KJSEA 2025 Maths Q29 — Total Surface Area of a Cone

KJSEA 2025 Grade 9 Measurement

Published

The Question

“A learner molded a solid cone whose slant height was 20 cm and whose base radius was 4.2 cm. Calculate the total surface area of the cone, correct to two decimal places.”

1

Write down the total surface area formula

A solid cone is bounded by two surfaces: the flat circular base and the sloping curved surface. The base is a circle so its area is pi times the radius squared. The curved surface has area pi times the radius times the slant height. Adding them gives the total surface area.

TSA=πr2+πrlTSA = \pi r^{2} + \pi r l
2

Factor out the common term

Both terms share the factor pi times the radius. Taking this factor outside the bracket leaves the radius plus the slant height inside. This makes the arithmetic quicker because you only do one multiplication at the end.

TSA=πr(r+l)TSA = \pi r (r + l)
3

Substitute the known values

The radius is 4.2 cm and the slant height is 20 cm. Use the value of pi as twenty-two over seven, which pairs neatly with 4.2 because 4.2 divides exactly by 7. Add the radius and slant height inside the bracket to get 24.2.

TSA=227×4.2×(4.2+20)TSA = \frac{22}{7} \times 4.2 \times (4.2 + 20)
TSA=227×4.2×24.2TSA = \frac{22}{7} \times 4.2 \times 24.2

4.2 + 20 = 24.2, the value inside the bracket.

4

Simplify pi times the radius, then multiply out

First work out twenty-two over seven times 4.2, which gives 13.2. Then multiply this by 24.2 to reach the final area. The units are square centimetres because area is a two-dimensional measure.

227×4.2=13.2\frac{22}{7} \times 4.2 = 13.2
TSA=13.2×24.2=319.44TSA = 13.2 \times 24.2 = 319.44

Final Result

The total surface area of the cone is 319.44 cm², correct to two decimal places.

Why this method works

The total surface area works because a cone is made of exactly two pieces of surface, and area is additive: the whole is the sum of its parts. The base contributes the area of a circle, pi times the radius squared, while the slanting side unrolls into a sector whose area equals pi times the radius times the slant height. Factoring out pi times the radius is not a new rule but simple distributive-law algebra run in reverse, which reduces the number of separate multiplications and lowers the chance of a slip. Choosing twenty-two over seven for pi is deliberate: since 4.2 is a multiple of 0.7, the seven in the denominator cancels cleanly and keeps the working exact rather than rounded until the very last step.

✓

Check separately: base = 22/7 × 4.2² = 55.44 and curved = 22/7 × 4.2 × 20 = 264; 55.44 + 264 = 319.44 cm², which agrees.