Question

Create a cylinder that has the same diameter as the height. Find its surface area.

Answer 1

ANSWER

$S=\frac{3}{2}\pi d^2$

EXPLANATION

The surface area is the sum of the areas of the sides.

The areas of the top and the base are the areas of circles with diameter d:

$A_{\text{circle}}=\pi r^2=\pi\mleft(\frac{d}{2}\mright)^2=\frac{\pi d^2}{4}$

The area of the rest of the cylinder is the area of a rectangle, with height d and width the circumference of the circle:

$C=\pi d$$A_{\text{rectangle}}=C\cdot d=\pi d^2$

The surface area is:

$\begin{gathered} S=A_{\text{rectangle}}+2A_{\text{circle}} \\ S=\pi d^2+2\frac{\pi d^2}{4} \\ S=\pi d^2+\frac{\pi d^2}{2} \\ S=\pi d^2(1+\frac{1}{2}) \\ S=\frac{3}{2}\pi d^2 \end{gathered}$