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Intermediate Geometry : How to find the surface area of a prism

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Example Questions

Example Question #1 : How To Find The Surface Area Of A Prism

Find the surface area of the rectangular prism:

The_surface_area_of_a_prism

Possible Answers:

$720\:units^2$ $\displaystyle 720\:units^2$

$258\:units^2$ $\displaystyle 258\:units^2$

$336\:units^2$ $\displaystyle 336\:units^2$

$180\:units^2$ $\displaystyle 180\:units^2$

$516\:units^2$ $\displaystyle 516\:units^2$

Correct answer:

$516\:units^2$ $\displaystyle 516\:units^2$

Explanation:

The_surface_area_of_a_prism

To find the surface area of a prism, the problem can be approached in one of two ways.

1. Through an equation that uses lateral area
2. Through finding the area of each side and taking the sum of all the faces

Using the second method, it's helpful to realize rectangular prisms contain $\displaystyle 6$ faces. With that, it's helpful to understand that there are $\displaystyle 3$ pairs of sides. That is, there are two faces with the same dimensions. Therefore, we really only have three sides for which we need to calculate areas:

Faces 1 & 2:

$15\cdot8=120$ $\displaystyle 15\cdot8=120$

Faces 3 & 4:

$6\cdot8=48$ $\displaystyle 6\cdot8=48$

Faces 5 & 6:

$15\cdot6=90$ $\displaystyle 15\cdot6=90$

Now, we can add up the areas of all six sides:

$\displaystyle 120+120+48+48+90+90=516$

The surface area is $516\:units^2$ $\displaystyle 516\:units^2$.

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Example Question #1 : How To Find The Surface Area Of A Prism

Prism_1_454590

The height of a ramp is $\displaystyle 3$ meters and spans an $\displaystyle 8$ meter walkway. Sara wants to paint the ramp to match her house, but needs to know the surface area.

What is the surface area of the ramp to the nearest meter?

Possible Answers:

81 m $\displaystyle 81 m$

91 m $\displaystyle 91 m$

115 m $\displaystyle 115 m$

80 m $\displaystyle 80 m$

90 m $\displaystyle 90 m$

Correct answer:

91 m $\displaystyle 91 m$

Explanation:

Since the ramp forms a 45-45-90 triangle, the base of the ramp is equal to the height. So the area of the triangle is $\frac{3\cdot 3}{2}=4.5$ $\displaystyle \frac{3\cdot 3}{2}=4.5$ meters. The area of the two triangles would be $4.5\cdot2=9$ $\displaystyle 4.5\cdot2=9$ meters. The other sides of the ramp are rectangles. Two of the rectangles are the same with one having a different length due to the hypotenuse of the triangle. The two that are the same have a length of 3 and a width of 8. The area for each of these is $3\cdot8=24$ $\displaystyle 3\cdot8=24$ meters. Since there are 2 of these, we mulitply 24 by 2. For the last triangle, we must find the hypotenuse of the triangle. Since it is a 45-45-90, the hypotenuse is the base multiplied by $\sqrt{2}$ $\displaystyle \sqrt{2}$. Therefore the last rectangle's is $3\sqrt{2}\cdot8=33.9$ $\displaystyle 3\sqrt{2}\cdot8=33.9$ meters. The find the surface area, all of the areas must be added together. Triangles+rectangles=$\displaystyle 9+48+33.9=90.9$ meters. To the nearest whole meter, the answer is 91 meters.

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Example Question #3 : How To Find The Surface Area Of A Prism

Kate has an open top box that has the following dimensions: $\displaystyle 6$ inches tall, $\displaystyle 7$ inches wide, and $\displaystyle 12$ inches long.

In square inches, how much wrapping paper would it take to cover the box?

Possible Answers:

400 $\displaystyle 400$ in^2 $\displaystyle in^2$

312 $\displaystyle 312$ $in^{2}$ $\displaystyle in^{2}$

390 $\displaystyle 390$ in^2 $\displaystyle in^2$

168 $\displaystyle 168$ $in^{2}$ $\displaystyle in^{2}$

396 $\displaystyle 396$ $in^{2}$ $\displaystyle in^{2}$

Correct answer:

312 $\displaystyle 312$ $in^{2}$ $\displaystyle in^{2}$

Explanation:

Since the box has an open top, the surface area is calculated by finding the four sides plus the floor of the box.

The short sides' areas are calculated by multiplying height times width times two, for the two sides: $6\cdot7\cdot2=84$ $\displaystyle 6\cdot7\cdot2=84$.

The longer sides' areas are calculated by multiplying height times length times two, for the two sides: $6\cdot12\cdot2=144$ $\displaystyle 6\cdot12\cdot2=144$.

Now, we calculate the area of the floor of the box by multiplying length times width times one, for the only floor and no top: $7\cdot12\cdot1=84$ $\displaystyle 7\cdot12\cdot1=84$.

Lastly, we add the areas together to calculate the total surface area of the open-top box: $\displaystyle 84+84+144=312 in^2$,

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Example Question #1 : How To Find The Surface Area Of A Prism

Find the surface area of the rectangular prism:

Find_the_surface_area

Possible Answers:

$452\:units^2$ $\displaystyle 452\:units^2$

$226\:units^2$ $\displaystyle 226\:units^2$

$485\:units^2$ $\displaystyle 485\:units^2$

$504\:units^2$ $\displaystyle 504\:units^2$

$379\:units^2$ $\displaystyle 379\:units^2$

Correct answer:

$452\:units^2$ $\displaystyle 452\:units^2$

Explanation:

Surface area means the entire area that all the sides of a prism take up.

The surface area can be calculated in one of two ways. One way involves using an equation for lateral area. The other method involves taking the area of all the sides and summing the areas.

Using the latter of the two methods:

It's helpful to understand that rectangular prisms have three pairs of sides with the same dimensions, making up the total of six faces. This means that only three novel calculations for individual areas of faces need to be calculated.

Faces 1 & 2:
$4 \cdot 7 = 28$ $\displaystyle 4 \cdot 7 = 28$
$4 \cdot 7 = 28$ $\displaystyle 4 \cdot 7 = 28$

Faces 3 & 4:
$7 \cdot 18 = 126$ $\displaystyle 7 \cdot 18 = 126$
$7 \cdot 18 = 126$ $\displaystyle 7 \cdot 18 = 126$

Faces 5 & 6:
$4 \cdot 18 = 72$ $\displaystyle 4 \cdot 18 = 72$
$4 \cdot 18 = 72$ $\displaystyle 4 \cdot 18 = 72$

$Surface \:area = 28 + 28 + 126 + 126 + 72 + 72$ $\displaystyle Surface \:area = 28 + 28 + 126 + 126 + 72 + 72$
$Surface \:area = 2(28) + 2(126) + 2(72)$ $\displaystyle Surface \:area = 2(28) + 2(126) + 2(72)$
$Surface \:area = 56 + 252 + 144$ $\displaystyle Surface \:area = 56 + 252 + 144$
$Surface \:area = 452 \:units^2$ $\displaystyle Surface \:area = 452 \:units^2$

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Example Question #1 : How To Find The Surface Area Of A Prism

A small rectangular jewelry box has two square ends with areas of 36 square centimeters, and a width of 10 centimeters. What is the surface area of the outside of the jewelry box.

Possible Answers:

$72 \; cm^2$ $\displaystyle 72 \; cm^2$

$312 \; cm^2$ $\displaystyle 312 \; cm^2$

$216 \; cm^2$ $\displaystyle 216 \; cm^2$

$360 \; cm^2$ $\displaystyle 360 \; cm^2$

$240 \; cm^2$ $\displaystyle 240 \; cm^2$

Correct answer:

$312 \; cm^2$ $\displaystyle 312 \; cm^2$

Explanation:

To find the surface area of the rectangular box we just need to add up the areas of all six sides. We know that two of the sides are 36 square centimeters, that means we need to find the areas of the four mising sides. To find the area of the missing sides we can just multiply the side of one of the squares (6 cm) by the width of the box:

$6 \times 10 = 60 \; cm^2$ $\displaystyle 6 \times 10 = 60 \; cm^2$

But remember we have four of these rectangular sides:

$4 \times 60 = 240 \; cm^2$ $\displaystyle 4 \times 60 = 240 \; cm^2$

Now we just add the two square sides and four rectangular sides to find the total surface area of the jewelry box:

$36 \; cm^2 + 36 \; cm^2 + 240 \; cm^2 = 312 \; cm^2$ $\displaystyle 36 \; cm^2 + 36 \; cm^2 + 240 \; cm^2 = 312 \; cm^2$

That is the total surface area!

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Example Question #2 : Solve For Surface Area

Alice is wrapping a rectangular box that measures $5\textup {in}\times 6\textup {in}\times 12\textup {in}$ $\displaystyle 5\textup {in}\times 6\textup {in}\times 12\textup {in}$. How many square feet of wrapping paper does she need?

Possible Answers:

$2.25\textup{ ft}^2$ $\displaystyle 2.25\textup{ ft}^2$

$3.25\textup{ ft}^{2}$ $\displaystyle 3.25\textup{ ft}^{2}$

$2.75 \textup{ ft}^{2}$ $\displaystyle 2.75 \textup{ ft}^{2}$

$2.00 \textup{ ft}^{2}$ $\displaystyle 2.00 \textup{ ft}^{2}$

$1.75\textup{ ft}^{2}$ $\displaystyle 1.75\textup{ ft}^{2}$

Correct answer:

$2.25\textup{ ft}^2$ $\displaystyle 2.25\textup{ ft}^2$

Explanation:

The surface area of a rectangular prism is given by

$\textup {SA=2lw+2lh+2wh}$ $\displaystyle \textup {SA=2lw+2lh+2wh}$ where $\displaystyle l$ is the length, $\displaystyle w$ is the width, and $\displaystyle h$ is the height.

Let $\textup l=5\textup { in}$ $\displaystyle \textup l=5\textup { in}$, $\textup {w}=6\textup{ in}$ $\displaystyle \textup {w}=6\textup{ in}$, and $\textup {h}=12\textup { in}$ $\displaystyle \textup {h}=12\textup { in}$

So the equation to solve becomes $\textup SA=2\cdot 5\cdot 6+2\cdot 5\cdot 12+2\cdot 6\cdot 12$ $\displaystyle \textup SA=2\cdot 5\cdot 6+2\cdot 5\cdot 12+2\cdot 6\cdot 12$ or $324\textup{ in}^{2}$ $\displaystyle 324\textup{ in}^{2}$

However the question asks for an answer in square feet. Knowing that $144\textup { in}^{2}=1\textup { ft}^{2}$ $\displaystyle 144\textup { in}^{2}=1\textup { ft}^{2}$ we can convert square inches to square feet. It will take $2.25\textup{ ft}^{2}$ $\displaystyle 2.25\textup{ ft}^{2}$ of paper to wrap the present.

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Example Question #7 : How To Find The Surface Area Of A Prism

Find the surface area of the regular hexagonal prism.

Possible Answers:

1119.08 $\displaystyle 1119.08$

1239.62 $\displaystyle 1239.62$

1248.67 $\displaystyle 1248.67$

1339.05 $\displaystyle 1339.05$

Correct answer:

1239.62 $\displaystyle 1239.62$

Explanation:

In order to find the surface area, you will need to add up the areas of each face of the prism.

This hexagonal prism has two regular hexagons as its bases.

Recall how to find the area of a regular hexagon:

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side}^2)$ $\displaystyle \text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side}^2)$

Now, since we have two hexagons, we can multiply the area by $\displaystyle 2$ to get the area of both bases.

$\text{Area of Bases}=3\sqrt3(\text{side})^2$ $\displaystyle \text{Area of Bases}=3\sqrt3(\text{side})^2$

Next, this prism has $\displaystyle 6$ rectangles that make up its sides.

Recall how to find the area of a rectangle:

$\text{Area of Rectangle}=\text{length}\times\text{width}$ $\displaystyle \text{Area of Rectangle}=\text{length}\times\text{width}$

In this prism, the length is also the side of the hexagon, and the width of the rectangle is the height of the prism.

Now, multiply the area of the rectangle by $\displaystyle 6$ to find the total area of all of the sides of the prism.

$\text{Area of Prism Sides}=6(\text{side})(\text{height})$ $\displaystyle \text{Area of Prism Sides}=6(\text{side})(\text{height})$

Add together the area of the sides and of the bases to find the surface area of the prism.

$\text{Surface Area of Prism}=\text{Area of Bases}+\text{Area of Sides}$ $\displaystyle \text{Surface Area of Prism}=\text{Area of Bases}+\text{Area of Sides}$

$\text{Surface Area of Prism}=3\sqrt3(\text{side})^2+6(\text{side})(\text{height})$ $\displaystyle \text{Surface Area of Prism}=3\sqrt3(\text{side})^2+6(\text{side})(\text{height})$

Plug in the given values to find the surface area of the prism.

$\text{Surface Area of Prism}=3\sqrt3(10)^2+6(10)(12)=1239.62$ $\displaystyle \text{Surface Area of Prism}=3\sqrt3(10)^2+6(10)(12)=1239.62$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #8 : How To Find The Surface Area Of A Prism

Find the surface area of the regular hexagonal prism.

Possible Answers:

991.20 $\displaystyle 991.20$

913.68 $\displaystyle 913.68$

875.67 $\displaystyle 875.67$

908.55 $\displaystyle 908.55$