Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 9: the two shorter sides of a right-angled triangle are 1.5 cm and 2 cm. How long is the longest side?
Answer
2.5 cm. 1.5² + 2² = 2.25 + 4 = 6.25 and √6.25 = 2.5. - From Year 9: what is the area of a circle of radius 4 cm, as a multiple of π?
Answer
16π cm², because π × 4² = 16π. - From Year 8: a sector of a circle has an angle of 60°. What fraction of the whole circle is it?
Answer
1/6, because 60 ÷ 360 = 1/6.
Words used in this lesson
- slant height
- the length along the sloping surface, from the rim to the apex. For example: s = 17 for r = 8 and h = 15.
- vertical height
- the straight-up distance from the apex to the centre of the base. For example: the 36 in a cone with s = 39 and r = 15.
- sector
- a slice of a circle between two radii. For example: a quarter circle.
- curved surface
- the part of a cone's surface that is not the base. For example: πrs.
- apex
- the point of a cone or pyramid opposite its base. For example: the tip of a hat.
The hat that opens flat
A hat-maker in Hue shapes a cone-shaped hat from a flat sector of woven palm leaf. Opened out, the curved surface is not a mystery shape. It is a slice of a circle whose radius is the slanting edge of the hat, and the size of the hat decides how much of the circle is used.
Predict first: a tall, thin cone and a short, wide cone have the same slanting edge. Which one needs the bigger slice of the circle? Then use the figure to test it, and to see why the area of the curved surface comes out as it does.
Radius 4 cm and vertical height 7 cm: slant height 8.1 cm, curved surface 101.3 cm², sector angle 178.6 degrees.
Show me the steps
- Set r = 3 and h = 4: the slant height is 5, the curved surface is 15π ≈ 47.1 cm², and the sector angle is 216°.
- Keep r = 3 and raise h to 11: the slant height rises to 11.4 and the sector narrows to 94.7°. A taller cone of the same radius uses a thinner slice of the circle.
- Set r = 9 and h = 1: the cone is almost flat, the slant height is 9.1, and the sector angle is 357.8°, nearly the whole circle.
Slant height. The radius r, the vertical height h and the slant height s of a cone make a right-angled triangle, so s² = r² + h². The slant height is the longest side, and it is not the vertical height.
Why the curved surface is πrs. Opened out, it is a sector of a circle of radius s. Its arc is the edge of the base, 2πr, so the sector is the fraction 2πr ÷ 2πs = r ÷ s of the whole circle, whose area is πs². The area is (r ÷ s) × πs² = πrs, and the angle of the sector is 360 × r ÷ s degrees.
Total surface. A cone is πr² + πrs. A pyramid is its base plus its triangular faces, each ½ × (base edge) × (slant height of that face), and the face's slant height comes from a right-angled triangle made from the vertical height and half of the base edge.
Use the slanting edge, not the height. A cone with radius 16 cm and vertical height 30 cm has slant height 34 cm, so its curved surface is π × 16 × 34 = 544π cm², not π × 16 × 30 = 480π cm². On a pyramid the slant height of a face is not the length of the sloping edge where two faces meet.
Worked examples
1. A cone has radius 8 cm and vertical height 6 cm. Find its total surface area.
- Slant height: s² = 8² + 6² = 100, so s = 10 cm.
- Curved surface: π × 8 × 10 = 80π. Base: π × 8² = 64π.
- Total: 80π + 64π = 144π ≈ 452.4 cm².
2. A square pyramid has base side 6 cm and vertical height 4 cm. Find its total surface area.
- The triangle of the face: half the base is 3 cm, so the slant height of a face is √(4² + 3²) = 5 cm.
- Each face: ½ × 6 × 5 = 15 cm², and there are four, so 60 cm².
- Base: 6 × 6 = 36 cm². Total: 96 cm².
3. A cone is made from a sector of radius 25 cm and angle 216°. Find its base radius and its curved area.
- The arc of the sector is 216/360 of the circle of radius 25: 0.6 × 50π = 30π cm.
- That arc becomes the edge of the base: 2πr = 30π, so r = 15 cm.
- Curved surface: π × 15 × 25 = 375π ≈ 1178.1 cm². (The vertical height is √(25² − 15²) = 20 cm.)
Your go. The first step is given. A cone has base radius 3 cm and slant height 7 cm. Find the angle of the sector its curved surface opens into, to one decimal place.
- The sector is the fraction r ÷ s = 3 ÷ 7 of a whole circle.
- Now multiply that fraction by 360°.
Check the answer
360 × 3 ÷ 7 = 154.28…, so the angle is 154.3°. It is less than 180° because the slant height is more than twice the radius.
With a GDC (optional)
A GDC can find the slant height in one line, √(r² + h²), and then the curved area π × r × s. Store the slant height in memory (or use the Ans key, which holds the last answer) so that it is not rounded halfway through. A table of s for several heights shows how the cone and its sector change together.
Keep the exact form, such as a multiple of π, until the end, and round once. Everything here can be done by hand.
Check before you go on
1. A cone has radius 9 cm and vertical height 40 cm. What fraction of a full circle is the sector that its curved surface opens into?
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 conical lampshade in Marrakech has radius 20 cm and vertical height 21 cm. Find its slant height, in centimetres.
3. A lampshade in Hue is a frustum of a cone, with radius 10 cm at the top, radius 22 cm at the bottom and vertical height 35 cm. Find the length of its sloping side, in centimetres.
Level 2 · Building
4. A cone has radius 9 cm and slant height 15 cm. Find its total surface area, in square centimetres, to one decimal place.
5. A paper cup is a cone made from a sector of a circle of radius 12 cm. The cup's rim has radius 3 cm. Find the angle of the sector, in degrees.
6. A cone has base radius 5 cm and a curved surface area of 65π cm². Find its vertical height, in centimetres.
Level 3 · Stretch
7. A cone's slant height is three times its radius, and its curved surface area is 48π cm². Find the radius, in centimetres.
8. A square pyramid has a base of side 14 cm and a total surface area of 896 cm². Find its vertical height, in centimetres.
Something to think about
Keep a cone's radius the same and make it taller and taller. What happens to the angle of the sector it opens into? What would the angle be for a cone with no height at all, and what does the sector look like?
One way to think about it
The angle is 360 × r ÷ s, and s grows as h grows while r stays fixed, so the angle shrinks towards 0°: a very tall cone is a thin slice, almost a spike. With no height at all, s = r, the angle is 360°, and the sector is the whole circle: the cone has become a flat disc, and its curved surface is the same disc as its base.
Exit ticket
9. Two cones have the same slant height, 10 cm. One has base radius 3 cm and the other 6 cm. How many times larger is the curved surface of the second?