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This page sets out the method for area and circumference of circles in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Area and circumference of circles topic page.
When you see a circle, first decide whether you were given the radius or the diameter.
When the shape is half a circle, halve the area and halve the curved edge — then add the diameter for perimeter.
The diameter is twice the radius.
If the question gives the diameter, halve it before you use r².
C = πd = 2πr.
It is a length, so the unit is cm or m, not a square unit.
A = πr².
Square the radius, then multiply by π. Do not square π.
If the question asks for an exact answer, keep π in the answer.
The coefficient is the number you worked out. 2 × 5 × π is 10π.
A decimal answer uses the π key, not 3.14, unless the question says to use 3.14.
Round only at the end, to the accuracy asked for.
A semicircle circumference is half the circumference plus the diameter.
The curved part is half. The straight edge is extra.
The area of a circle is A=π r^2. Square the radius first, then multiply by π. For r=5cm, r^2=25 and A=π× 25, which is 78.5cm^2 to 1 decimal place. For r=6cm, r^2=36 and A=π× 36, which is 113.1cm^2 to 1 decimal place. Notice that doubling the radius does not double the area. From 5 to 10 the radius doubles, but the area is multiplied by 4 because 10^2=100 and 5^2=25.
A common swap is to calculate 2π r and call it the area, or π r^2 and call it the circumference. The circumference grows like the radius. The area grows like the radius squared, which is why the units are cm^2. If you know the diameter only, halve it before you square. Diameter 10cm means r=5, so the area is π× 5^2, not π× 10^2.
Method: 1. Square the radius, then multiply by π: A=π r^2. 2. Radius 5cm gives π× 25=78.5cm^2 to 1 decimal place. 3. Radius 6cm gives π× 36=113.1cm^2 to 1 decimal place.
The perimeter of a semicircle is the curved edge plus the diameter. For a diameter of 10cm the radius is 5cm, the curved length is π× 5, and the perimeter is π× 5+10. That is 25.7cm to 1 decimal place. The area of the same semicircle would be half of π r^2, which is a different question. Read whether the question wants the distance around the shape or the surface inside it.
Round once, at the end. Keep more decimal places on the calculator, or keep π exact, until the final answer. If you round π to 3.1 too early you can miss the required decimal place. To round to 1 decimal place, look at the second decimal digit. The circumference 2×π× 6 is 37.699ldots, and the second decimal is 9, so 37.6 rounds up to 37.7cm. The area π× 36 is 113.097ldots, so to 1 decimal place it is 113.1cm^2.
Method: 1. Add the curved half-circumference to the full diameter. 2. Diameter 10cm gives perimeter π× 5+10=25.7cm to 1 decimal place. 3. Round once at the end: 2×π× 6=37.7cm to 1 decimal place.
Non-calculator papers often say give your answer in terms of π. This means leave π as a symbol and do not multiply it out. A circle with radius 5 cm has area π× 5^2=25π cm^2 and circumference 2×π× 5=10π cm. The number goes in front of the π. This is often the neatest and most accurate way to give an answer, because nothing has been rounded.
When a diameter is given, halve it first. A diameter of 18 cm gives a radius of 9 cm and an area of π× 9^2=81π cm^2. Square the radius, not π. If the question mixes straight edges and curves, as in a perimeter, write the answer as 6π+24 with the multiple of π first and the plain number after it.
Method: 1. Write the radius, halving the diameter if needed. 2. Substitute into A=π r^2 or C=2π r without using the calculator button for π. 3. Tidy up so the number comes before π. 4. Check you have not squared π or forgotten the units.
Tip: Use the calculator only when the question says give your answer to a number of decimal places or significant figures.
A semicircle is half a circle and a quarter circle is a quarter. The curved part is the matching fraction of the circumference: for a quarter circle of radius 12 cm the arc is tfrac14× 2π× 12=6π cm. The perimeter also includes the straight edges, which here are the two radii, 12+12=24. So the perimeter is 6π+24.
Compound shapes combine circles and rectangles. A running track has two straight sides and two semicircular ends, and the two semicircles together make one full circle. With straight sides of 90 m and ends of diameter 60 m, the distance round is 2× 90+π× 60, which is 368.5 m to 1 decimal place. For the area, add the rectangle and the circle.
Method: 1. Identify the fraction of the circle: half, a quarter, or whole when two halves join. 2. Work out the curved length or the area of the circular part. 3. Add any straight edges for a perimeter, or the rectangle for an area. 4. Round only at the end, and give the units.
Some questions give the circumference and ask for the diameter or radius. Since C=π d, divide the circumference by π to get the diameter. A circumference of 40 cm gives d=40÷π=12.7 cm to 1 decimal place. Halve it for the radius if needed. Check your answer by putting it back into the formula and seeing that you recover the number you were given.
From an area, undo the formula A=π r^2 in two steps: divide by π and then take the square root. For an area of 154 cm^2 with π≈tfrac227, dividing by π gives 49 and the square root gives r=7. When the circumference or area is given in terms of π, such as 14π or 49π, divide by π on paper and no calculator is needed.
Method: 1. Write the formula and put in the number you were given. 2. Divide both sides by π or by 2π. 3. For an area, take the square root to get the radius. 4. Use the new radius in the next part of the question and check it makes sense.
Area depends on the square of the radius, so doubling the radius makes the area four times as big, not twice. Tripling the radius makes the area nine times as big. The circumference depends on the radius directly, so it only doubles when the radius doubles. These facts appear in explain or show that questions.
Percentage questions combine the idea with a multiplier. If the radius is increased by 20%, the radius multiplier is 1.2. The area multiplier is 1.2^2=1.44, so the area increases by 44%. Never say the area increases by 20%. Square the multiplier, then convert back to a percentage change by subtracting 1.
Method: 1. Write the multiplier for the radius, such as 1.2 for a 20% increase. 2. Square it to get the multiplier for the area. 3. Subtract 1 and convert to a percentage for the change in area. 4. For circumference, use the same multiplier as the radius, not the square.
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