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Lesson 7.2.1Lesson 7.2.1

VolumesVolumes

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Lesson

7.2.1VolumesVolumes

California Standards:Measurement and Geometry 2.1Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

Mathematical Reasoning 2.2Apply strategies and results from simpler problems to more complex problems.

Mathematical Reasoning 3.2Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.

What it means for you:You’ll be reminded what volume is, and then see how you can work out the volume of prisms and cylinders.

Key words:• volume• cubic units• prism• cylinder

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Lesson

7.2.1VolumesVolumes

The volume of a 3-D object, like a box, a swimming pool, or a can, is a measure of the amount of space that’s contained inside it.

Volume is measured in units like cubic feet (ft3) or cubic centimeters (cm3).

This Lesson, you’ll learn how to find the volume of prisms and cylinders.

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Volume Measures Space Inside a Figure

Lesson

7.2.1VolumesVolumes

The amount of space inside a 3-D figure is called the volume.Volume is measured in cubic units.

One cubic unit is the volume of a unit cube — a cube with a side length of 1 unit.

The number of unit cubes that could fit inside a solid shape and fill it completely is the volume in cubic units.

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1

1

4 unit cubes

4 unit cubes

8 unit cubes

Volume = 8 cubic units

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The Volume of a Prism is a Multiple of its Base Area

Lesson

7.2.1VolumesVolumes

When the prism’s height is 3 units, it has a volume of 15 cubic units.

You can work out the volume of a prism from the area of its base.

Height = 1 unitHeight = 2 unitsHeight = 3 units

BaseThe base is made of 5 unit squares. So it has an area of 5 square units.

When the prism’s height is 1 unit, it has a volume of 5 cubic units because it would take 5 unit cubes to make it.When the prism’s height is 2 units, it has a volume of 10 cubic units because it would take 10 unit cubes to make it.

Every time you increase the height by 1 unit you add an extra 5 unit cubes.

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Guided Practice

Solution follows…

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7.2.1VolumesVolumes

1. The figure below is constructed from unit cubes. What is its volume?

21 unit3

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Guided Practice

Solution follows…

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7.2.1VolumesVolumes

A prism is one yard high. It has volume 4 yd3.

2. What is the area of the prism’s base?

3. A prism with an identical base has a volume of 16 yd3. How tall is this prism?

4 yd2

4 yards

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Area Formulas Help Work Out the Volume of a Prism

Lesson

7.2.1VolumesVolumes

When you count the number of unit cubes that make a shape, you find the number of unit cubes that make up the base layer,and multiply it by the height in units.

You can’t always count the number of unit cubes that are inside a shape, because not all shapes can fit an exact number of unit cubes inside them.

Volume of prism = Base area × Height

Instead you can work out the volume of any prism by multiplying the area of the base by the height.

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Example 1

Solution follows…

Lesson

7.2.1VolumesVolumes

What is the volume of this prism?

Solution

Volume of prism = base area × height = 3 × 7 = 21 in3.

The base of this prism is a triangle. So use the area of a triangle formula to work out its area.

Then just multiply that area by the height of the prism.

Area of base = bh = × 2 × 3 = 3 in2.1

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1

2

3 in

7 in

2 in

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7.2.1VolumesVolumes

It doesn’t matter if the prism looks like it is lying down — the same method of finding volume can still be used.

The volume is the same whichever way the prism stands.

The base is always the shape that is the same through the entire prism — it’s just not always at the bottom.

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Example 2

Solution follows…

Lesson

7.2.1VolumesVolumes

What is the volume of this prism?

Solution

So, volume of prism = base area × height = 1 × 5 = 5 yd3.

Treat the triangle as the base of the prism, and the length of 5 yards as the height. Remember, the base is always the shape that is the same through the entire prism.

Area of base = bh = × 1 × 2 = 1 yd2.12

12

5 yd

2 yd

1 yd

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Guided Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

Work out the volumes of the figures in Exercises 4–6.

4. 5. 6.

1 ft

1 ft

3 ft

4 in

2 in

9 in

3 m

11 m

3 m½ × 1 × 1 = 0.50.5 × 3 = 1.5 ft3

½ × 4 × 2 = 44 × 9 = 36 in3 3 × 3 × 11 = 99 m3

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Find the Volume of a Cylinder in the Same Way

Lesson

7.2.1VolumesVolumes

Circular cylinders are similar to prisms — the only difference is that the base is a circle instead of a polygon.

You use the area of a circle formula to get the base area of a cylinder.

So you can work out the volumes of cylinders in the same way as the volumes of prisms — by multiplying the base area by the height.

Area of a circle = × radius2

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Example 3

Solution follows…

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7.2.1VolumesVolumes

What is the volume of this cylinder? Use = 3.14.

Solution

Area of base = r2 = × 42 = × 16 = 50.24 cm2.

Height = 15 cm.

So volume of cylinder = 50.24 × 15 = 753.6 cm3.

4 cm

15 cm

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Guided Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

Work out the volumes of the figures in Exercises 7–9. Use = 3.14.

7. 8. 9. 4 in

10 in

1 yd

8 yd 2 cm

5 cm

3.14 × 42 = 50.2450.24 × 10 = 502.4 in3

3.14 × 12 = 3.143.14 × 8 = 25.12 yd3

3.14 × 22 = 12.5612.56 × 5 = 62.8 cm3

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Rectangular Prisms and Cubes are Special Cases

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7.2.1VolumesVolumes

The area of the base of a rectangular prism is:

length (l) × width (w)

Volume = length (l) × width (w) × height (h)

If you multiply that by height to get the volume then you get:

V (rectangular prism) = lwh

lw

h

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Example 4

Solution follows…

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7.2.1VolumesVolumes

What is the volume of this rectangular prism?

Solution

Volume = lwh = 13 × 20 × 6 = 1560 ft3.

13 ft

20 ft

6 ft

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Lesson

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All sides of a cube are the same length.

For a cube with side length s, the base area is s × s = s2, and the height is also s, so the volume is s2 × s = s3.

s

V (cube) = s3 where s is the side length.

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Example 5

Solution follows…

Lesson

7.2.1VolumesVolumes

What is the volume of this cube?

Solution

Volume = s3 = 73 = 343 in3.

7 in

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Guided Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

Work out the volumes of the figures in Exercises 10–12. Figures with only one side length shown are cubes.

10. 11.

12.

8 cm 2 cm

2 cm8 in

5 yd

8 yd

3 yd

8 × 2 × 2 = 32 cm383 = 512 in3

8 × 5 × 3 = 120 yd3

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Independent Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

1. The figure on the right is constructed from cubes with a volume of 1 in3. What is its volume?

2. How many unit cubes can you fit inside a figure with dimensions 3 units × 3 units × 5 units?

34 in3

45

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Independent Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

3. What is the volume of the prism shown on the right?

4. A cylinder of volume 32 in3 is cut in half. What is the volume of each half?

35,000 ft3

Base area = 500 ft2 Height = 70 ft

16 in3

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Independent Practice

Solution follows…

Lesson

7.2.1VolumesVolumes

Work out the volumes of the figures shown in Exercises 5–7. Use = 3.14.5. 6. 7.

8. What is the volume of a cube with side length 3 yd?

70 in3 2000 cm3 62.8 yd3

27 yd3

5 in 2 in

7 in8 cm

10 cm50 cm

2 yd

5 yd

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Round UpRound Up

Lesson

7.2.1VolumesVolumes

Volume is the amount of space inside a 3-D figure, and it’s measured in cubed units.

For cylinders and prisms, you can multiply the base area by the height of the shape to find the volume.