1. Volumes
Lesson 7.2.1

2. Volumes 7.2.1 California Standards: What it means for you: Key words:
Lesson
7.2.1
Volumes
California Standards:
Measurement and Geometry 2.1
Use 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 2.2
Apply strategies and results from simpler problems to more complex problems.
Mathematical Reasoning 3.2
Note 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

3. Lesson
7.2.1
Volumes
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.

4. Volumes 7.2.1 Volume Measures Space Inside a Figure
Lesson
7.2.1
Volumes
Volume Measures Space Inside a Figure
The amount of space inside a 3-D figure is called the volume.
Volume is measured in cubic units. 1
One cubic unit is the volume of a unit cube — a cube with a side length of 1 unit.
4 unit cubes
8 unit cubes
Volume = 8 cubic units
The number of unit cubes that could fit inside a solid shape and fill it completely is the volume in cubic units.

5. Volumes 7.2.1 The Volume of a Prism is a Multiple of its Base Area
Lesson
7.2.1
Volumes
The Volume of a Prism is a Multiple of its Base Area
You can work out the volume of a prism from the area of its base.
Base
The base is made of 5 unit squares. So it has an area of 5 square units.
Height = 3 units
Height = 2 units
Height = 1 unit
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.
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 3 units, it has a volume of 15 cubic units.
Every time you increase the height by 1 unit you add an extra 5 unit cubes.

6. Volumes 7.2.1 Guided Practice
Lesson
7.2.1
Volumes
Guided Practice
1. The figure below is constructed from unit cubes. What is its volume?
21 unit3
Solution follows…

7. Volumes 7.2.1 Guided Practice
Lesson
7.2.1
Volumes
Guided Practice
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
Solution follows…

8. Volumes 7.2.1 Area Formulas Help Work Out the Volume of a Prism
Lesson
7.2.1
Volumes
Area Formulas Help Work Out the Volume of a Prism
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.
Instead you can work out the volume of any prism by multiplying the area of the base by the height.
Volume of prism = Base area × Height

9. Volumes 7.2.1 3 in What is the volume of this prism? 7 in Solution
Lesson
7.2.1
Volumes
Example 1
3 in
7 in
2 in
What is the volume of this prism?
Solution
The base of this prism is a triangle. So use the area of a triangle formula to work out its area.
Area of base = bh = × 2 × 3 = 3 in2. 1 2
Then just multiply that area by the height of the prism.
Volume of prism = base area × height = 3 × 7 = 21 in3.
Solution follows…

10. The volume is the same whichever way the prism stands.
Lesson
7.2.1
Volumes
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.

11. What is the volume of this prism?
Lesson
7.2.1
Volumes
Example 2
5 yd
2 yd
1 yd
What is the volume of this prism?
Solution
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. 1 2
So, volume of prism = base area × height = 1 × 5 = 5 yd3.
Solution follows…

12. Volumes 7.2.1 Guided Practice
Lesson
7.2.1
Volumes
Guided Practice
Work out the volumes of the figures in Exercises 4–6.
3 m
11 m
1 ft
3 ft
4 in
2 in
9 in
½ × 1 × 1 = 0.5
0.5 × 3 = 1.5 ft3
½ × 4 × 2 = 4
4 × 9 = 36 in3
3 × 3 × 11 = 99 m3
Solution follows…

13. Volumes 7.2.1 Find the Volume of a Cylinder in the Same Way
Lesson
7.2.1
Volumes
Find the Volume of a Cylinder in the Same Way
Circular cylinders are similar to prisms — the only difference is that the base is a circle instead of a polygon.
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.
You use the area of a circle formula to get the base area of a cylinder.
Area of a circle = p × radius2

14. Volumes 7.2.1 What is the volume of this cylinder? Use p = 3.14.
Lesson
7.2.1
Volumes
Example 3
What is the volume of this cylinder? Use p = 3.14.
4 cm
15 cm
Solution
Area of base = pr2 = p × 42 = p × 16 = cm2.
Height = 15 cm.
So volume of cylinder = × 15 = cm3.
Solution follows…

15. Volumes 7.2.1 Guided Practice
Lesson
7.2.1
Volumes
Guided Practice
Work out the volumes of the figures in Exercises 7–9. Use p = 3.14.
4 in
10 in
1 yd
8 yd
2 cm
5 cm
3.14 × 42 = 50.24
50.24 × 10 = in3
3.14 × 12 = 3.14
3.14 × 8 = yd3
3.14 × 22 = 12.56
12.56 × 5 = 62.8 cm3
Solution follows…

16. Volumes 7.2.1 Rectangular Prisms and Cubes are Special Cases
Lesson
7.2.1
Volumes
Rectangular Prisms and Cubes are Special Cases
The area of the base of a rectangular prism is:
length (l) × width (w)
If you multiply that by height to get the volume then you get:
Volume = length (l) × width (w) × height (h) l w h
V (rectangular prism) = lwh

17. Volumes 7.2.1 What is the volume of this rectangular prism? Solution
Lesson
7.2.1
Volumes
Example 4
What is the volume of this
rectangular prism?
13 ft
20 ft
6 ft
Solution
Volume = lwh = 13 × 20 × 6 = 1560 ft3.
Solution follows…

18. Volumes 7.2.1 All sides of a cube are the same length.
Lesson
7.2.1
Volumes
All sides of a cube are the same length. s
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.
V (cube) = s3
where s is the side length.

19. Volumes 7.2.1 What is the volume of this cube? Solution
Lesson
7.2.1
Volumes
Example 5
7 in
What is the volume of this cube?
Solution
Volume = s3 = 73 = 343 in3.
Solution follows…

20. Volumes 7.2.1 Guided Practice
Lesson
7.2.1
Volumes
Guided Practice
Work out the volumes of the figures in Exercises 10–12. Figures with only one side length shown are cubes.
12.
8 cm
2 cm
8 in
8 × 2 × 2 = 32 cm3
83 = 512 in3
5 yd
8 yd
3 yd
8 × 5 × 3 = 120 yd3
Solution follows…

21. Volumes 7.2.1 Independent Practice
Lesson
7.2.1
Volumes
Independent Practice
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
Solution follows…

22. Volumes 7.2.1 Independent Practice
Lesson
7.2.1
Volumes
Independent Practice
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
Solution follows…

23. Volumes 7.2.1 Independent Practice
Lesson
7.2.1
Volumes
Independent Practice
Work out the volumes of the figures shown in Exercises 5–7. Use p =
8. What is the volume of a cube with side length 3 yd?
5 in
2 in
7 in
8 cm
10 cm
50 cm
2 yd
5 yd
70 in3
2000 cm3
62.8 yd3
27 yd3
Solution follows…

24. Lesson
7.2.1
Volumes
Round Up
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.