Area, Surface Area and Volume: Question 6

Worked solution

Part (a): Volume of the cylinder

The volume of a cylinder is given by

$V = \pi r^2 h$

Here the radius is $r = 4.5$ cm and the height is $h = 11$ cm. Substituting these values:

$V = \pi \times (4.5)^2 \times 11$

First square the radius:

$(4.5)^2 = 20.25$

Then multiply through:

$V = \pi \times 20.25 \times 11 = 222.75\pi$

$V = 699.789\ldots \text{ cm}^3$

Rounding to 3 significant figures:

$V = 700 \text{ cm}^3$

Part (b): Total surface area of the cylinder

The cylinder is closed, so its surface is made of the curved side plus the two flat circular ends.

Curved surface area:

$A_{\text{curved}} = 2\pi r h = 2\pi \times 4.5 \times 11 = 99\pi = 311.017\ldots \text{ cm}^2$

Two circular ends:

$A_{\text{ends}} = 2\pi r^2 = 2\pi \times (4.5)^2 = 2\pi \times 20.25 = 40.5\pi = 127.234\ldots \text{ cm}^2$

Total surface area:

$A = 2\pi r h + 2\pi r^2 = 2\pi r (r + h)$

$A = 2\pi \times 4.5 \times (4.5 + 11) = 9\pi \times 15.5 = 139.5\pi$

$A = 438.252\ldots \text{ cm}^2$

Rounding to 3 significant figures:

$A = 438 \text{ cm}^2$

Summary

• Volume: $\mathbf{700}$ cm$^3$ (3 s.f.)
• Total surface area: $\mathbf{438}$ cm$^2$ (3 s.f.)

Notice how using the factorised form $2\pi r(r+h)$ for the surface area saves work and helps avoid missing one of the ends. Keep full accuracy in your calculator and only round at the very end.