyle 94.3	Evaluate the addition

The surface area of the candle is 94.3\text{ in}^2. This means that Sabrina will need 94.3\text{ in}^2 of wrapping paper.

Idea summary

We can use the concept of surface area and volume in real-world problems.

The surface area is the sum of the areas of all surfaces of a figure.

The volume of a three-dimensional figure is a measure of capacity and is measured in cubic units.

Problems involving calculations regarding the outside of figure require calculating surface area. Problems involving how much a figure can hold or the interior capacity of a figure require calculating volume.

Outcomes

7.MG.1

The student will investigate and determine the volume formulas for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.

7.MG.1a

Develop the formulas for determining the volume of right cylinders and solve problems, including those in contextual situations, using concrete objects, diagrams, and formulas.

7.MG.1b

Develop the formulas for determining the surface area of rectangular prisms and right cylinders and solve problems, including those in contextual situations, using concrete objects, two- dimensional diagrams, nets, and formulas.

7.MG.1c

Determine if a problem in context, involving a rectangular prism or right cylinder, represents the application of volume or surface area.

7.05 Solve problems with volume and surface area

Ideas

Solve problems with volume and surface area

Examples

Example 1

Example 2

Example 3

Outcomes

7.MG.1

7.MG.1a

7.MG.1b

7.MG.1c

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