# Intermediate Geometry : Prisms

## Example Questions

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### Example Question #1 : How To Find The Length Of An Edge Of A Prism

The volume of a prism is  .

Given the length is  and the height is , find the width of the prism.

Explanation:

The volume of a prism is length times width times height.

We are given the length is 7m and the width is 5m.

So, we plug these into our formula:

.

We then solve for height:

.

The height is therefore 6m.

### Example Question #2 : How To Find The Length Of An Edge Of A Prism

Find the missing edge of the prism when its volume is .

Explanation:

The goal is to find the height of the rectangular prism with the given information of its width and length. The volume of a rectangular prism is , where  is width and  is height.

Because we're given the final volume and two of the three variables, we can substitute in the information we know and solve for the missing variable.

Therefore, the height of the prism is .

### Example Question #3 : How To Find The Length Of An Edge Of A Prism

A rectangular prism has a volume of  cubic meters. Its length is twice its width, and its height is twice its length. What is the length of the prism?

Explanation:

If we let  be the width of our prism, then since the length is twice the width, our length would be .  Since the height is twice the length, our height would be .

Since the volume of a rectangular prism is simply the product of width, length, and height, we get

We then simply solve for the width.

Therefore, our width is 3.  Since our length is twice the width, the length is 6.

### Example Question #4 : How To Find The Length Of An Edge Of A Prism

A right, rectangular prism has a volume of  cubic inches. Its width is  inches and its height is  inches. What is its length?

Explanation:

The formula for the volume of a right, rectangular prism is , so substitute the known values and solve for lenght.

.  So the length is 8 inches.

### Example Question #5 : How To Find The Length Of An Edge Of A Prism

The surface area of a right rectangular prism is  square cm. Its length is  cm and its height is  cm. Find the width of the prism.

Explanation:

Use the formula for finding the surface area of a right, rectangular prism,

and substitute the known values, and then solve for w.

So,

So, the width is 7 cm.

### Example Question #6 : How To Find The Length Of An Edge Of A Prism

The height of a retangular prism is three times its width; its length is twice its width. Its surface area is 1,078 square inches. Give the width of the box.

Explanation:

Since the length of the box is twice the width and the height is three times the width,  and

The surface area of the box is

To find the width:

### 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:

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