GCSE Maths Perimeter, Area and Circles (Foundation): check-up and exam-style questions
Part of the GCSE Maths perimeter, area and circles chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Exercise 18Z Check-up (non-calculator)
Non-calculatorKey words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Area counts the unit squares inside the shape. A rectangle has length squares in each row and width rows, so we multiply the two.
- Write the formula. There are length squares in every row and width rows, so the number of squares is length × width.Area = length × width
- Substitute the numbers. Both lengths must be in the same unit before you multiply.Area = 9 × 5
- Multiply. Work out the product.9 × 5 = 45
- Write the units. Each little square is 1 cm by 1 cm, so the area is measured in square cm.Area = 45 cm²
Check. Add the rows instead: 9 + 9 + 9 + 9 + 9 = 45. ✓
Watch out. Area is in square units (cm²), not cm. Do not add the sides: adding is for perimeter.
Key words. Perimeter is the distance all the way round the outside of a shape (a length, measured in cm, m ...). Area is the amount of flat space inside a shape (measured in square units, cm², m² ...).
Method. Perimeter is the distance all the way round. A rectangle has two lengths and two widths, so add all four sides.
- Mark the four sides. Opposite sides of a rectangle are equal, so the sides are length, width, length, width.9, 5, 9, 5
- Add the four sides. The perimeter is the total length of the boundary.9 + 5 + 9 + 5 = 28
- Write the units. Perimeter is a length, so we use cm, not square units.Perimeter = 28 cm
Check. Shortcut: 2 × (length + width) = 2 × 14 = 28. ✓
Watch out. Perimeter is a length (cm). Do not give it in square units.
Key words. The base is any side you choose. The perpendicular height is the distance from the base to the opposite corner, measured at 90° to the base.
Method. The area of a triangle is half of base × perpendicular height. The height must be at right angles to the base.
- Pick the base and the perpendicular height. The height is the straight-up distance from the base to the top corner (the dotted line at a right angle). A slanting side is not the height.base = 10 cm, height = 7 cm
- Write the formula. A triangle is half of the rectangle with the same base and height.Area = ½ × base × height
- Multiply base by height. This is the area of the rectangle round the triangle.10 × 7 = 70
- Halve it. The triangle is half of that rectangle.70 ÷ 2 = 35
- Write the units. Area is in square cm.Area = 35 cm²
Check. Use half the base instead: 5 × 7 = 35. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
Method. A parallelogram can be cut and re-arranged into a rectangle with the same base and perpendicular height, so area = base × height.
- Pick the base and the perpendicular height. The height is the distance between the two parallel sides, measured at a right angle. The slanting side is not the height. Ignore the slant length.base = 8 cm, height = 4 cm
- Write the formula. Cut off the triangle at one end and slide it to the other end: you get a rectangle.Area = base × perpendicular height
- Multiply. Work out the product.8 × 4 = 32
- Write the units. Area is in square cm.Area = 32 cm²
Check. Swap the order: 4 × 8 = 32. ✓
Watch out. Use the perpendicular height (dotted line), not the slanting side. There is no half in the parallelogram formula.
Key words. A trapezium is a four-sided shape with exactly one pair of parallel sides (sides that stay the same distance apart).
Method. A trapezium has one pair of parallel sides. Its area is the average of the two parallel sides times the height: ½ × (a + b) × h.
- Pick out the two parallel sides and the height. The parallel sides are the two sides that never meet (a and b). The height is the perpendicular distance between them (the dotted line). The slanting sides are not used.a = 6 cm, b = 10 cm, h = 4 cm
- Write the formula. Two copies of the trapezium fit together into a parallelogram of base (a + b) and height h; the trapezium is half of it.Area = ½ × (a + b) × h
- Add the parallel sides first. Do the bracket first (BIDMAS).a + b = 6 + 10 = 16
- Multiply by the height. This gives the area of the double (parallelogram).16 × 4 = 64
- Halve it. The trapezium is half of the parallelogram.64 ÷ 2 = 32
- Write the units. Area is in square cm.Area = 32 cm²
Check. The average of the parallel sides is 16 ÷ 2 = 8; 8 × 4 = 32. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. Split the shape into two rectangles, find each area, then add.
- Draw a line to split the shape. A vertical cut makes a tall rectangle on the left and a shorter rectangle on the right. Mark the sides of each.Left rectangle: width 8 − 3 = 5 cm, height 6 cm
Right rectangle: width 3 cm, height 6 − 2 = 4 cm - Area of the left rectangle. Area = length × width.5 × 6 = 30 cm²
- Area of the right rectangle. Do the same for the second piece.3 × 4 = 12 cm²
- Add the two areas. The whole shape is made of the two pieces.30 + 12 = 42 cm²
Check. Subtract method: whole rectangle 8 × 6 = 48, take away the notch 3 × 2 = 6: 48 − 6 = 42. ✓
Watch out. Work out the length of each missing side before multiplying. Do not use the labelled lengths of the whole shape for the small pieces.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: m² to cm² gives a bigger number. Each m² is 10,000 cm², so multiply.2.5 × 10,000 = 25,000 cm²
Check. Go back: 25,000 ÷ 10,000 = 2.5 m². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Key words. The circumference is the distance round a circle. The diameter is the distance across through the centre; the radius is half of that. π (pi) is the number 3.14159… you get by dividing any circumference by its diameter.
Method. The circumference of a circle is π × diameter, or 2 × π × radius. Find the diameter or radius, substitute, and calculate.
- Write down what is given. The radius is the distance from the centre to the edge. The diameter goes right across the circle through the centre, so it is twice the radius.radius = 4 cm
- Choose the formula. The circumference is the distance round the circle. It is always about 3.14 times the diameter, and that number is π. Since d = 2r, this can also be written 2πr.C = 2πr
- Substitute the number. Replace the letter with the value you know.C = 2 × π × 4
- Multiply the numbers and keep the π. The question wants an exact answer in terms of π, so do not use a calculator for π. Put the number in front of π.2 × 4 = 8
C = 8π
Check. With π ≈ 3.14: 8 × 3.14 = 25.12, a sensible circumference for a circle of diameter 8 cm. ✓
Watch out. If you are given the diameter use πd. If you are given the radius, double it (2πr). Do not mix them up.
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 = 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 - Write the answer in terms of π. Leave the π as a symbol and put the number in front.A = 25π cm²
Check. With π ≈ 3.14: 25 × 3.14 = 78.50 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.
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 × 12 = 6π cm
- Straight edge. The diameter closes the shape.diameter = 12 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 = 6π + 12 = (12 + 6π) cm
Check. Estimate with π ≈ 3: the answer should be about 30 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. The area of a sector is (θ ÷ 360) × πr2: the fraction of the circle it covers times the whole area.
- Write the angle as a fraction of a full turn. A sector with angle θ is the same fraction of the whole circle as θ is of 360.60360 = 16
- Find the area of the whole circle. Use A = πr2. Square the radius first.A = π × 62 = 36π
- Multiply the fraction by the area. The sector is that fraction of the circle.sector area = 16 × 36π
- Simplify and keep π. Multiply the number parts together.sector area = 6π cm²
Check. The whole circle has area 36π. The sector is 16 of it, and 16 × 36 = 6. ✓
Watch out. Use πr2 (area) for a sector’s area and 2πr (circumference) for an arc length. Do not mix them.
Total 40 marks. Allow about 40 minutes.
Exercise 18X Exam-style questions
Non-calculatorKey words. The base is any side you choose. The perpendicular height is the distance from the base to the opposite corner, measured at 90° to the base.
Method. The area of a triangle is half of base × perpendicular height. The height must be at right angles to the base.
- Pick the base and the perpendicular height. The height is the straight-up distance from the base to the top corner (the dotted line at a right angle). A slanting side is not the height. The slanting sides are not used for the area.base = 12 cm, height = 8 cm
- Write the formula. A triangle is half of the rectangle with the same base and height.Area = ½ × base × height
- Multiply base by height. This is the area of the rectangle round the triangle.12 × 8 = 96
- Halve it. The triangle is half of that rectangle.96 ÷ 2 = 48
- Write the units. Area is in square cm.Area = 48 cm²
Check. Use half the base instead: 6 × 8 = 48. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
Key words. A trapezium is a four-sided shape with exactly one pair of parallel sides (sides that stay the same distance apart).
Method. A trapezium has one pair of parallel sides. Its area is the average of the two parallel sides times the height: ½ × (a + b) × h.
- Pick out the two parallel sides and the height. The parallel sides are the two sides that never meet (a and b). The height is the perpendicular distance between them (the dotted line). The slanting sides are not used.a = 5 cm, b = 13 cm, h = 6 cm
- Write the formula. Two copies of the trapezium fit together into a parallelogram of base (a + b) and height h; the trapezium is half of it.Area = ½ × (a + b) × h
- Add the parallel sides first. Do the bracket first (BIDMAS).a + b = 5 + 13 = 18
- Multiply by the height. This gives the area of the double (parallelogram).18 × 6 = 108
- Halve it. The trapezium is half of the parallelogram.108 ÷ 2 = 54
- Write the units. Area is in square cm.Area = 54 cm²
Check. The average of the parallel sides is 18 ÷ 2 = 9; 9 × 6 = 54. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. To find the perimeter, first work out every missing side, then add all six sides.
- Find the missing horizontal side. The bottom side W is as long as the top side plus the horizontal part of the notch. So the notch part is the bottom minus the top.9 − 5 = 4 cm
- Find the missing vertical side. The left side H is as long as the lower right side plus the vertical part of the notch. So the notch part is the left minus the lower right.7 − 4 = 3 cm
- Add all six sides. Go round the shape once, ticking each side.9 + 4 + 4 + 3 + 5 + 7 = 32 cm
Check. The steps of the notch add up to the same as the sides they replace, so the perimeter equals the perimeter of the big rectangle: 2 × (9 + 7) = 32. ✓
Watch out. Do not forget the missing sides: an L-shape has six sides, not four.
Method. The border is the big outer rectangle with the inner rectangle taken away. Find both areas and subtract.
- Area of the outer rectangle. length × width.24 × 18 = 432 cm²
- Find the sides of the inner rectangle. The border is 2 cm wide on both sides, so each inside length is 2 × 2 = 4 cm shorter.24 − 4 = 20 cm
18 − 4 = 14 cm - Area of the inner rectangle. length × width.20 × 14 = 280 cm²
- Subtract. The border is what is left.432 − 280 = 152 cm²
Check. Four strips: two of 24 × 2 = 48 and two of 14 × 2 = 28. 96 + 56 = 152. ✓
Watch out. Take 2 × 2 off each length, not just 2: the border is on both sides.
Method. Find the perimeter of the field, then multiply by the cost per metre.
- Perimeter. The fence goes all the way round.80 + 45 + 80 + 45 = 250 m
- Cost. Each metre costs £5.50.250 × 5.50 = £1,375
(250 × 5 = 1250 and 250 × 0.5 = 125; 1250 + 125 = 1375)
Check. Estimate: 250 × 6 = 1500, close to 1375. ✓
Watch out. Marks: 1 for the perimeter, 1 for multiplying by 5.50, 1 for the correct answer with the pound sign.
Method. Find the area, double it for two coats, work out the litres and tins, then the cost.
- Area of the wall. width × height.7 × 3 = 21 m²
- Two coats. The paint must cover the wall twice.21 × 2 = 42 m²
- Litres needed. One litre covers 6 m².42 ÷ 6 = 7 litres
- Tins needed. Each tin is 2.5 litres, so round up.7 ÷ 2.5 = 2.8, so 3 tins
- Cost. Each tin costs £16.80.3 × 16.80 = £50.40
Check. 3 tins hold 7.5 litres, which is at least 7 litres. ✓
Watch out. Marks: 1 area, 1 two coats and litres, 1 tins rounded up, 1 cost.
Method. To change units of area, work out how many small squares make one big square: it is the length conversion squared.
- Compare the lengths. Area units come from squares, so first see how the lengths compare.1 m = 100 cm
- Square it to compare the areas. 1 m² is a square 1 m by 1 m. Measured in cm it is 100 cm by 100 cm, which is 100 × 100 small squares.1 m² = 100 × 100 = 10,000 cm²
- Convert: m² to cm² gives a bigger number. Each m² is 10,000 cm², so multiply.0.35 × 10,000 = 3500 cm²
Check. Go back: 3500 ÷ 10,000 = 0.35 m². ✓
Watch out. 1 m = 100 cm, but 1 m² = 10,000 cm², not 100. Areas need the conversion squared.
Key words. The circumference is the distance round a circle. The diameter is the distance across through the centre; the radius is half of that. π (pi) is the number 3.14159… you get by dividing any circumference by its diameter.
Method. The circumference of a circle is π × diameter, or 2 × π × radius. Find the diameter or radius, substitute, and calculate.
- Write down what is given. The diameter goes right across the circle through the centre. (The radius is half of it.)diameter = 14 cm
- Choose the formula. The circumference is the distance round the circle. It is always about 3.14 times the diameter, and that number is π. Using the radius it is 2πr.C = πd
- Substitute the number. Replace the letter with the value you know.C = π × 14
- Multiply the numbers and keep the π. The question wants an exact answer in terms of π, so do not use a calculator for π. Put the number in front of π.C = 14π
Check. With π ≈ 3.14: 14 × 3.14 = 43.96, a sensible circumference for a circle of diameter 14 cm. ✓
Watch out. If you are given the diameter use πd. If you are given the radius, double it (2πr). Do not mix them up.
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. 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 = 5 m, radius = 5 ÷ 2 = 2.5 m
- Area of the rectangle. length × width.5 × 4 = 20 m²
- Area of the semicircle. Half of πr2, using the radius you found.½ × π × 2.52 = ½ × π × 6.25 = 9.8175… 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 = 20 + 9.8175… = 29.8175… m²
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Area = 29.8 m²
Check. Estimate with π ≈ 3: the answer should be about 29.375 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 = 5 m
- Two side edges. Two sides, each of height 4 m.2 × 4 = 8 m
- Curved top. Half of the circumference of a circle with diameter 5 m.½ × π × 5 = 7.8540… 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 = 5 + 8 + 7.8540… = 20.8540… m
- Round to 1 decimal place. Look at the next digit: 5 or more rounds up.Perimeter = 20.9 m
Check. Estimate with π ≈ 3: the answer should be about 20.5 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. Use C = πd and rearrange: divide the circumference by π to find the diameter (then halve it for the radius).
- Write the formula. Circumference = π × diameter.C = πd
- Substitute the circumference. Put the number in for C.40 = π × d
- Divide both sides by π. To undo “multiply by π” we divide by π (use the calculator button).d = 40 ÷ π = 12.7324…
- Halve it to get the radius. The radius is half the diameter.r = 12.7324… ÷ 2 = 6.3662…
- Round to 1 decimal place. Round only at the end.r = 6.4 cm
Check. Check: 2 × π × 6.4 = 40.21, close to 40. ✓
Watch out. Divide by π, do not multiply by it. Check that your answer is about a third of the circumference.
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 = 8 ÷ 2 = 4 cm
- Area of the square. side × side.8 × 8 = 64 cm²
- Area of the circle. The diameter is 8 cm, so the radius is half of it.π × 42 = π × 16 = 16π 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 = 64 − 16π = (64 − 16π) cm²
Check. Estimate with π ≈ 3: the answer should be about 16 cm², which is close to the value found. ✓
Watch out. Radius is half the side of the square, not the whole side.
Key words. An arc is part of the circumference. A sector is a slice of a circle between two radii and an arc. The angle θ is measured at the centre.
Method. An arc is part of the circumference. Its length is (θ ÷ 360) × 2πr, the fraction of the circle it covers times the whole circumference.
- Write the angle as a fraction of a full turn. A whole circle is 360°. An arc with angle θ is the same fraction of the whole circumference as θ is of 360.80360 = 29
- Find the circumference of the whole circle. Use C = 2πr.C = 2 × π × 9 = 56.5487…
- Multiply the fraction by the circumference. The arc is that fraction of the whole circumference.arc length = 29 × 56.5487…
- Round to 1 decimal place. Look at the next digit.arc length = 12.6 cm
Check. The arc should be 29 of the full circumference 56.55 cm: 29 × 56.55 = 12.57. ✓
Watch out. Divide the angle by 360, not by 180 or 100. Use 2πr (the circumference), not πr2 (the area).