Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9 (spheres): a hemisphere has radius 3 cm. What is its volume, in terms of π?
Answer
18π cm³. The sphere is 4/3 × 27 = 36π, and half of it is 18π. - From Year 9 (cones): a cone has radius 3 cm and slant height 5 cm. What is its curved surface area, in terms of π?
Answer
15π cm². It is π × 3 × 5. - Change 2500 cm³ into litres.
Answer
2.5 litres. 1 litre is 1000 cm³, so divide by 1000.
Words used in this lesson
- composite solid
- a solid made of two or more simple solids joined together. For example: a capsule.
- capacity
- how much a container holds. For example: 1 litre.
- litre
- a unit of capacity, 1000 cm³. For example: a bottle of water.
- hemisphere
- half of a sphere. For example: a bowl.
- surface area
- the total area of the outside faces. For example: 84π cm².
- design
- a choice of shape and size to do a job. For example: a one-litre tin.
One litre of olive oil
A company in Crete is deciding the shape of a metal tin for exactly one litre of olive oil. One litre is 1000 cm³. The metal costs money, so the less of it the better. Designer Eleni proposes a tall, thin tin and designer Kofi a short, wide one.
Predict first: does the tall thin tin or the short wide tin use less metal? Or is the answer somewhere in the middle? Use the first figure to try seven radii, and then the second figure to see how a solid made of two or three parts is measured.
Radius 6 cm: the height must be 1000 ÷ (π × 6²) = 8.8 cm, and the metal needed is 2π × 6² + 2π × 6 × 8.8 = 559.5 cm².
Show me the steps
- Radius 6 cm: the height must be 1000 ÷ (π × 6²) = 8.8 cm, and the metal needed is 2π × 6² + 2π × 6 × 8.8 = 559.5 cm².
- Radius 3 cm: a tall, thin tin, 35.4 cm high. The metal needed is 723.2 cm², much more.
- Radius 5 cm: 12.7 cm high, and the metal needed is 557.1 cm², the smallest of the seven.
- Radius 9 cm: a wide, flat tin, only 3.9 cm high. The metal needed is 731.2 cm², much more again. Both extremes cost more than the middle.
- Press Show all seven bars to draw every bar at once: they fall from 723.2 cm² at radius 3 cm to 557.1 cm² at radius 5 cm, then rise to 731.2 cm² at radius 9 cm.
A composite solid is split into solids you know. Its volume is the sum of the volumes of the parts. Its surface area is the sum of the areas of the faces that are on the outside: where two parts meet, the flat circle is inside and is not counted.
Capacity is volume. 1 litre = 1000 cm³, and 1 millilitre = 1 cm³. To compare designs that hold the same amount, compare their surface areas: the smaller the surface, the less material.
The best shape is often in the middle. Very tall and very wide tins both need more metal than a tin that is about as tall as it is wide.
Capsule: a cylinder of radius 3 cm and length 8 cm with a hemisphere at each end. Volume 72π + 36π = 108π cm³ ≈ 339.3 cm³; surface 48π + 36π = 84π cm² ≈ 263.9 cm².
Show me the steps
- Capsule: a cylinder of radius 3 cm and length 8 cm, with a hemisphere at each end. Volume 72π + 36π = 108π ≈ 339.3 cm³. Surface 48π + 36π = 84π ≈ 263.9 cm². The two flat circles where the parts join are inside, so they are not counted.
- Ice-cream cone: a cone of radius 3 cm and height 4 cm (slant height 5 cm) with a hemisphere on top. Volume 12π + 18π = 30π ≈ 94.2 cm³. Surface 15π + 18π = 33π ≈ 103.7 cm². The circle where the scoop sits on the cone is inside.
- Grain silo: a cylinder of radius 5 m and height 8 m with a cone roof 12 m high (slant height 13 m). Volume 200π + 100π = 300π ≈ 942.5 m³. Surface: floor 25π + wall 80π + roof 65π = 170π ≈ 534.1 m². The circle between the wall and the roof is inside.
A common mistake: adding the surface area of each part as if it were a whole solid. A capsule is not two hemispheres and a closed cylinder: the circles where they join are inside, so the surface is the cylinder's curved side plus the two curved hemispheres, 48π + 36π = 84π for the capsule in the figure. Adding a flat circle would count a face that nobody can touch.
Worked example: a grain bin
A grain bin is a cylinder of radius 2 m and height 3 m, with a hemispherical dome of radius 2 m on top. Find its volume and the area of its outside wall and dome, to one decimal place.
- Cylinder volume: π × 2² × 3 = 12π m³. Dome volume: 2/3 × π × 2³ = 16π/3 m³.
- Total volume: 12π + 16π/3 = 52π/3 m³ = 54.5 m³ to one decimal place.
- Wall: 2π × 2 × 3 = 12π m². Dome: 2π × 2² = 8π m². The circle between them is inside, so it is not counted.
- Outside area: 12π + 8π = 20π m² = 62.8 m² to one decimal place.
Your go. The first step is given. A toy rocket is a cylinder of radius 3 cm and height 10 cm with a cone on top, of radius 3 cm and height 4 cm. Find its volume to one decimal place.
- The cylinder: π × 3² × 10 = 90π cm³.
- Now find the cone's volume, add the two, and use the π key.
Check the answer
The cone is ⅓ × π × 3² × 4 = 12π cm³. The total is 90π + 12π = 102π, which is 320.4 cm³ to one decimal place.
With a calculator
Write each part in terms of π, add the parts, and press the π key once at the end. For a cylinder with radius 2 m and height 3 m, topped by a hemisphere of radius 2 m: 12π + 16π/3 = 52π/3, and 52 × π ÷ 3 = 54.45…, which is 54.5 m³ to one decimal place.
Litres: 1 litre is 1000 cm³, so divide a volume in cm³ by 1000 to get litres. A volume of 37 700 cm³ is 37.7 litres. Estimate first: a number of thousands of cm³ is that many litres.
Check before you go on
1. A capsule is a cylinder with a hemisphere stuck on each end. Which parts make up the outside surface of the capsule?
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 solid is a cylinder of radius 6 cm and height 10 cm with a hemisphere of radius 6 cm on top. Its volume is ?π cm³. What number goes in front of π?
3. A cylindrical water tank has radius 50 cm and height 120 cm. How many litres does it hold, to one decimal place? (1 litre is 1000 cm³.)
Level 2 · Building
4. An ice-cream cone has radius 6 cm and height 8 cm (slant height 10 cm), and a hemisphere of ice cream of radius 6 cm sits on top. The volume of the cone and the ice cream together is ?π cm³. What number goes in front of π?
5. The same ice-cream cone has radius 6 cm, height 8 cm and slant height 10 cm, with a hemisphere on top. Find the area of its outside surface (the cone's curved side and the hemisphere's curved surface), in cm², to one decimal place.
6. A medicine capsule is a cylinder of radius 2 cm and length 10 cm with a hemisphere of radius 2 cm at each end. Find its outside surface area in cm², to one decimal place.
Level 3 · Stretch
7. Two drinks cans hold the same amount, 108π cm³. Can A has radius 3 cm and height 12 cm. Can B has radius 4 cm and height 6.75 cm. How many cm² less metal does Can B use (closed cans), to one decimal place?
8. A silo is a cylinder of radius 4 m and height 10 m, with a cone roof of radius 4 m and height 3 m (slant height 5 m). Find its volume in m³, to one decimal place.
Something to think about
In the first figure the least metal is near a radius of 5 cm, where the height is about 12.7 cm. Find the radius and height of a tin holding 1000 cm³ whose height is exactly equal to its diameter, and say whether it is close to the best.
One way to think about it
If h = 2r, the volume is πr² × 2r = 2πr³ = 1000, so r³ = 1000 ÷ (2π) = 159.15… and r = 5.42 cm, with h = 10.8 cm. Its surface is 2πr² + 2πr × 2r = 6πr² = 553.6 cm², which is a little under the 557.1 cm² at radius 5 cm. The best tin is about as tall as it is wide.
Exit ticket
9. Two closed cartons each hold 128π cm³. Carton P has radius 4 cm and height 8 cm. Carton Q has radius 8 cm and height 2 cm. Which uses less card, and why?