Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9 (cylinders): a cylinder has radius 3 cm and height 4 cm. What is its volume, in terms of π?
Answer
36π cm³. The base area is π × 3² = 9π, and 9π × 4 = 36π. - From Year 9 (Pythagoras): a right-angled triangle has legs 5 cm and 12 cm. What is the hypotenuse?
Answer
13 cm. 5² + 12² = 25 + 144 = 169, and the square root of 169 is 13. - From Year 8 (circles): a circle has radius 4 cm. What is its circumference, in terms of π?
Answer
8π cm. The circumference is 2 × π × 4.
Words used in this lesson
- cone
- a solid with a circular base and one point at the top. For example: a paper cup for chestnuts.
- apex
- the point at the top of a cone. For example: the tip of a party hat.
- perpendicular height
- the height straight up from the base to the apex. For example: 12 cm.
- slant height
- the length from the apex to the edge of the base, along the surface. For example: 13 cm.
- curved surface
- the part of a cone that is not the circular base. For example: the leaf covering of a hat.
- sector
- a slice of a circle, like a piece of pie. For example: the flat shape of a cone's side.
Cones and tubs of chestnuts
A street vendor in Istanbul sells roasted chestnuts in paper cones. She also has a cylindrical tub with exactly the same radius and the same height as a cone. A customer asks: “If I pour a cone of chestnuts into the tub, how many cones will it take to fill it?”
Predict first: is it two, three or four? Is the cone half of the tub, or less? Then use the first figure to pour. The second figure deals with the slanting side, which is the new part.
Radius 3 cm and height 6 cm: the cylinder holds π × 3² × 6 = 54π cm³. Nothing has been poured in yet.
Show me the steps
- Radius 3 cm and height 6 cm, nothing poured yet: the cylinder holds 54π cm³. Pour in one cone-ful and it fills the cylinder to one third of its height.
- Pour in all three cone-fuls: 3 × 18π = 54π cm³, exactly the cylinder. Three cones fill the cylinder.
- Change the radius to 6 cm and the height to 3 cm: the cylinder holds π × 6² × 3 = 108π cm³ and the cone 36π cm³. It is still one third of the cylinder.
- Set the cone-fuls to 2: the water is 2/3 of the way up the cylinder and 2 × 18π = 36π cm³ has been poured in (with radius 3 cm and height 6 cm).
A cone holds one third of the cylinder with the same base and height: V = ⅓πr²h, with h the perpendicular height.
The slanting side is a curved surface. Its length from the apex to the edge is the slant height l, and l² = r² + h². The curved surface area is πrl, and the whole surface adds the circle: πr² + πrl.
The curved side of a cone opens out flat into part of a circle with radius l. Its edge is the same length as the circle round the base, 2πr, which makes the area ½ × l × 2πr = πrl. Press the button under the second figure to open it out.
Radius 3 cm, height 4 cm: the slant height is √(3² + 4²) = 5 cm, the volume is ⅓ × π × 9 × 4 = 12π cm³, and the whole surface is 9π + 15π = 24π ≈ 75.4 cm².
Show me the steps
- Radius 3 cm, height 4 cm: l = √(3² + 4²) = 5 cm. Volume ⅓ × π × 9 × 4 = 12π cm³. Curved surface π × 3 × 5 = 15π cm². Whole surface 9π + 15π = 24π ≈ 75.4 cm².
- Radius 5 cm, height 12 cm: l = √(5² + 12²) = 13 cm. Volume ⅓ × π × 25 × 12 = 100π cm³. Curved surface π × 5 × 13 = 65π cm². Whole surface 25π + 65π = 90π cm².
- Radius 12 cm, height 5 cm: l = √(12² + 5²) = 13 cm. A wide, flat cone has the same slant height as the tall one above. Volume ⅓ × π × 144 × 5 = 240π cm³. Curved surface π × 12 × 13 = 156π cm².
- Radius 9 cm, height 12 cm: l = √(9² + 12²) = 15 cm. Volume ⅓ × π × 81 × 12 = 324π cm³. Curved surface π × 9 × 15 = 135π cm². Whole surface 81π + 135π = 216π cm².
- Press Open out the curved surface with radius 3 cm and height 4 cm: the side lies flat as a sector of radius l = 5 cm. Its arc is as long as the circle round the base, 2π × 3 = 6π cm, and the sector is 360 × 3 ÷ 5 = 216° of a full circle.
A common mistake: putting the slant height l into the volume, where only the perpendicular height h belongs, or the height h into the curved surface, where only l belongs. A volume uses the height inside; an area on the surface uses the slant.
Worked example: a cone-shaped roof
A cone has radius 8 cm and height 15 cm. Find its volume and its curved surface area, in terms of π and to one decimal place.
- The volume is ⅓ × π × 8² × 15 = ⅓ × 64 × 15 × π = 320π cm³, which is 1005.3 cm³ to one decimal place.
- The slant height: l² = 8² + 15² = 64 + 225 = 289, so l = 17 cm.
- The curved surface area is π × 8 × 17 = 136π cm², which is 427.3 cm² to one decimal place.
Your go. The first step is given. A cone has radius 12 cm and height 16 cm. Find its slant height and its curved surface area, in terms of π.
- l² = 12² + 16² = 144 + 256 = 400.
- Now find l, then the curved surface area πrl, then check.
Check the answer
l = 20 cm, so the curved surface area is π × 12 × 20 = 240π cm², which is 754.0 cm² to one decimal place.
With a calculator
Find the slant height first, then leave π as a symbol until the end. For a cone with radius 5 cm and height 12 cm, l = √(5² + 12²) = √169 = 13 cm. The whole surface is π × 5² + π × 5 × 13 = 25π + 65π = 90π cm². Press 90 × the π key: 282.74…, so 282.7 cm² to one decimal place.
When the square root is not a whole number, keep it on the calculator display and use it straight away, without rounding in the middle. Estimate first, with 3 for π: 3 × (25 + 65) = 270, so an answer near 283 is sensible.
Check before you go on
1. A cone has base radius 6 cm, perpendicular height 8 cm and slant height 10 cm. Which expression gives its volume in cm³?
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 cone has radius 3 cm and height 7 cm. Its volume is ?π cm³. What number goes in front of π?
3. A cone has radius 8 cm and height 6 cm. What is its slant height, in cm?
Level 2 · Building
4. A cone has radius 7 cm and slant height 25 cm. Its curved surface area is ?π cm². What number goes in front of π?
5. A cone has radius 10 cm and perpendicular height 24 cm. Find its whole surface area, including the circular base, in cm², to one decimal place.
6. A cone has volume 96π cm³ and radius 4 cm. What is its perpendicular height, in cm?
Level 3 · Stretch
7. A conical hat from Vietnam has a radius of 20 cm and a height of 15 cm. How much leaf covers its curved surface, in cm², to one decimal place?
8. A conical heap of grain has a radius of 4.5 m and a height of 2 m. Find its volume in m³, to one decimal place.
Something to think about
A cone's radius is doubled and its height is halved. What happens to its volume? Check with the cone in the second figure that has radius 3 cm and height 4 cm.
One way to think about it
Doubling the radius multiplies the volume by 2² = 4, and halving the height multiplies it by 1/2, so the volume is multiplied by 2. Check: radius 6 cm and height 2 cm gives ⅓ × π × 36 × 2 = 24π, and the original is ⅓ × π × 9 × 4 = 12π. The new cone holds twice as much.
Exit ticket
9. A cone has radius 9 cm and perpendicular height 40 cm. Which is its curved surface area?