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

## Example Questions

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

Find the surface area of the rectangular prism:

Explanation:

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  faces. With that, it's helpful to understand that there are  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:

Faces 3 & 4:

Faces 5 & 6:

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

The surface area is .

### Example Question #2 : How To Find The Surface Area Of A Prism

The height of a ramp is  meters and spans an  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?

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  meters. The area of the two triangles would be  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  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 . Therefore the last rectangle's is  meters. The find the surface area, all of the areas must be added together. Triangles+rectangles= meters. To the nearest whole meter, the answer is 91 meters.

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

Kate has an open top box that has the following dimensions:  inches tall,  inches wide, and  inches long.

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

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: .

The longer sides' areas are calculated by multiplying height times length times two, for the two sides: .

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: .

Lastly, we add the areas together to calculate the total surface area of the open-top box: ,

### Example Question #4 : How To Find The Surface Area Of A Prism

Find the surface area of the rectangular prism:

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:

Faces 3 & 4:

Faces 5 & 6:

### 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.

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:

But remember we have four of these rectangular sides:

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

That is the total surface area!

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

Alice is wrapping a rectangular box that measures .  How many square feet of wrapping paper does she need?

Explanation:

The surface area of a rectangular prism is given by

where is the length, is the width, and is the height.

Let , , and

So the equation to solve becomes  or

However the question asks for an answer in square feet.  Knowing that  we can convert square inches to square feet.  It will take  of paper to wrap the present.

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

Find the surface area of the regular hexagonal prism.

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:

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

Next, this prism has  rectangles that make up its sides.

Recall how to find the area of a rectangle:

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  to find the total area of all of the sides of the prism.

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

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

Make sure to round to  places after the decimal.

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

Find the surface area of the regular hexagonal prism.

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:

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

Next, this prism has  rectangles that make up its sides.

Recall how to find the area of a rectangle:

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  to find the total area of all of the sides of the prism.

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

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

Make sure to round to  places after the decimal.

### Example Question #9 : How To Find The Surface Area Of A Prism

Find the surface area of the regular hexagonal prism.

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:

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

Next, this prism has  rectangles that make up its sides.

Recall how to find the area of a rectangle:

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  to find the total area of all of the sides of the prism.

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

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

Make sure to round to  places after the decimal.

### Example Question #10 : How To Find The Surface Area Of A Prism

Find the surface area of the regular hexagonal prism.

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:

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

Next, this prism has  rectangles that make up its sides.

Recall how to find the area of a rectangle:

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  to find the total area of all of the sides of the prism.

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

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

Make sure to round to  places after the decimal.

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