Year 10 Extended · Unit 5 Solids, similarity and transformations · Lesson 2 of 6

Three solids, one cylinder

A pyramid and a cone fill a fraction of the prism or cylinder around them, and a sphere fills a fraction of its cylinder too. This lesson shows why the fractions are what they are and uses them on joined solids.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: the volume of a cylinder of radius 3 cm and height 10 cm, as a multiple of π.
    Answer90π cm³, because π × 3² × 10 = 90π.
  2. From Year 8: a prism has a base area of 25 cm² and a height of 9 cm. What is its volume?
    Answer225 cm³, because 25 × 9 = 225.
  3. From Year 9: solve r³ = 125.
    Answerr = 5, because 5 × 5 × 5 = 125.
Words used in this lesson
volume
the amount of space a solid fills, in cubic units. For example: a box 3 by 4 by 5 has volume 60.
hemisphere
half of a sphere. For example: a bowl of radius 2.5 cm.
capacity
the volume of liquid a container holds. For example: 1 cm³ = 1 mL.
recast
melted down and made into a new shape with the same volume. For example: a candle melted into a ball.
composite solid
a solid made by joining simpler solids. For example: an ice-cream cone with a scoop.

The sphere that fits its tin

A ceramics studio in Jingdezhen packs a round ball of clay in a snug cylindrical tin, so that the ball touches the sides, the lid and the base. A cone of clay with the same base and the same height as the tin fits inside it too. The three solids share a radius, and their volumes are related in a simple way.

Predict first: what fraction of the tin do you think the ball fills, and what fraction the cone? Then use the figure and compare the three volumes with each other.

2
16.8cone (cm³)
33.5sphere (cm³)
50.3cylinder (cm³)

Radius 2 cm, with the cylinder 2 × 2 = 4 cm tall: cone 16.8 cm³, sphere 33.5 cm³, cylinder 50.3 cm³.

Show me the steps
  1. Set r = 3: the cone is 56.5 cm³, the sphere 113.1 cm³ and the cylinder 169.6 cm³. Divide each by 56.5 and you get 1, 2 and 3.
  2. Set r = 5: the cone is 261.8 cm³, the sphere 523.6 cm³ and the cylinder 785.4 cm³. Again the sphere is twice the cone and the cylinder is three times the cone.
  3. Set r = 1: the volumes are 2.1, 4.2 and 6.3 cm³. The sphere is two thirds of the cylinder whatever the radius is.

Pointed solids. A pyramid is ⅓ × base area × height, and a cone is ⅓πr²h, where h is the vertical height. Why a third? A cube of side a splits from its centre into six identical square pyramids, each with base a² and height a ÷ 2, so each has volume a³ ÷ 6, which is ⅓ × a² × (a ÷ 2).

Sphere. V = 4⁄3πr³. A sphere that exactly fits a cylinder has two thirds of the cylinder's volume: cylinder πr² × 2r = 2πr³, and ⅔ of that is 4⁄3πr³.

Joined solids. Break the solid into parts you know, find each volume, then add them (or take one away if a part is cut out). A capacity in cm³ converts directly: 1 cm³ = 1 mL.

Vertical height, not the slant, and r cubed, not r squared. A cone with radius 8 cm and slant height 10 cm has vertical height 6 cm, so its volume is ⅓ × π × 8² × 6 = 128π cm³. Using 10 would give 640π ÷ 3. And doubling a sphere's radius multiplies its volume by 2³ = 8, not by 2.

Worked examples

1. A cone has radius 5 cm and slant height 13 cm. Find its volume.

  1. Vertical height: h² = 13² − 5² = 144, so h = 12 cm.
  2. V = ⅓ × π × 5² × 12 = 100π ≈ 314.2 cm³.

2. A hemispherical bowl has radius 12 cm. How many litres does it hold when full?

  1. A hemisphere is half a sphere: ½ × 4⁄3 × π × 12³ = ⅔ × π × 1728 = 1152π.
  2. That is about 3619.1 cm³ = 3619.1 mL, and 1000 mL = 1 L, so the bowl holds about 3.6 litres.

3. A pencil-shaped solid is a cylinder of radius 3 cm and height 8 cm with a cone of height 4 cm on one end. Find its volume.

  1. Cylinder: π × 3² × 8 = 72π. Cone: ⅓ × π × 3² × 4 = 12π.
  2. Add the parts: 72π + 12π = 84π ≈ 263.9 cm³.

Your go. The first step is given. A cylinder of radius 2 cm and height 6 cm has a hemisphere of radius 2 cm scooped out of its top. Find the volume that is left.

  1. The whole cylinder is π × 2² × 6 = 24π cm³.
  2. Now take away the hemisphere, ½ × 4⁄3 × π × 2³.
Check the answer

The hemisphere is ⅔ × π × 8 = 16π ÷ 3. The volume left is 24π − 16π ÷ 3 = 56π ÷ 3 ≈ 58.6 cm³.

With a GDC (optional)

A GDC can hold a long product such as ⅓ × π × r² × h in one line, so the only rounding is at the end. Use the cube key (or the power key) for r³, and store a volume in memory to reuse it in the next step of a joined solid.

Leave results as multiples of π while you work and round once. Everything here can be done by hand.

Check before you go on

1. A square pyramid has base side 10 cm, and each triangular face has slant height 13 cm. Which is its volume?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A square pyramid has a base of side 12 cm and a vertical height of 7 cm. Find its volume, in cubic centimetres.

3. A steel ball has a diameter of 12 cm. Find its volume, in cubic centimetres, to one decimal place.

Level 2 · Building

4. A cylindrical grain bin has radius 5 m and straight height 12 m, with a hemispherical roof of radius 5 m. Find its total volume, in cubic metres, to one decimal place.

5. A solid cone of radius 6 cm and height 3 cm is melted and recast as a sphere. Find the radius of the sphere, in centimetres.

6. A sphere fits exactly inside a cylinder, touching the sides, the top and the base. The cylinder has volume 540 cm³. Find the volume of the sphere, in cubic centimetres.

Level 3 · Stretch

7. A cone and a hemisphere have the same radius r and the same volume. The cone's vertical height is k times r. Find k.

8. A sphere fits exactly inside a cube, touching all six faces. What percentage of the cube's volume does the sphere fill? Give your answer to one decimal place.

Something to think about

A sphere of radius r fits snugly inside a cylinder. Compare the sphere's surface area, 4πr², with the whole surface of the cylinder, including both ends. What do you notice, and does it match the volumes?

One way to think about it

The cylinder's surface is 2πr × 2r + 2 × πr² = 6πr², and 4πr² ÷ 6πr² = 2/3. The sphere has two thirds of the cylinder's surface as well as two thirds of its volume. Archimedes was so pleased with this that he is said to have asked for a sphere in a cylinder on his tomb.

Exit ticket

9. A toy is a cone of radius 4 cm and vertical height 10 cm with a hemisphere of radius 4 cm fixed on its base. Which calculation gives its volume?