Volume

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1. Volume of prisms and cylinders

Warm-up

Three quick questions from earlier lessons.

Discover

A stack of identical coins forms a cylinder. If each coin has area AA and the stack is hh high, what is its volume?

  1. A×hA \times h
  2. A+hA + h

Key idea

Base area times height.

For any prism or cylinder: V=B⋅hV = B \cdot h, with BB the area of the base. Cylinder: V=πr2hV = \pi r^2 h.

Worked example

A cylinder with r=3r = 3, h=10h = 10.

Not to scale

B=9πB = 9\pi
Base area.
V=90π≈282.7V = 90\pi \approx 282.7
Times height.

Your turn, guided

Your turn: a rectangular prism 4 by 5 by 6.

120120
4 × 5 × 6.

Practice ★

  1. Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
  2. Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale
  1. A prism has a triangular base with base 2 cm and height 7 cm, and the prism is 7 cm long. Find its volume (cm³).Not to scale

Watch out!

Practice ★★

  1. A cylinder of radius 5 cm has volume 175π175\pi cm³. Find its height (cm).Not to scale
  2. A cylinder of radius 6 cm has volume 360π360\pi cm³. Find its height (cm).Not to scale

Exit ticket

Three questions to check you have got it.

  1. Find the volume of a cylinder with radius 3 cm and height 4 cm (exact, in terms of π).Not to scale
  2. Find the volume of a cylinder with radius 8 cm and height 11 cm (exact, in terms of π).Not to scale
  3. Find the volume of a cylinder with radius 5 cm and height 5 cm (exact, in terms of π).Not to scale

Summary card

Prisms and cylinders

V=BhV = Bh; cylinder V=πr2hV = \pi r^2 h.

Watch out: base area, not circumference.

2. Volume of pyramids and cones

Warm-up

Three quick questions from earlier lessons.

Discover

A cone and a cylinder have the same base and height. How many cones of water fill the cylinder?

Not to scale
  1. 3
  2. 2
Not to scale

Tilt them to compare. How many full cones of water fill the cylinder?

  1. 3
  2. 2

Key idea

Pointed solids hold one third.

For a pyramid or cone: V=13BhV = \tfrac{1}{3}Bh. Cone: V=13πr2hV = \tfrac{1}{3}\pi r^2 h.

Worked example

A square pyramid, base 6 m, height 10 m.

B=36B = 36
Base area.
V=13⋅36⋅10=120V = \tfrac{1}{3} \cdot 36 \cdot 10 = 120 m³
One third of Bh.

Your turn, guided

Your turn: a cone with r=3r = 3, h=4h = 4 (in terms of π).

12π12\pi
⅓ · 9π · 4.

Practice ★

  1. Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
  2. Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale
  1. A pyramid has a square base of side 4 m and height 15 m. Find its volume (m³).Not to scale

Watch out!

Practice ★★

  1. A cone and a cylinder have the same radius 5 and the same height. What fraction of the cylinder's volume is the cone's?
  1. Boss problem. An ice cream cone has radius 3 cm and height 8 cm, topped with a hemisphere of ice cream of the same radius. Give exact answers in terms of π.

Exit ticket

Three questions to check you have got it.

  1. Find the volume of a cone with radius 6 cm and height 6 cm (exact, in terms of π).Not to scale
  2. Find the volume of a cone with radius 6 cm and height 12 cm (exact, in terms of π).Not to scale
  3. Find the volume of a cone with radius 4 cm and height 9 cm (exact, in terms of π).Not to scale

Summary card

Pyramids and cones

V=13BhV = \tfrac{1}{3}Bh.

Watch out: do not forget the ⅓.

3. Volume of spheres; density

Warm-up

Three quick questions from earlier lessons.

Discover

A ball of radius rr fits exactly in a cylinder of radius rr and height 2r2r. The ball fills 23\tfrac{2}{3} of the cylinder. What is the ball's volume?

Not to scale
  1. 43πr3\tfrac{4}{3}\pi r^3
  2. 4πr24\pi r^2

Key idea

Spheres, and how heavy a solid is.

Sphere: V=43πr3V = \tfrac{4}{3}\pi r^3 (surface area 4πr24\pi r^2). Density = mass ÷ volume, so mass = density × volume.

Worked example

A steel ball of radius 3 cm, density 7.8 g/cm³.

V=36π≈113.1V = 36\pi \approx 113.1 cm³
(4/3)π · 27.
m≈7.8×113.1≈882m \approx 7.8 \times 113.1 \approx 882 g
Mass = density × volume.

Your turn, guided

Your turn: mass 200 g, volume 50 cm³. The density.

44 g/cm³
200 ÷ 50.

Practice ★

  1. Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
  2. Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
  1. A block has mass 526 g and volume 27 cm³. Find its density (g/cm³).

Watch out!

Practice ★★

  1. Find the volume of a hemisphere (half a ball) of radius 3 cm (exact, in terms of π).
  2. Find the volume of a hemisphere (half a ball) of radius 5 cm (exact, in terms of π).

Exit ticket

Three questions to check you have got it.

  1. Find the volume of a sphere of radius 5 cm (exact, in terms of π).Not to scale
  2. Find the volume of a sphere of radius 9 cm (exact, in terms of π).Not to scale
  3. Find the volume of a sphere of radius 8 cm (exact, in terms of π).Not to scale

Summary card

Spheres and density

V=43πr3V = \tfrac{4}{3}\pi r^3; density = mass/volume.

Watch out: r3r^3 for volume, r2r^2 for area.

4. Percent error

Warm-up

Three quick questions from earlier lessons.

Discover

A 50 cm board is measured as 52 cm. Which describes the error?

  1. A 4% error
  2. A 2% error

Key idea

Percent error compares the error with the actual value.

Percent error =∣measured−actual∣actual×100= \dfrac{|\text{measured} - \text{actual}|}{\text{actual}} \times 100. A tolerance of p%p\% allows values within p%p\% of the target, above or below.

Worked example

A 400 g bag with at most 5% error.

2020 g
5% of 400.
From 380 g to 420 g
Range.

Your turn, guided

Your turn: actual 40, measured 38. The percent error.

55
2 ÷ 40 × 100.

Practice ★

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  1. A box really holds 400 cups, but a label says 520. What is the percent error?

Watch out!

Practice ★★

  1. A bag should hold 600 g of rice, with at most 7% error. What is the smallest allowed mass (g)?
  2. A bag should hold 400 g of rice, with at most 8% error. What is the smallest allowed mass (g)?

Exit ticket

Three questions to check you have got it.

  1. A board is really 36 cm long, but Ana measures 40 cm. What is her percent error?
  2. A board is really 73 cm long, but Ana measures 76 cm. What is her percent error?
  3. A board is really 52 cm long, but Ana measures 53 cm. What is her percent error?

Summary card

Percent error

Error ÷ actual × 100.

Watch out: divide by the actual value, not the measured one.

Answers: Volume

1. Volume of prisms and cylinders

A

  1. 36π36 \pi
  2. 704π704 \pi

B

  1. 4949

C

  1. 77
  2. 1010

D

  1. 36π36 \pi
  2. 704π704 \pi
  3. 125π125 \pi

2. Volume of pyramids and cones

A

  1. 72π72 \pi
  2. 144π144 \pi

B

  1. 8080

C

  1. 13\frac{1}{3}

D

  1. 42π42 \pi

E

  1. 72π72 \pi
  2. 144π144 \pi
  3. 48π48 \pi

3. Volume of spheres; density

A

  1. 500π3\frac{500 \pi}{3}
  2. 972π972 \pi

B

  1. 52627\frac{526}{27}

C

  1. 18π18 \pi
  2. 250π3\frac{250 \pi}{3}

D

  1. 500π3\frac{500 \pi}{3}
  2. 972π972 \pi
  3. 2048π3\frac{2048 \pi}{3}

4. Percent error

A

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}

B

  1. 3030

C

  1. 558558
  2. 368368

D

  1. 1009\frac{100}{9}
  2. 30073\frac{300}{73}
  3. 2513\frac{25}{13}