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Volume of Pyramids

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Volume of Pyramids
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Have you ever seen a candle that is shaped like a pyramid? Take a look at this dilemma.

Brianna bought the candle below for her friend’s birthday. The package says that the candle burns one hour for every 20 cubic centimeters of wax. For how many hours will it take for the entire candle to burn?

Pay attention and this Concept will teach you all that you need to know about volume and pyramids.

Guidance

What is volume?

Volume is the measure of how much space a three-dimensional figure takes up or holds.

Imagine a funnel. Its size determines how much water the funnel will hold. If we fill it with water, the amount of water tells the volume of the funnel. Volume is often what we think of when we talk about measuring liquid or liquid capacity.

We measure volume in three dimensions: length, width, and height. We therefore measure volume in cubic units. We can use unit cubes to represent volume. Have a look at the cube below.

You can see that the cube is 3 \times 3 \times 3 . If we wanted to find the volume of this cube, we could find the area of the base and then multiply it by the height.

V & =Bh\\V & =(s^2 )3\\V & =3^2 (3)\\V & =27 \ cubic \ units

Notice that we measure volume in cubic units.

Pyramids are unusual, however, because they are so much smaller at the top than they are at their base. It becomes very difficult to use unit cubes to measure the volume of these solids because we would be calculating parts of unit cubes.

The important thing to remember is that measuring volume involves filling up a solid figure.

A pyramid has exactly one-third the volume of a cube.

Here is the formula for finding the volume of pyramids.

V= \frac{1}{3} Bh

Pyramids can be tricky, however, because they can have bases of any shape. Pyramids can have triangular, rectangular, or square bases. That means we need to choose the appropriate formula for finding the area of the base, or B . Here are the common area formulas:

Square: A  =  s^2

Rectangle: A  =  lw

Triangle: A  =  \frac{1}{2} bh

When given a pyramid, the first thing we need to do is determine the shape of the base. Then we’ll know which formula to use to find the base area. Once we have the base area, we put it into the volume formula along with the height of the pyramid and then solve for V .

Let’s give it a try.

What is the volume of the pyramid below?

First, let’s decide what shape the base of the pyramid is. There are two pairs of parallel sides that meet at right angles, so it must be a rectangle. We need to use the area formula for rectangles to find B , the base area.

B & = lw\\B & =  11 (6.3)\\B & =  69.3 \ cm^2

The area of this pyramid’s base is 69.3 square centimeters. Now we multiply this by the height and \frac{1}{3} , according to the formula.

V & =  \frac{1}{3} Bh\\V & =  \frac{1}{3} (69.3) (15)\\V & =  23.1 (15)\\V & =  346.5 \ cm^3

The volume of this pyramid is 346.5 \ cm^3 . Remember that volume is always measured in cubic units and that is why our exponent is a three.

Find the volume of each pyramid.

Example A

A square pyramid with a base of 5.5 in and a height of 4 in.

Solution: 40.33 \ in^3

Example B

A square pyramid with a base of 8 cm and a height of 6 cm.

Solution: 128 \ cm^3

Example C

A rectangular pyramid with a length of 10 cm, a width of 8 cm and a height of 9 cm.

Solution: 240 \ cm^3

Now let's go back to the dilemma from the beginning of the Concept.

First, let’s determine what the problem is asking us to find. We need to find the number of hours the candle will burn. This depends on how big the candle is, so first we need to find its volume. The volume of the candle is the amount of wax it holds.

What information have we been given? We know the dimensions of the base, which is a square, so let’s use the area formula for squares to find the base area.

B & =  s^2\\B & = (12^2)\\B & =  144 \ cm^2

The base area of the pyramid is 144 square centimeters. We can put this information into the formula and solve for V , volume.

V & = \frac{1}{3} Bh\\V & = \frac{1}{3} (144) (24)\\V & = 48 (24)\\V & = 1,152 \ cm^3

Now we know that the candle contains 1,152 cubic centimeters of wax.

But we’re not done yet! Remember, we need to find how many hours the candle will burn. Look back at the problem. It tells us that the candle burns one hour for every 20 cubic centimeters of wax. To find how many hours the candle will burn, we need to divide the total volume of wax by 20.

1,152 \div 20 = 57.6

The candle will burn for 57.6 hours. This is our answer.

Vocabulary

Pyramid
a solid figure with a polygon base and triangular side faces that meet in a single vertex.
Base Area
the area of the base of a solid figure.
Height
the measurement that is perpendicular to the base of a solid figure.

Guided Practice

Here is one for you to try on your own.

A triangular pyramid has a volume of 266 cubic feet and a base area of 42 square feet. What is its height?

Solution

What do we need to find? We need to solve for the height, h . We have been given the volume and the base area so we simply put this information into the formula. Volume the capacity inside a solid figure or the amount of space a solid figure can hold.

V & = \frac{1}{3} Bh\\266 & = \frac{1}{3} (42)h\\266 & = 14h\\266 \div 14 & =  h\\19 & = h

The height of this pyramid is 19 feet.

Video Review

Practice

  1. What is the formula that you would need for finding the volume of a pyramid?
  2. When you see a capital B in a formula it means that you are looking for the perimeter or area of the base?
  1. What is the name of the figure pictured below?
  2. What is the shape of the base?
  3. What is the volume of the figure?

  1. What is the name of the figure pictured above?
  2. What is the difference between this figure and the last figure?
  3. What is the volume of this figure?
  4. What is the shape of the base?

Directions: Use what you have learned to solve each of the following problems.

  1. A square pyramid has a base with sides of 6 yards each and a volume of 175 cubic yards. What is its height?
  2. Claire has a perfume bottle shaped like a triangular pyramid. Its base area is 48 square centimeters, and its height is 28 centimeters. How much does the bottle hold when it is exactly half full?
  3. Find the volume of a square pyramid with a base of 4 inches and a height of 6 inches.
  4. Find the volume of a rectangular pyramid with a length of 5 inches, a width of 7 inches and a height of 8 inches.
  5. Find the volume of a square pyramid with a base of 8 meters and a height of 12 meters.
  6. Find the volume of a square pyramid with a base of 13.5 meters and a height of 15 meters.

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