### Topics

## Introduction to :

For prisms, the fastest way to find the volume is to multiply the area of the base (whatever it is) times the height.

*V* = *Bh*

That's all volume is: base times height

### Sample Problem

What is the volume of this prism?

Step 1: Find the area of the base. It's a triangle, so we know that the area equals half the base times the height. We only know the height.

Step 0: We know the height is 9 centimeters and the hypotenuse is 14 centimeters, but what about the base? Pythagorean theorem, here we come.

*a*^{2} + *b*^{2} = *c*^{2}

(9 cm)^{2} + *b*^{2} = (14 cm)^{2}

*b* ≈ 10.72 cm

Step 1: Find the area of the base...take two.

*B* = ½*bh*

*B* = ½(9 cm)(10.72 cm)

*B* ≈ 48.26 cm^{2}

Step 2: Find the volume. In this case, the height is the length of the prism, not the height of the triangle.

*V* = *Bh*

*V* = (48.26 cm^{2})(12 cm)

Prepare for centimeters *cubed*.

*V* ≈ 579 cm^{3}

Awww, yeah.

To mix up things a little, here is a video on the surface area and volume of a cube:

#### Example 1

Find the volume of the prism.
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The basic formula for a prism is the area of the base times the height. *V* = *Bh*
Since the base is a triangle, the first thing we gotta do is find the area of that triangle. *B* = ½*bh*
*B* = ½(10 in)(7 in)
*B* = 35 in^{2}
Now, we plug that value into the volume formula. *V* = *Bh*
*V* = (35 in^{2})(8 in)
*V* = 280 in^{3}
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#### Example 2

Find the mass of this enormous block of gold if the density of gold is 19.3 grams per cubic centimeter. One meter equals 100 centimeters.
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In order to calculate the mass of the gold block, we need to know the volume. To calculate the volume, we need to know the area of the prism's base. Well, we have to start somewhere. The trapezoidal base can be split up into two identical right triangles and a rectangle. The area of the rectangle is length times width, or 14 meters times 16 meters. *A*_{rectangle} = *l* × *w* = (14 m)(16 m) = 224 m^{2}
The base of each triangle is 3 meters and the height is 16 meters.
Each triangle is 24 m^{2}, so two of them make 48 m^{2}. Adding 224 m^{2} and 48 m^{2} gives us 272 m^{2} for the area of the base. Phew. The basic volume formula is: *V* = *Bh*
You don't have to be a rocket scientist to plug in the numbers. It might help, but you don't *have* to be one. *V* = (272 m^{2})(35 m) = 9520 m^{3}
Next is finding the mass. Since the density is given in grams per cubic centimeter, we need to convert cubic meter to cubic centimeter. That's done like this:
Now we can multiply by the volume by the density to get mass. Here we go. Mass = *V* × *d* = (9,520,000,000 cm^{3})(19.3 g/cm^{3}) Mass = 183,736,000,000 g = 1.84 × 10^{8} kg That's a lot of gold. We love gold. | |

#### Exercise 1

Find the volume of the oblique prism in cubic inches.

Hint

Pythagoras and Cavalieri are important names to remember. And not just because they're fun to say.

#### Exercise 2

How many liters of water will fit into this swimming pool? Assume that 1 cubic foot equals about 28.3 liters.

Hint

Think of the pool as two rectangular prisms. Add the volumes of both prisms and then convert feet to liters.

Answer

The pool can fit 23,232 liters of water. Or liters of anything we want, really.

#### Exercise 3

A regular right pentagonal prism has a perimeter of 10 cm^{3} and a height of 8 cm^{3}. What is its density of the prism if it weighs 110 grams?

Hint

The pentagon is made up of 5 triangles with bases equal to the edges and heights from the center of the pentagon down perpendicular to the edges. Density is just mass over volume.

Answer

The density of the prism is about 2 grams per cubic centimeter.