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Surface Area of Prisms

Area of base multiplied by height

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Surface Area of Prisms

Crystal has a fragile gift that she is packing in a box before wrapping it. To figure out how much wrapping paper she will need she has drawn the box with all its dimensions. How can Crystal calculate how much gift-wrap she will need using the above dimensions?

In this concept, you will learn to find the surface area of prisms.

Guidance

A prism is a three–dimensional figure with two parallel congruent polygons as bases. The side faces of a prism are rectangular in shape.

One measure when working with three–dimensional figures is called surface area. Surface area is the total of the areas of each face of a solid figure. Imagine you could wrap one of these figures in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, you must be able to calculate the area of each face and then add these areas together.

There are several different ways to calculate surface area. One way is to use a net. A net is a two-dimensional diagram of a three-dimensional figure. Imagine you could unfold a box so that it is completely flat. You would have something that looks like this.

If you folded this up, you could see that it would form a cube. A cube is made up of faces that are squares. If you wanted to figure out the surface area or measurement of the outer covering of this cube, then you could find the area of each surface of the cube and then add the products together.

You could also look at a net of a rectangular prism.

A rectangular prism is made up of rectangles. To find the surface area of a prism, you would need to calculate the area of each of the faces and then add them together.

Let’s begin by calculating the surface area of a rectangular prism.

First, let’s break apart the shape so you can organize the six area calculations.

Next, calculate the area of each face.

Abottom face===l×w12×784

Alonger side===l×w12×336

Atop face===l×w12×784

Ashorter side===l×w7×321

Alonger side===l×w12×336

Ashorter side===l×w7×321

Surface AreaSurface Area==84+84+36+36+21+21282

The surface area of the prism is 282 in2\begin{align*}282 \ in^2\end{align*}.

Now you can see each face so that you can calculate their area and then add them together. However, you can also use a formula to represent the faces as you find their area. The formula gives a nice short cut that you can use for any kind of prism, no matter what shape its base is.

Take a look at the formula below.

SA=Ph+2B

Let’s look at the first part of the formula. P\begin{align*}P\end{align*} represents the perimeter of the base, and h\begin{align*}h\end{align*} represents the height of the prism. By multiplying the perimeter and height, you are finding the area of all of the side faces at once. This will be very useful if the prism that you are working with isn’t just a cube or a rectangular prism.

The second part of the formula represents the area of the top and bottom faces. B\begin{align*}B\end{align*} represents the area of one base, which you find using whichever area formula is appropriate for the shape of the base. Then you multiply it by 2 to show the area of the top and bottom faces at once.

Let’s look at an example.

Find the surface area of this figure using a formula.

First, find the perimeter of the base.

PPP===2l+2w2×21+2×1470

Next, calculate the area of the base.

AAA===l×w21×14294

Then, knowing the height is 5 cm, determine the surface area.

SASASASA====Ph+2B70×5+2×294350+588938

The surface area of the prism is 938 cm2\begin{align*}938 \ cm^2\end{align*}.

Guided Practice

What is the surface area of the figure below?

First, find the perimeter of the base.

PPP===2a+b2×5+818

Next, calculate the area of the base.

AAA===12bh128×312

Then, knowing the height is 15 in, determine the surface area.

SASASASA====Ph+2B18×15+2×12270+24294

The surface area of the prism is 294 in2\begin{align*}294 \ in^2\end{align*}.

Examples

Example 1

True or false: The surface area includes the inside of a prism.

The surface area is the measurement of the outer covering of a prism.

Example 2

True or false: A net shows all three dimensions of a prism.

A net does show all dimensions of a prism.

Example 3

True or false: You know a figure is a prism because the faces are rectangles.

A figure with all rectangular faces is a prism.

Credit: Personal Creations
Source: https://www.flickr.com/photos/personalcreations/18495564258/in/photolist-uboASW-quL2jJ-e8LSng-7m4JNS-imbaWw-5LZtNu-i2TDMK-4fwjAq-imbgcG-DDhS4-kFxMRp-4gyeSL-7woThM-qsmWRY-5XLYBC-6r5pja-6wWPTo-dzEJAk-9a3RKJ-93vuU1-dbqjHZ-o9p5JB-7DRXAB-dBCC2j-6N2UGx-izL7kj-5NfoQj-5KW1xn-7tLDtG-s75K8E-6E3ThJ-9naX8S-7t46mb-7ngvNx-ayZn7U-qhfpb4-pkEGbF-jQDMHR-81envi-jnL4eH-7sZ8hZ-dDHPhz-83KTMe-q6Q5uH-7kZMig-95cw32-scKGU-HrpBB-btwBNu-dvcL2y

Remember Crystal and her gift box?

The box to be wrapped has a length of 12\begin{align*}12^{\prime \prime}\end{align*}, a wide of 9\begin{align*}9^{\prime \prime}\end{align*} and a height of 6\begin{align*}6^{\prime \prime}\end{align*}.

First, find the perimeter of the base.

PPP===2l+2w2×12+2×942

Next, calculate the area of the base.

AAA===l×w12×9108

Then, knowing the height is 6 in., determine the surface area.

SASASASA====Ph+2B42×6+2×108252+216468

Crystal needs 468 in.2\begin{align*}468 \ in.^2\end{align*} of wrapping paper.

Explore More

Look at each figure and then answer the following questions.

1. What is the name of the figure pictured above?

2. What is the surface area of this figure?

3. What is the shape of it’s base?

4. What is the height of this figure?

5. What is the area of this figure’s base?

6. What is the name of this figure?

7. What is the surface area of this figure?

8. What is the shape of it’s base?

9. What is the height of this figure?

10. What is the area of this figure’s base?

11. What is the name of this figure?

12. What is the shape of the base?

13. How many bases does this figure have?

14. How many side faces are there?

15. What is the surface area of this figure?

Vocabulary Language: English

Net

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Prism

Prism

A prism is a three-dimensional object with two congruent parallel bases that are polygons.
Surface Area

Surface Area

Surface area is the total area of all of the surfaces of a three-dimensional object.
Three – Dimensional

Three – Dimensional

A figure drawn in three dimensions is drawn using length, width and height or depth.