GCSE Maths: Area of a circle, semicircles and composite shapes (Higher)
Part of the GCSE Maths perimeter, area and circles chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Area of a circle, semicircles and composite shapes
To see why the area of a circle is πr2, cut the circle into thin sectors, like slices of pizza, and lay them alternately point up and point down. The pieces fit together to make a shape that is almost a rectangle. The more slices you cut, the closer it is to a rectangle.
The long side of this rectangle is half of the circumference, ½ × 2πr = πr. The short side is the radius r. So the area is πr × r:
Area of a circle A = πr2
Always use the radius. If you are given the diameter, halve it first. Then square the radius first (BIDMAS: powers before multiplying) and multiply by π. As before, you may be asked for an answer with a calculator (rounded) or in terms of π, such as 36π cm².
Semicircles and quarter circles
A semicircle is half a circle, so its area is ½πr2. A quarter circle has area ¼πr2.
Composite shapes
As in Section 18.4, split the shape into pieces whose area you know (rectangles, triangles, semicircles...) and add, or subtract a piece that has been cut out (a hole, a corner, a middle circle). Keep full calculator values until the very end and round once. If the question says “in terms of π”, keep π as a symbol and add numbers to numbers and π-terms to π-terms.
Working backwards from the area
If you know the area: divide by π to get r2, then take the square root to get r.
Key ideas
- Area of a circle A = πr2. Use the radius; halve a diameter first.
- Square the radius first, then multiply by π.
- Semicircle: ½πr2. Quarter circle: ¼πr2.
- Composite shapes: split and add, or subtract a piece that is removed.
- Do not round until the final answer. “In terms of π” means keep the π symbol.
- From the area: divide by π, then square root, to find the radius.
Watch out
The most common mistakes are using the diameter instead of the radius, and multiplying π by the radius before squaring. For a circle of diameter 10 cm the area is π × 52 = 25π cm² = 78.5 cm², not π × 102.Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Write down the radius. The radius is the distance from the centre to the edge.radius = 6 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 62
62 = 6 × 6 = 36
A = π × 36 - Write the answer in terms of π. Leave the π as a symbol and put the number in front.A = 36π cm²
Check. With π ≈ 3.14: 36 × 3.14 = 113.04 cm², a sensible size. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Find the radius. The formula for area needs the radius. The radius is half the diameter.radius = 10 ÷ 2 = 5 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 52
52 = 5 × 5 = 25
A = π × 25 - Use the π button on your calculator. Multiply the squared radius by π.A = 78.5398… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.A = 78.5 cm²
Check. Estimate: π is about 3, so A is about 3 × 25 = 75 cm², close to 78.5. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Method. A quarter circle is one quarter of a full circle, so its area is ¼ × πr2.
- Write down the radius. Both straight edges of a quarter circle are radii, so the radius is the length shown.radius = 8 cm
- Area of the quarter circle. A quarter of πr2. Square the radius first.¼ × π × 82 = ¼ × π × 64 = 16π cm²
- Write the answer with its units. Leave π as a symbol; the units are square units for an area and plain units for a length.Area = 16π cm²
Check. Estimate with π ≈ 3: the answer should be about 48 cm², which is close to the value found. ✓
Watch out. Quarter means divide by 4 (not by 2).
Method. Split the shape into a rectangle and a semicircle, find each area and add.
- Find the radius of the semicircle. The semicircle sits across the top, so its diameter is the width of the rectangle. The radius is half of that.diameter = 6 m, radius = 6 ÷ 2 = 3 m
- Area of the rectangle. length × width.6 × 8 = 48 m²
- Area of the semicircle. Half of πr2, using the radius you found.½ × π × 32 = ½ × π × 9 = 14.13717… m²
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Area = 48 + 14.13717… = 62.13717… m²
- Round to 2 decimal places. Look at the next digit: 5 or more rounds up.Area = 62.14 m²
Check. Estimate with π ≈ 3: the answer should be about 61.5 m², which is close to the value found. ✓
Watch out. The semicircle’s diameter is the width of the rectangle: halve it to get the radius.
Method. The shaded ring is the big circle with the small circle taken away: subtract the two areas.
- Area of the big circle. Use πR2 with the outer radius.π × 102 = π × 100 = 100π cm²
- Area of the small circle. Use the inner radius.π × 62 = π × 36 = 36π cm²
- Combine the pieces. Subtract the parts that are taken away. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Area = 100π − 36π = 64π cm²
Check. Estimate with π ≈ 3: the answer should be about 192 cm², which is close to the value found. ✓
Watch out. Square each radius separately and then subtract. (R − r)2 is not the same as R2 − r2.
Method. Use A = πr2 and rearrange: divide the area by π, then take the square root.
- Write the formula. Area = π × radius squared.A = πr2
- Substitute the area. Put the number in for A.100 = π × r2
- Divide both sides by π. To undo “multiply by π” divide by π.r2 = 100 ÷ π = 31.8310…
- Take the square root. The opposite of squaring is the square root (use the calculator).r = √31.8310… = 5.6419…
- Double the radius. The diameter is twice the radius.diameter = 2 × 5.6419… = 11.2838…
- Round to 1 decimal place. Round only at the end.diameter = 11.3 cm
Check. Check: π × 5.62 = 98.5, close to 100. ✓
Watch out. Divide by π first, then take the square root. Taking the square root first will give the wrong answer.
Method. The shaded corners are the square with the circle taken away. The circle touches all four sides, so its diameter equals the side of the square.
- Find the radius of the circle. The circle touches opposite sides, so its diameter is the side of the square.radius = 10 ÷ 2 = 5 cm
- Area of the square. side × side.10 × 10 = 100 cm²
- Area of the circle. The diameter is 10 cm, so the radius is half of it.π × 52 = π × 25 = 25π cm²
- Combine the pieces. Subtract the parts that are taken away. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Area = 100 − 25π = (100 − 25π) cm²
Check. Estimate with π ≈ 3: the answer should be about 25 cm², which is close to the value found. ✓
Watch out. Radius is half the side of the square, not the whole side.
Exercise 18G Area of a circle, semicircles and composite shapes
Calculator allowedKey words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Write down the radius. The radius is the distance from the centre to the edge.radius = 5 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 52
52 = 5 × 5 = 25
A = π × 25 - Use the π button on your calculator. Multiply the squared radius by π.A = 78.5398… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.A = 78.5 cm²
Check. Estimate: π is about 3, so A is about 3 × 25 = 75 cm², close to 78.5. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Find the radius. The formula for area needs the radius. The radius is half the diameter.radius = 12 ÷ 2 = 6 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 62
62 = 6 × 6 = 36
A = π × 36 - Use the π button on your calculator. Multiply the squared radius by π.A = 113.0973… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.A = 113.1 cm²
Check. Estimate: π is about 3, so A is about 3 × 36 = 108 cm², close to 113.1. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Write down the radius. The radius is the distance from the centre to the edge.radius = 9 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 92
92 = 9 × 9 = 81
A = π × 81 - Write the answer in terms of π. Leave the π as a symbol and put the number in front.A = 81π cm²
Check. With π ≈ 3.14: 81 × 3.14 = 254.34 cm², a sensible size. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Write down the radius. The radius is the distance from the centre to the edge.radius = 9 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 92
92 = 9 × 9 = 81
A = π × 81 - Use the π button on your calculator. Multiply the squared radius by π.A = 254.4690… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.A = 254.5 cm²
Check. Estimate: π is about 3, so A is about 3 × 81 = 243 cm², close to 254.5. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Key words. The area of a circle is the space inside it, in square units. The radius is the distance from the centre to the edge.
Method. The area of a circle is πr2. Make sure you use the radius, square it first, then multiply by π.
- Write down the radius. The radius is the distance from the centre to the edge.radius = 3.7 cm
- Choose the formula. The area of a circle is π times the radius squared. “Squared” means radius × radius.A = πr2
- Substitute and square the radius first. BIDMAS: powers come before multiplying. Square the radius, then multiply by π.A = π × 3.72
3.72 = 3.7 × 3.7 = 13.69
A = π × 13.69 - Use the π button on your calculator. Multiply the squared radius by π.A = 43.0084… cm²
- Round to 3 significant figures. Look at the next digit: 5 or more rounds up.A = 43.0 cm²
Check. Estimate: π is about 3, so A is about 3 × 13.69 = 41.07 cm², close to 43.0. ✓
Watch out. Square the radius first: πr2 is π × r × r. It is not (π × r) squared, and it is not 2 × r.
Method. The area of a semicircle is half the area of the full circle: ½ × πr2.
- Find the radius. The area formula needs the radius, and the radius is half the diameter.10 ÷ 2 = 5 cm
- Area of the semicircle. Half of πr2. Square the radius first.½ × π × 52 = ½ × π × 25 = 39.2699… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Area = 39.3 cm²
Check. Estimate with π ≈ 3: the answer should be about 37.5 cm², which is close to the value found. ✓
Watch out. Do not forget to halve the full-circle area, and do not use the diameter in the formula.
Method. The perimeter of a semicircle is the curved edge (half a circumference) plus the straight edge (the diameter). Do not forget the straight edge.
- Curved edge. Half of the circumference πd.½ × π × 10 = 15.7080… cm
- Straight edge. The diameter closes the shape.diameter = 10 cm
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Perimeter = 15.7080… + 10 = 25.7080… cm
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Perimeter = 25.7 cm
Check. Estimate with π ≈ 3: the answer should be about 25 cm, which is close to the value found. ✓
Watch out. The straight edge is part of the perimeter. Half a circle’s curve alone is not enough.
Method. A quarter circle is one quarter of a full circle, so its area is ¼ × πr2.
- Write down the radius. Both straight edges of a quarter circle are radii, so the radius is the length shown.radius = 8 cm
- Area of the quarter circle. A quarter of πr2. Square the radius first.¼ × π × 82 = ¼ × π × 64 = 16π cm²
- Write the answer with its units. Leave π as a symbol; the units are square units for an area and plain units for a length.Area = 16π cm²
Check. Estimate with π ≈ 3: the answer should be about 48 cm², which is close to the value found. ✓
Watch out. Quarter means divide by 4 (not by 2).
Method. The perimeter of a quarter circle is the curved edge (a quarter of the circumference) plus the two straight edges (two radii).
- Curved edge. A quarter of the circumference 2πr.¼ × 2 × π × 8 = ¼ × 16π = 4π cm
- Two straight edges. Both are radii.2 × 8 = 16 cm
- Combine the pieces. Add. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Perimeter = 4π + 16 = (16 + 4π) cm
Check. Estimate with π ≈ 3: the answer should be about 28 cm, which is close to the value found. ✓
Watch out. Add the two radii. The perimeter has three edges: one curved and two straight.
Method. Split the shape into a rectangle and a semicircle, find each area and add.
- Find the radius of the semicircle. The semicircle sits across the top, so its diameter is the width of the rectangle. The radius is half of that.diameter = 6 m, radius = 6 ÷ 2 = 3 m
- Area of the rectangle. length × width.6 × 8 = 48 m²
- Area of the semicircle. Half of πr2, using the radius you found.½ × π × 32 = ½ × π × 9 = 14.1372… m²
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Area = 48 + 14.1372… = 62.1372… m²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Area = 62.1 m²
Check. Estimate with π ≈ 3: the answer should be about 61.5 m², which is close to the value found. ✓
Watch out. The semicircle’s diameter is the width of the rectangle: halve it to get the radius.
Method. The perimeter is the distance round the outside: the bottom, the two sides and the curved top. The dotted line across the middle is inside the shape, so leave it out.
- Bottom edge. The width of the rectangle.bottom = 6 m
- Two side edges. Two sides, each of height 8 m.2 × 8 = 16 m
- Curved top. Half of the circumference of a circle with diameter 6 m.½ × π × 6 = 9.4248… m
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Perimeter = 6 + 16 + 9.4248… = 31.4248… m
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Perimeter = 31.4 m
Check. Estimate with π ≈ 3: the answer should be about 31 m, which is close to the value found. ✓
Watch out. The dotted line is not on the outside edge, so it is not part of the perimeter.
Method. The shaded ring is the big circle with the small circle taken away: subtract the two areas.
- Area of the big circle. Use πR2 with the outer radius.π × 102 = π × 100 = 100π cm²
- Area of the small circle. Use the inner radius.π × 62 = π × 36 = 36π cm²
- Combine the pieces. Subtract the parts that are taken away. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Area = 100π − 36π = 64π cm²
Check. Estimate with π ≈ 3: the answer should be about 192 cm², which is close to the value found. ✓
Watch out. Square each radius separately and then subtract. (R − r)2 is not the same as R2 − r2.
Method. The two curved ends together make one full circle, so add the circumference of that circle to the two straight sides.
- The two straight sides. Each has length 10 m.2 × 10 = 20 m
- The two curved ends. Together they are one circle of diameter 6 m, so use πd.π × 6 = 6π m
- Combine the pieces. Add. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Perimeter = 20 + 6π = (20 + 6π) m
Check. Estimate with π ≈ 3: the answer should be about 38 m, which is close to the value found. ✓
Watch out. The dotted lines are inside the shape. Only the outside edge counts.
Method. The two semicircular ends fit together to make one whole circle. The area is a rectangle plus a circle.
- Find the radius of the ends. The ends are semicircles whose diameter is the width of the rectangle.radius = 6 ÷ 2 = 3 m
- Area of the rectangle. length × width, where the width is the diameter of the ends.10 × 6 = 60 m²
- Area of the circle (two semicircles). Use πr2 with the radius.π × 32 = π × 9 = 9π m²
- Combine the pieces. Add. Whole numbers go with whole numbers and the π terms go with π terms. A number and a π term cannot be merged into one, so leave them as a sum.Area = 60 + 9π = (60 + 9π) m²
Check. Estimate with π ≈ 3: the answer should be about 87 m², which is close to the value found. ✓
Watch out. Two semicircles make one whole circle, not two circles.
Method. The shaded area is the whole rectangle with the circular hole taken away.
- Area of the rectangle. length × width.12 × 8 = 96 cm²
- Area of the circular hole. Use πr2.π × 22 = π × 4 = 12.5664… cm²
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Area = 96 − 12.5664… = 83.4336… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Area = 83.4 cm²
Check. Estimate with π ≈ 3: the answer should be about 84 cm², which is close to the value found. ✓
Watch out. Subtract the hole. Adding it would give an area bigger than the rectangle.
Method. The shaded corners are the square with the circle taken away. The circle touches all four sides, so its diameter equals the side of the square.
- Find the radius of the circle. The circle touches opposite sides, so its diameter is the side of the square.radius = 10 ÷ 2 = 5 cm
- Area of the square. side × side.10 × 10 = 100 cm²
- Area of the circle. The diameter is 10 cm, so the radius is half of it.π × 52 = π × 25 = 78.5398… cm²
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Area = 100 − 78.5398… = 21.4602… cm²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Area = 21.5 cm²
Check. Estimate with π ≈ 3: the answer should be about 25 cm², which is close to the value found. ✓
Watch out. Radius is half the side of the square, not the whole side.
Method. The shaded area is the rectangle with the quarter circle cut out.
- Area of the rectangle. length × width.12 × 8 = 96 cm²
- Area of the quarter circle. A quarter of πr2.¼ × π × 52 = ¼ × π × 25 = 19.6350… cm²
- Combine the pieces. Add (or subtract) the full calculator values. Do not round the pieces first, or the answer can be a little out.Area = 96 − 19.6350… = 76.3650… cm²
- Round to 3 significant figures. Look at the next digit: 5 or more rounds up.Area = 76.4 cm²
Check. Estimate with π ≈ 3: the answer should be about 77.25 cm², which is close to the value found. ✓
Watch out. A quarter circle is ¼ of πr2. Remember to subtract it from the rectangle.
Method. Use A = πr2, divide by π, then take the square root.
- Write the formula. Area = π × radius squared.A = πr2
- Substitute the area. The area is given in terms of π, so both sides contain π.64π = π × r2
- Divide both sides by π. The π’s cancel.r2 = 64
- Square root. The opposite of squaring is the square root. The radius is a length so it is positive.r = √64 = 8 cm
Check. π × 82 = 64π. ✓
Watch out. Take the square root of r2. Do not halve it.
Method. Use A = πr2 and rearrange: divide the area by π, then take the square root.
- Write the formula. Area = π × radius squared.A = πr2
- Substitute the area. Put the number in for A.100 = π × r2
- Divide both sides by π. To undo “multiply by π” divide by π.r2 = 100 ÷ π = 31.8310…
- Take the square root. The opposite of squaring is the square root (use the calculator).r = √31.8310… = 5.6419…
- Double the radius. The diameter is twice the radius.diameter = 2 × 5.6419… = 11.2838…
- Round to 1 decimal place. Round only at the end.diameter = 11.3 cm
Check. Check: π × 5.62 = 98.5, close to 100. ✓
Watch out. Divide by π first, then take the square root. Taking the square root first will give the wrong answer.
Method. Find the area of each pizza, then work out the price per square centimetre (in pence) for each. The lower price per cm² is the better value.
- Radius of each pizza. Halve the diameters.30 ÷ 2 = 15 cm and 40 ÷ 2 = 20 cm
- Area of the medium pizza. A = πr2.π × 152 = π × 225 = 706.9 cm²
- Area of the large pizza. A = πr2.π × 202 = π × 400 = 1256.6 cm²
- Price per cm² (in pence). £8 = 800p and £12 = 1200p. Divide the price by the area.Medium: 800 ÷ 706.9 = 1.13p
Large: 1200 ÷ 1256.6 = 0.95p - Compare. Less pence per square centimetre means better value.0.95p is less than 1.13p, so the large pizza is better value.
Check. The large pizza has area about 1.78 times the medium, but costs only 1.5 times as much. ✓
Watch out. Compare area, not diameter: doubling the diameter quadruples the area.
Method. Find the area of the lawn, round up to whole square metres, then multiply by the price.
- Area of the lawn. A = πr2.π × 42 = π × 16 = 50.27 m²
- Round up. Turf is sold in whole square metres, and 50 m² would not be enough.51 m²
- Cost. Each square metre costs £3.50.51 × 3.50 = £178.50
Check. Estimate: 3 × 16 = 48 m², so about 50 m² × £3.50 ≈ £175. ✓
Watch out. You need to cover the whole lawn, so round up rather than to the nearest.
Method. Check which length Kai put in the formula.
- Spot the mistake. The formula A = πr2 needs the radius. Kai used the diameter, 10 cm.The radius is 10 ÷ 2 = 5 cm.
- Correct the working. Square the radius, then multiply by π.A = π × 52 = π × 25 = 78.5 cm²
Check. Kai’s answer is exactly 4 times too big, because he doubled the radius. ✓
Watch out. Halve the diameter before you square.
Method. Work out both areas in terms of π and compare.
- Area with radius 5 cm. A = πr2.π × 52 = 25π cm²
- Area with radius 10 cm. A = πr2.π × 102 = 100π cm²
- Compare. Divide the larger by the smaller.100π ÷ 25π = 4
- Conclude. So the larger area is 4 times the smaller one.
Check. Doubling a length multiplies the area by 2 × 2 = 4. ✓
Watch out. Doubling the radius makes the area 4 times bigger, not 2 times.
Method. Write the area and the circumference using r, make them equal, and solve.
- Write both formulas. Area = πr2. Circumference = 2πr.πr2 = 2πr
- Divide both sides by π. The π’s cancel.r2 = 2r
- Divide both sides by r. The radius is not 0, so we can divide.r = 2
Check. Radius 2 cm: area = π × 4 = 4π; circumference = 2 × π × 2 = 4π. They match. ✓
Watch out. Divide both sides by r to leave r = 2, rather than leaving r2 = 2r.
Method. Use A = πr2 with the algebraic radius, divide by π, and take the square root.
- Write the equation. Area = πr2.π(x + 2)2 = 49π
- Divide by π. The π’s cancel.(x + 2)2 = 49
- Square root. The radius is positive.x + 2 = 7
- Solve. Subtract 2.x = 5
Check. Radius 7 cm: π × 49 = 49π. ✓
Watch out. Take the square root of both sides before you subtract 2.
Method. Big circle area minus small circle area.
- Outer circle. πr2 with r = 3x.π(3x)2 = 9πx2
- Inner circle. r = x.πx2
- Subtract. Collect like terms.9πx2 − πx2 = 8πx2
Check. With x = 1: π(9 − 1) = 8π. ✓
Watch out. (3x)2 = 9x2, not 3x2.
Method. Set 8πx2 = 72π and solve.
- Make the equation. The area is 72π.8πx2 = 72π
- Divide by 8π. The π’s cancel and 72 ÷ 8 = 9.x2 = 9
- Square root. Length is positive.x = 3
Check. 8π × 9 = 72π. ✓
Watch out. Divide by 8 first, then take the square root.