KCSE 2025 Maths P1 Q8 — Cone from a Sector

KCSE 2025 Form 4 Measurement

Published

The Question

“A sector of a circle with area 550 square centimetres is curved round to form an open cone with base radius 7 centimetres. Find the height of the cone.”

1

Link the sector to the cone's curved surface

When the flat sector is rolled up into an open cone, the area of the sector does not change — it simply becomes the sloping outside of the cone, which is the curved surface. So the area of the sector equals the curved surface area of the cone. The formula for the curved surface area of a cone is pi times the base radius times the slant height, so set that equal to 550.

πrl=550\pi r l = 550
2

Solve for the slant height

Substitute pi as 22 over 7 and the base radius r as 7. The 7 in the denominator of pi cancels with the radius of 7, which tidies things up and leaves just 22 times the slant height equal to 550. Divide both sides by 22 to find the slant height.

227×7×l=550\frac{22}{7} \times 7 \times l = 550
22 l=55022\,l = 550
l=55022=25 cml = \frac{550}{22} = 25 \text{ cm}

The slant height is the distance down the sloping side of the cone, not the vertical height.

3

Set up the right-angled triangle

Inside the cone, the base radius, the vertical height, and the slant height form a right-angled triangle. The slant height is the longest side, the hypotenuse, because it runs from the tip down to the edge of the base. By Pythagoras' theorem, the slant squared equals the radius squared plus the height squared.

l2=r2+h2l^{2} = r^{2} + h^{2}
4

Rearrange and find the height

Make the height squared the subject by subtracting the radius squared from the slant squared. Put in 25 for the slant and 7 for the radius, work out each square, then take the square root of the result to get the height.

h2=l2−r2=252−72h^{2} = l^{2} - r^{2} = 25^{2} - 7^{2}
h2=625−49=576h^{2} = 625 - 49 = 576
h=576=24 cmh = \sqrt{576} = 24 \text{ cm}

Final Result

The height of the cone is 24 centimetres.

Why this method works

The method works because rolling a flat sector into a cone conserves area — no material is added or lost, so the sector's 550 cm² becomes exactly the cone's curved surface. That single fact lets you use the curved-surface formula to unlock the slant height. Once the slant height is known, the cone's radius, height and slant height are locked together as the three sides of a right-angled triangle whose right angle sits at the centre of the base, so Pythagoras' theorem converts the slant and radius into the vertical height.

✓

Check with Pythagoras: 7² + 24² = 49 + 576 = 625 = 25², so the slant height of 25 cm is consistent with radius 7 cm and height 24 cm.