Cones & spheres · Ages 14–15
Surface area of pyramids and cones
Cone's curved surface πrl; l² = r² + h².
How to do it
Method
Cone's curved surface πrl; l² = r² + h²
r = 5, h = 12: l = 13, 65π + 25π = 90π
Another way
Open it into a sector
513 × 169π = 65π
In full
A pyramid's surface = its base + the triangles round it. A square base of side 10 cm, each triangle's slant height 13 cm: four triangles 4 × 10 × 13 ÷ 2 = 260, plus the base 100, 360 cm² in all. A cone's curved surface opens out into a sector whose radius is the slant height l and whose arc is the base's circumference 2πr: its area = 12 × 2πr × l = πrl. Total surface area = πr² + πrl. Radius 5 cm, height 12 cm: the slant height by Pythagoras' theorem first, l = 52 + 122 = 13; curved π × 5 × 13 = 65π, base 25π, 90π cm² in all. Or work it as the sector: its area = (2πr ÷ 2πl) × πl² = (513) × 169π = 65π, the same. The commonest slip: the height used as the slant height; the slant height is always the longer.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The cone in the picture has a base radius of 30 cm and a slant height of 34 cm. What is its curved surface area, in terms of π?
Show the answer and working
Answer: 1020 π cm²
Curved surface = πrl = π × 30 × 34 = 1020π cm².
Example 2
The solid cone in the picture has a base radius of 6 cm, a height of 8 cm and a slant height of 10 cm. What is its total surface area?
- A. 60π cm²
- B. 132π cm²
- C. 84π cm²
- D. 96π cm²
Choose the right answer
Show the answer and working
Answer: D. 96π cm²
πr² + πrl = π × 62 + π × 6 × 10 = 96π cm².
Example 3
Taking π as 3.14, a cone has a base radius of 3 cm and a slant height of 19 cm. What is its curved surface area, in cm²?
Show the answer and working
Answer: 178.98 cm²
πrl = 3.14 × 3 × 19 = 178.98 cm².
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The solid cone in the picture has a base radius of 3 cm, a height of 4 cm and a slant height of 5 cm. What is its total surface area?
- A. 15π cm²
- B. 24π cm²
- C. 33π cm²
- D. 21π cm²
Choose the right answer
Answer:
2.The solid cone in the picture has a base radius of 72 cm and a height of 21 cm. What is its total surface area, in terms of π?
Answer:
3.A cone of base radius 3 cm and height 4 cm has a curved surface area of 12π cm².
True or false?
Answer:
4.The cone in the picture has a base radius of 12 cm and a curved surface area of 156π cm². How high is it, in cm?
Answer:
5.A cone has a base radius of 3 cm and a slant height of 8 cm. Cut its curved surface along a slanting line and lay it flat: it is a sector. What is the sector's angle at the centre, in degrees?
Answer:
All the answers
- 1. B. 24π cm²
- 2. 10584 π cm²
- 3. False
- 4. 5 cm
- 5. 135°
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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