Circumference and Area of a Circle

Worked example

1. 1

   $r=4.5\text{ cm}$ — The question gives the radius. $C=2\pi r$ uses the radius, so it goes straight in.

2. 2

   $C=2\times \pi \times 4.5$ — Put 4.5 in place of r. Write this line down: it earns a mark on its own.

3. 3

   $C=9\pi$ — 2 × 4.5 = 9. If the question says “in terms of π”, stop here.

4. 4

   $C=\mathrm{28.274...}$ — Press 9 × π on the calculator. Write down at least 4 figures before you round.

5. 5

   $C=28.3\text{ cm}$ — Round to 3 significant figures: the 4th figure is 7, so 28.2 rounds up to 28.3. It is a length, so the unit is cm.

Worked example

1. 1

   $d=16\text{ cm}$ — The question gives a diameter. $\pi r^2$ needs the radius, so change it first.

2. 2

   $r=16÷2=8$ — Halve the diameter. The radius only goes half way across.

3. 3

   $r^2=8^2=64$ — Square the radius first: 8 × 8 = 64. Only r is squared, not π.

4. 4

   $A=\pi \times 64=64\pi$ — Now multiply by π. 64π is the exact answer, the one to give “in terms of π”.

5. 5

   $A=\mathrm{201.06...}$ — Press 64 × π on the calculator. Keep the full value for now.

6. 6

   $A=201{\text{ cm}}^2$ — 3 significant figures: the 4th figure is 0, so it stays 201. Area is measured in square units, so cm².

Worked example

1. 1

   $r=20÷2=10$ — Halve the diameter to get the radius, because $\pi r^2$ needs r.

2. 2

   $r^2={10}^2=100$ — Square the radius: 10 × 10 = 100.

3. 3

   $A=100\pi {\text{ cm}}^2$ — Write π after the number. 100π is the full answer. Stop: no calculator step.

Worked example

1. 1

   $\pi r^2=81\pi$ — The area formula equals the area given. Write it as an equation.

2. 2

   $r^2=81$ — Divide both sides by π. The π on each side cancels.

3. 3

   $r=\sqrt{81}=9$ — Square root to undo the square. A radius is a length, so take the positive root, 9.

4. 4

   $C=2\times \pi \times 9$ — Now use the circumference formula with r = 9.

5. 5

   $C=18\pi \text{ cm}$ — 2 × 9 = 18, so the answer is 18π. Again, no calculator step.

Worked example

1. 1

   $\pi r^2=50$ — The area formula equals the area given.

2. 2

   $r^2=50÷\pi =\mathrm{15.915...}$ — r² is being multiplied by π, so divide both sides by π.

3. 3

   $r=\sqrt{\mathrm{15.915...}}=\mathrm{3.989...}$ — r is being squared, so square root both sides. Use the full value, not 15.9.

4. 4

   $r=3.99\text{ cm}$ — Round to 3 significant figures at the very end.

Worked example

1. 1

   $2\pi r=40$ — The circumference formula equals the length given. Use 2πr because the question asks for r.

2. 2

   $r=40÷2\pi$ — r is multiplied by 2 and by π, so divide by both. Put 2π in brackets on a calculator.

3. 3

   $r=\mathrm{6.366...}$ — 40 ÷ (2 × π) on the calculator.

4. 4

   $r=6.37\text{ cm}$ — 3 significant figures.

Worked example

1. 1

   $\pi r^2=\pi \times 6^2=36\pi$ — Find the area of the whole circle first: 6² = 36.

2. 2

   $A=\frac{1}{2}\times 36\pi =18\pi$ — Halve it, because a semicircle is half the circle.

3. 3

   $A=\mathrm{56.548...}=56.5{\text{ cm}}^2$ — 18 × π on the calculator, then 3 significant figures.

Worked example

1. 1

   $\pi \times 11=\mathrm{34.557...}$ — The whole circumference is πd.

2. 2

   $\mathrm{34.557...}÷2=\mathrm{17.278...}$ — The curved edge is only half of it.

3. 3

   $\mathrm{17.278...}+11=\mathrm{28.278...}$ — Add the straight edge. The perimeter goes all the way round, and the diameter is part of the edge.

4. 4

   $P=28.3\text{ cm}$ — 3 significant figures.

Worked example

1. 1

   $\text{square area}=64{\text{ cm}}^2$ — The question gives the square's area, not its side.

2. 2

   $\text{side}=\sqrt{64}=8$ — A square's area is side × side, so square root it to get the side.

3. 3

   $d=8$ — The circle touches all four sides, so it stretches right across: diameter = side.

4. 4

   $C=\pi \times 8=\mathrm{25.132...}$ — The circle's length is its circumference, πd. Keep the full value.

5. 5

   $4\times 8=32$ — The square's edge is its perimeter: four sides of 8.

6. 6

   $32+\mathrm{25.132...}=\mathrm{57.132...}$ — The wire makes both shapes, so add the two lengths.

7. 7

   $57.1\text{ cm}$ — 3 significant figures.

Worked example

1. 1

   $C=2\times \pi \times 9=18\pi$ — The circle's length is its circumference. Keep π, because the answer must be in terms of π.

2. 2

   $P=4x$ — A square has 4 equal sides of x.

3. 3

   $4x=18\pi$ — The question says they are equal, so put an equals sign between them.

4. 4

   $x=\frac{18\pi }{4}=\frac{9\pi }{2}$ — Divide both sides by 4, then simplify 18/4 to 9/2.

Worked example

1. 1

   $\pi \times 2^2=4\pi$ — Area of one small circle: 2² = 4.

2. 2

   $25\times 4\pi =100\pi$ — There are 25 of them, so multiply by 25.

3. 3

   $\pi R^2=100\pi$ — The large circle's area is the same, so write an equation.

4. 4

   $R^2=100$ — Divide both sides by π. The π cancels.

5. 5

   $R=\pm 10$ — Square root both sides. Both 10 and −10 square to 100.

6. 6

   $R=10$ — R is a radius, and a length cannot be negative, so reject −10.

Worked example

1. 1

   $C=\pi \times 60=\mathrm{188.495...}\text{ cm}$ — One turn moves the wheel one circumference. The diameter is given, so use πd.

2. 2

   $1\text{ km}=1000\text{ m}=100000\text{ cm}$ — The units must match. 1 km = 1000 m, and 1 m = 100 cm.

3. 3

   $100000÷\mathrm{188.495...}=\mathrm{530.5...}$ — Turns = distance ÷ distance per turn.

4. 4

   $530\text{ complete turns}$ — Round down: the 531st turn is not finished, and the question asks for complete turns.