Alliance Form 3 P1 Q9 — Surface Area of a Cylinder (Find the Diameter)

KCSE (Form 3) Form 3 Measurement

Published

The Question

“A closed cylinder has its radius equal to its height. Its total surface area is 154 square centimetres. Taking pi as 3.142, find the diameter of the cylinder, giving your answer to two decimal places.”

1

Write the total surface area formula

A closed cylinder is made of three surfaces: a top circle, a bottom circle, and the curved side that wraps around it. The two circular ends together contribute two lots of pi r squared, and the curved side contributes 2 pi r h. Adding these gives the total surface area.

A=2πr2+2πrhA = 2\pi r^{2} + 2\pi r h
2

Use the fact that the radius equals the height

The question tells us the radius and the height are equal, so wherever we see h we can write r instead. This turns the curved-surface term into another 2 pi r squared, which we then add to the first term. Two identical terms of 2 pi r squared combine into 4 pi r squared, so the whole surface area depends on r alone.

h=rh = r
A=2πr2+2πr(r)=2πr2+2πr2=4πr2A = 2\pi r^{2} + 2\pi r (r) = 2\pi r^{2} + 2\pi r^{2} = 4\pi r^{2}
3

Substitute the known surface area and solve for r squared

We set the simplified expression equal to the given surface area of 154 square centimetres. To isolate r squared we divide both sides by 4 pi. Using pi as 3.142, four times pi is about 12.57, and dividing 154 by that gives r squared.

4πr2=1544\pi r^{2} = 154
r2=1544×3.142=15412.568=12.253r^{2} = \frac{154}{4 \times 3.142} = \frac{154}{12.568} = 12.253
4

Take the square root to find the radius

Taking the positive square root of r squared gives the radius. We keep only the positive value because a length cannot be negative.

r=12.253=3.5 cmr = \sqrt{12.253} = 3.5 \text{ cm}
5

Convert the radius to a diameter

The question asks for the diameter, not the radius. The diameter is simply twice the radius, so we double our answer to get the final value to two decimal places.

d=2r=2×3.5=7.00 cmd = 2r = 2 \times 3.5 = 7.00 \text{ cm}

Final Result

The diameter of the cylinder is 7.00 cm.

Why this method works

The method works because the total surface area of a closed cylinder is fixed by its two ends and its curved side. Normally these depend on two separate measurements, the radius and the height, so a single surface-area value would not be enough to pin them down. The extra condition that the radius equals the height removes that second unknown: every term collapses into a multiple of r squared, leaving one equation in one unknown that we can solve directly. Once the radius is found, the diameter follows immediately because it is defined as twice the radius.

✓

Substitute back: 4 × 3.142 × 3.5² = 4 × 3.142 × 12.25 = 153.96 ≈ 154 cm², matching the given surface area.