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This page sets out the method for sectors and arcs 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 Sectors and arcs topic page.
When you have the angle and the radius and want an arc length or a sector area.
When the arc length or sector area is given and you need .
A sector with angle 60 degrees and radius 6 has area 6π. Take the fraction of the full circle: 60/360 = 1/6 of 36π.
The fraction is angle over 360.
The arc length is the fraction of the circumference 2πr.
The perimeter includes the arc plus two straight edges of length r.
Arc is curved only. Perimeter includes the two straight radii.
Always reduce the angle / 360 fraction to its simplest form before multiplying.
Every sector formula starts with the same fraction. Put the centre angle over 360. For 90^° the fraction is 90/360 = 1/4. For 60^° it is 60/360 = 1/6. For 120^° it is 120/360 = 1/3. Simplify that fraction before you multiply, because 1/3 of a quantity is easier to see than 120/360 of it.
The fraction is the same for the arc and for the area. Only the thing you take the fraction of changes. Students sometimes put the angle over 180, which is the sum of a triangle, or over π. Degrees stay over 360. If a question gives the angle in a pie-chart style diagram, read the centre angle, not an angle drawn somewhere else on the circumference.
Method: 1. Write the centre angle over 360, then simplify. 2. 90^° is 1/4, 60^° is 1/6, and 120^° is 1/3. 3. Use that same fraction for both the arc and the area.
The full circumference is 2π r. The arc is (θ)/360 of that. With radius 6 cm and angle 60^°, the arc is 60/360 × 2π × 6 = 2π cm. To 1 decimal place that is 6.3 cm. The exact form 2π is worth keeping if the question does not ask you to round.
You can also run the formula backwards. If you know the arc and the radius, the fraction of the circumference is arc divided by 2π r, and the angle is that fraction times 360. An arc of 8 cm on a circle of radius 10 cm gives θ = 8/(2π × 10) × 360, which is about 45.8^° to 1 decimal place. Do not replace π with 3.14 until the rounding line if you still need an exact area afterwards.
Method: 1. Take (θ)/360 of the circumference 2π r. 2. Radius 6cm and angle 60^° give 60/360× 2π× 6=2πcm. 3. To 1 decimal place that arc is 6.3cm.
The full circle has area π r^2. The sector has area (θ)/360 × π r^2. With radius 10 cm and angle 36^°, the fraction is 36/360 = 1/10, and the area is 1/10 × π × 100 = 10π cm^2, which is 31.4 cm^2 to 1 decimal place. Notice that r^2 is 100, not 20.
When the arc is known, the angle and the area are linked. The fraction (θ)/360 equals fracarc2π r. Substitute that fraction into the area formula and the π cancels: area = fracarc2r × r^2 = 1/2 × arc × r. For an arc of 8 cm and radius 10 cm the area is exactly 1/2 × 8 × 10 = 40 cm^2, even though the angle itself is not a tidy whole number.
Method: 1. Take (θ)/360 of π r^2. 2. Radius 10cm and angle 36^° give 1/10×π× 100=31.4cm^2 to 1 decimal place. 3. From an arc, use 1/2×arc× r, so arc 8cm and radius 10cm give 40cm^2.
Walk around a sector and you travel along the arc, then along a radius to the centre, then back out along the other radius. The perimeter is arc + 2r. For radius 8 cm and angle 120^°, the arc is 1/3 × 2π × 8 = (16π)/3 cm, about 16.8 cm. The two radii add 16 cm, so the perimeter is about 32.8 cm.
The area of that same sector is 1/3 × π × 8^2 = (64π)/3 cm^2, about 67.0 cm^2. Area and perimeter use the same angle, but they are different measurements. Round each answer only at the end, to the number of decimal places the question asks for, and attach the unit. Lengths are in cm. Areas are in cm^2.
Method: 1. Add the arc to both radii: arc +2r. 2. Radius 8cm and angle 120^° give arc (16π)/3≈ 16.8cm. 3. The perimeter is about 16.8+16=32.8cm, and the area is about 67.0cm^2.
When a question says "give your answer in terms of π", the answer keeps the symbol π and is not a decimal. This is common on non-calculator papers. Simplify the fraction of the circle first, then multiply the numbers and write π at the end, for example 60/360 × 2 × π × 12 = 4π. Leaving pi in is also how a non-calculator paper can ask for a sector area at all.
Cancelling is the key skill. A fraction such as 80/360 becomes 2/9, and a radius of 9 cancels the 9. Work in whole numbers wherever you can. If you meet a decimal on a calculator paper, round only the final answer, and do not swap π for 3.14 unless the question tells you to.
Method: 1. Write the fraction (θ)/360 and cancel it to its simplest form. 2. Write the formula with the numbers in, leaving π as a symbol. 3. Multiply and cancel the numbers. 4. Write the answer as a number times π with the correct unit.
Tip: A fraction of the circle such as 80/360 is much easier to handle once it is 2/9.
Exam questions often give the arc length or the area and ask for the centre angle or the radius. Put the numbers you know into the formula, leave the unknown as a letter, and solve the equation. For an angle x, the equation x/360 × 2 × π × 18 = 4π gives x = 40. The same method finds a radius if the angle and arc length are given.
To find a radius from an area, the radius is squared, so you take a square root last. For a sector with a known angle, divide the area by the fraction of the circle, divide by π, then square root. Check by substituting the answer back into the original formula to confirm you get the arc length or area given in the question.
Method: 1. Write the arc or area formula with the known numbers and the unknown as a letter. 2. Divide both sides by every known factor to isolate the unknown. 3. Take a square root last if the unknown radius was squared. 4. Substitute back to check the arc or area matches the question.
Many questions use a familiar fraction of a circle: a semicircle (1/2), a quadrant (1/4) or a sector of 120^° (1/3). A perimeter is the whole boundary, so you add the straight edges to the curved arc. A semicircle with diameter d has perimeter (π d)/2 + d, and a quadrant of radius r has perimeter (π r)/2 + 2r.
Running tracks and flower beds are built from straight lengths and semicircular ends. Two semicircles of the same diameter make one full circle, so a lap is π d plus the two straight sections. Draw the shape, label each curved part and each straight part, and add them at the end. Check that you have counted every part once.
Method: 1. Identify the fraction of a circle that each curved part is. 2. Work out each arc length using the radius, not the diameter. 3. Add every straight edge that forms part of the boundary. 4. Add the arcs and straight edges, then round and write the unit.
Tip: If the diameter is given, halve it for the radius before using (θ)/360 × π r^2.
A segment is the region between a chord and an arc. Its area is the area of the sector minus the area of the triangle formed by the two radii and the chord. For a quadrant (a 90^° sector), the triangle is right-angled with area 1/2r^2, so you can work without a sine.
For other angles, use the area of a triangle formula 1/2absin C with the two radii as a and b. This is a calculator question. Keep the unrounded sector and triangle areas on the calculator and subtract them at the end, so that the final answer is correct to the accuracy asked for.
Method: 1. Work out the area of the sector. 2. Work out the area of the triangle: 1/2r^2 for 90^°, or 1/2r^2sinθ in general. 3. Subtract the triangle from the sector. 4. Round at the end and write the unit as cm^2.
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