Sheet № 169 · Higher only · AQA · Edexcel · OCR
Surface Area of a Cone –
The surface area of a cone is a Higher tier topic that combines circle area, arc length ideas, and Pythagoras' theorem. You need to distinguish between the curved surface area and the total surface area, and between slant height and vertical height.
§Key definitions
Question:
A cone has a base radius of 5 cm and a slant height of 13 cm. Find the total surface area to 1 decimal place.
Answer:
282.7 cm²
Q1 (Foundation):
A cone has radius 4 cm and slant height 9 cm. Find the curved surface area in terms of π.
Q2 (Foundation):
A cone has radius 3 cm and slant height 10 cm. Find the total surface area to 1 decimal place.
Q3 (Higher):
A cone has radius 5 cm and vertical height 12 cm. Find the total surface area to the nearest whole number.
§Formulas to memorise
Curved surface area = πrl, where r is the base radius and l is the slant height
Total surface area = πrl + πr² (curved surface + base)
l = √(r² + h²), using Pythagoras to find the slant height from the radius and vertical height
Curved SA = πrl = π × 5 × 13 = 65π
Base area = πr² = π × 5² = 25π
Total SA = 65π + 25π = 90π = 282.7 (1 d.p.)
l = √(r² + h²) = √(36 + 64) = √100 = 10 cm
Curved SA = π × 6 × 10 = 60π
Base area = π × 6² = 36π
Total SA = 60π + 36π = 96π = 301.6 (1 d.p.)
Worked example
A cone has a base radius of 5 cm and a slant height of 13 cm. Find the total surface area to 1 decimal place.
Working: Curved SA = πrl = π × 5 × 13 = 65π Base area = πr² = π × 5² = 25π Total SA = 65π + 25π = 90π = 282.7 (1 d.p.)
⚠ Common mistakes
- ✗Confusing slant height and vertical height. The slant height l runs along the surface; the vertical height h is inside the cone from base centre to apex. Always check which one the question provides.
- ✗Forgetting the base. "Surface area" usually means total surface area including the base. If only the curved surface is required, the question will specify "curved surface area."
- ✗Using diameter instead of radius. The formulas use the radius. If the question gives the diameter, halve it first.
- ✗Not using Pythagoras when needed. When the slant height is not given, you must calculate it using l = √(r² + h²) before applying the surface area formula.
✦ Exam tips
- →The curved surface area formula πrl is given on the exam formula sheet — but you still need to remember to add πr² for the total.
- →Show the Pythagoras step clearly if you need to find the slant height — this earns method marks.
- →If the question asks for an exact answer, leave your answer in terms of π.