Formula sheet
Surface area of cone
The formula sheet's English line prints a typo ("πr² + 2πrs"); the correct formula, used here, is πr² + πrs. It is given, but you must still know πr² is the base circle, πrs is the curved surface, and that s is the slant height, not the vertical height.
What the symbols mean
- r Radius of the circular base (length unit).
- s Slant height, the sloping distance from the base edge to the apex (length unit).
- π Pi ≈ 3.142 or 22/7.
Given in the exam, or memorise?
The formula sheet's English line prints a typo ("πr² + 2πrs"); the correct formula, used here, is πr² + πrs. It is given, but you must still know πr² is the base circle, πrs is the curved surface, and that s is the slant height, not the vertical height.
Why it works
A cone is a base circle plus an unrolled curved fan.
- The flat base is a circle of area πr².
- Unrolling the curved surface gives a sector whose area works out to πrs.
- Adding them gives πr² + πrs.
Worked example 1
Find the total surface area of a cone with base radius 7 cm and slant height 10 cm. (Use π = 22/7)
- Surface area = πr² + πrs
- πr² = 22/7 × 7² = 154
- πrs = 22/7 × 7 × 10 = 220
- Total = 154 + 220 = 374
Worked example 2
A cone with base radius 7 cm has total surface area 594 cm². Find its slant height.
(Use π = 22/7)
- Surface area = πr² + πrs
- πr² = 22/7 × 49 = 154
- πrs = 594 − 154 = 440
- 22/7 × 7 × s = 22s = 440
- s = 440 ÷ 22 = 20
Where students go wrong
- Using the vertical height instead of the slant height s in πrs.
- Copying the formula-sheet typo 2πrs, the curved surface is πrs.
- Leaving out the base circle πr² when the cone is closed at the bottom.
- Using diameter instead of radius, or answering in cm instead of cm².
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What is slant height and how is it different from the height?
Slant height s runs along the sloping surface from the base rim up to the tip. Vertical height goes straight up the middle to the apex.
They differ, and the surface-area formula needs s. If a question gives the vertical height, use Pythagoras with the radius to find s first: s² = r² + height².
Why does the formula sheet show 2πrs but you use πrs?
The printed English line contains a known typo. The curved surface of a cone is πrs, so the correct total surface area is πr² + πrs.
Use πrs in your working. The unrolled curved surface is a sector, not a full circle, which is why it is πrs and not 2πrs.
Do I always include the base circle πr²?
Include πr² when the cone has a solid base, which gives the total surface area πr² + πrs. If the question asks only for the curved surface, or the cone is open like a party hat, use just πrs.
Read the wording carefully, total surface area means base plus curve, curved surface means the slope alone.