Area: $A = \pi r^2$
A circle has radius $9$ cm. Find its circumference and area, each to 1 d.p.
A circular pond has diameter $7$ m. Find its area, to 2 d.p.
A circle has area $200\text{ cm}^2$. Find its radius, to 2 d.p.
| Shape | Area | Perimeter |
|---|---|---|
| Semicircle | $\tfrac{1}{2}\pi r^2$ | $\pi r + 2r$ (curve $+$ diameter) |
| Quarter circle | $\tfrac{1}{4}\pi r^2$ | $\tfrac{1}{2}\pi r + 2r$ |
A semicircle has radius $8$ cm. Find its area and its perimeter, each to 1 d.p.
A shape is a rectangle $20$ cm by $12$ cm with a semicircle of diameter $12$ cm attached to one short end. Find its area, to 1 d.p.
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | $lwh$ | $2(lw + lh + wh)$ |
| Prism | cross-section $\times$ length | $2\times$cross-section $+$ perimeter $\times$ length |
| Cylinder | $\pi r^2 h$ | $2\pi r^2 + 2\pi r h$ |
| Cone | $\tfrac{1}{3}\pi r^2 h$ | $\pi r^2 + \pi r l$ ($l$ = slant height) |
| Sphere | $\tfrac{4}{3}\pi r^3$ | $4\pi r^2$ |
| Hemisphere (solid) | $\tfrac{2}{3}\pi r^3$ | $2\pi r^2 + \pi r^2 = 3\pi r^2$ |
| Pyramid | $\tfrac{1}{3} \times$ base area $\times h$ | base $+$ area of the triangular faces |
A sphere has radius $6$ cm. Find its volume and surface area, in terms of $\pi$ and as decimals to 1 d.p.
A cone has radius $5$ cm and vertical height $12$ cm. Find its total surface area, to 1 d.p.
A pyramid has a square base of side $8$ cm and vertical height $9$ cm. Find its volume.
- Break the solid into standard pieces.
- Find each volume separately and add (or subtract for a hole).
- For surface area, list only the faces that are actually visible.
- Remember: where two pieces join, both hidden faces disappear.
- Keep answers in terms of $\pi$ until the end.
A solid consists of a cylinder of radius $4$ cm and height $10$ cm with a hemisphere of the same radius on top. Find its volume and total surface area, both in terms of $\pi$.
A cuboid $10$ cm by $10$ cm by $6$ cm has a cylindrical hole of radius $2$ cm drilled all the way through its $6$ cm depth. Find the remaining volume, to 1 d.p.
Circle
$C = 2\pi r = \pi d$; $A = \pi r^2$. The squared one is the area.
Diameter first
Halve it to get $r$ before using any formula.
Semicircle
Area $\tfrac{1}{2}\pi r^2$; perimeter $\pi r + 2r$.
Cylinder
$V = \pi r^2 h$; $SA = 2\pi r^2 + 2\pi r h$.
Cone
$V = \tfrac{1}{3}\pi r^2 h$; curved surface $= \pi r l$.
Sphere
$V = \tfrac{4}{3}\pi r^3$; $SA = 4\pi r^2$.
Pyramid
$V = \tfrac{1}{3} \times$ base $\times$ height.
Composite
Add volumes; count only the visible faces for surface area.
Exact answers
Leave in terms of $\pi$ unless told otherwise.
A circle has radius $12$ cm. Find (a) its circumference, (b) its area, each in terms of $\pi$.
โถ Show solution
(a) $C = 2\pi r = 24\pi$ cm
(b) $A = \pi r^2 = 144\pi\text{ cm}^2$
A circular table has diameter $1.4$ m. Find its area, to 3 significant figures.
โถ Show solution
$r = 1.4 \div 2 = 0.7$ m
$A = \pi \times 0.7^2 = 0.49\pi = 1.539\ldots$
$1.54\text{ m}^2$ (3 s.f.)
A circle has circumference $50$ cm. Find its radius, to 2 d.p.
โถ Show solution
$2\pi r = 50$
$r = \dfrac{50}{2\pi} = \dfrac{25}{\pi} = 7.96$ cm
Find the perimeter of a semicircle of radius $10$ cm, to 1 d.p.
โถ Show solution
Curved part $= \pi r = 10\pi = 31.42$ cm
Straight part $= 2r = 20$ cm
Perimeter $= 31.42 + 20 = 51.4$ cm
A sphere has radius $3$ cm. Find its volume and surface area in terms of $\pi$.
โถ Show solution
$V = \tfrac{4}{3}\pi \times 27 = 36\pi\text{ cm}^3$
$SA = 4\pi \times 9 = 36\pi\text{ cm}^2$
(A curious coincidence: for $r = 3$ the two numbers match, but the units differ.)
A cone has radius $6$ cm and slant height $10$ cm. Find (a) its curved surface area, (b) its vertical height, (c) its volume. Give (a) and (c) in terms of $\pi$.
โถ Show solution
(a) $\pi r l = \pi \times 6 \times 10 = 60\pi\text{ cm}^2$
(b) $h^2 = l^2 - r^2 = 100 - 36 = 64$, so $h = 8$ cm.
(c) $V = \tfrac{1}{3}\pi \times 36 \times 8 = 96\pi\text{ cm}^3$
A cylinder has volume $1000\text{ cm}^3$ and radius $5$ cm. Find its height, to 2 d.p.
โถ Show solution
$\pi r^2 h = 1000$
$25\pi h = 1000$
$h = \dfrac{1000}{25\pi} = \dfrac{40}{\pi} = 12.73$ cm
A shape is a rectangle $14$ cm by $8$ cm with a semicircle of diameter $8$ cm attached to each short end. Find (a) the total area, (b) the total perimeter, each to 1 d.p.
โถ Show solution
Semicircle radius $= 4$ cm. Two semicircles make one full circle.
(a) Rectangle $= 14 \times 8 = 112\text{ cm}^2$. Full circle $= \pi \times 16 = 50.27\text{ cm}^2$.
Total $= 112 + 50.27 = 162.3\text{ cm}^2$
(b) The perimeter is the two long rectangle sides plus the two curves (which together make a full circumference).
$= 2 \times 14 + 2\pi \times 4 = 28 + 25.13 = 53.1$ cm
A solid hemisphere has radius $9$ cm. Find (a) its volume, (b) its total surface area (including the flat circular face), both in terms of $\pi$.
โถ Show solution
(a) $V = \tfrac{2}{3}\pi r^3 = \tfrac{2}{3} \times \pi \times 729 = 486\pi\text{ cm}^3$
(b) Curved part $= 2\pi r^2 = 162\pi$; flat circle $= \pi r^2 = 81\pi$.
Total $= 162\pi + 81\pi = 243\pi\text{ cm}^2$
A solid is made from a cylinder of radius $5$ cm and height $12$ cm with a cone of the same radius and slant height $13$ cm on top.
(a) Find the cone's vertical height. (b) Find the total volume in terms of $\pi$. (c) Find the total surface area in terms of $\pi$. (d) The solid is made of a metal of density $7.8\text{ g/cm}^3$. Find its mass, to the nearest gram.
โถ Show solution
(a) $h^2 = 13^2 - 5^2 = 169 - 25 = 144$, so $h = 12$ cm.
(b) Cylinder $= \pi \times 25 \times 12 = 300\pi$
Cone $= \tfrac{1}{3}\pi \times 25 \times 12 = 100\pi$
Total $= \mathbf{400\pi\text{ cm}^3}$
(c) Visible surfaces: the flat base, the cylinder's curved surface, the cone's curved surface. (The cylinder's top is covered by the cone.)
Base $= \pi \times 25 = 25\pi$
Cylinder curved $= 2\pi \times 5 \times 12 = 120\pi$
Cone curved $= \pi r l = \pi \times 5 \times 13 = 65\pi$
Total $= \mathbf{210\pi\text{ cm}^2}$
(d) Volume $= 400\pi = 1256.637\text{ cm}^3$
Mass $= 1256.637 \times 7.8 = 9801.8$, so about $\mathbf{9802}$ g (roughly $9.8$ kg).