GCSE Maths: Area of trapezia (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 trapezia
A trapezium is a four-sided shape with exactly one pair of parallel sides (sides that always stay the same distance apart). Think of the side view of a bucket, or a slice of a ramp. The two parallel sides have different lengths: call them a and b. The distance between them, measured at a right angle, is the perpendicular height h. The other two sides slant, and we do not need them for the area.
Where the formula comes from
Take two identical trapeziums and turn one upside down. Fitted together they make a parallelogram. Its base is a + b and its height is h, so the parallelogram has area (a + b) × h. Our trapezium is only half of that, so
area of a trapezium = ½ × (a + b) × h
Another way to remember it: (a + b) ÷ 2 is the average of the two parallel sides, so the area is the average width times the height. Always do the bracket (the addition) first.
The perimeter of a trapezium is the sum of all four sides. This time you do need the two slanting sides, but still not the dotted height.
Key ideas
- A trapezium has one pair of parallel sides, a and b. The height h is at 90° to them.
- Area = ½ × (a + b) × h. Add the parallel sides first.
- Do not use a slanting side as the height.
- Perimeter = sum of all four sides (including the two slanting sides).
- To find a missing measurement, put the numbers into the formula and undo it step by step.
Watch out
The two slanting sides are not the parallel sides. Parallel sides are the ones that look like a pair of railway lines. In a trapezium whose top is 8 cm and bottom is 22 cm, the 8 and 22 go in the bracket, whatever the slanting sides measure.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. Ignore the slant lengths.a = 8 cm, b = 22 cm, h = 12 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 = 8 + 22 = 30
- Multiply by the height. This gives the area of the double (parallelogram).30 × 12 = 360
- Halve it. The trapezium is half of the parallelogram.360 ÷ 2 = 180
- Write the units. Area is in square cm.Area = 180 cm²
Check. The average of the parallel sides is 30 ÷ 2 = 15; 15 × 12 = 180. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
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 = 7 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 = 7 + 10 = 17
- Multiply by the height. This gives the area of the double (parallelogram).17 × 4 = 68
- Halve it. The trapezium is half of the parallelogram.68 ÷ 2 = 34
- Write the units. Area is in square cm.Area = 34 cm²
Check. The average of the parallel sides is 17 ÷ 2 = 8.5; 8.5 × 4 = 34. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. Perimeter is the distance all the way round: add the four sides. The dotted height is not on the edge, so leave it out.
- List the four sides. Two parallel sides and two slanting sides.7, 10, 4, 5
- Add them. The perimeter is the total of the four sides.7 + 10 + 4 + 5 = 26
- Write the units. Perimeter is a length.Perimeter = 26 cm
Check. Add in pairs: (7 + 10) + (4 + 5) = 17 + 9 = 26. ✓
Watch out. Include both slanting sides. The dotted height is not part of the boundary.
Method. Put the area and the parallel sides into the trapezium formula, then undo each operation to find the height.
- Write the formula and substitute. Replace the letters with the numbers you know.90 = ½ × (7 + 11) × h
90 = ½ × 18 × h - Double the area. Undo the ½.90 × 2 = 180, so 18 × h = 180
- Divide by the sum of the parallel sides. Undo the multiplication by 18.h = 180 ÷ 18 = 10
- Write the answer. Give the unit.Height = 10 cm
Check. ½ × 18 × 10 = 90 cm². ✓
Watch out. Double the area first, then divide by (a + b), not by one of the sides.
Method. Put the area, one parallel side and the height into the formula. First find a + b, then take away the side you know.
- Write the formula and substitute. Replace the letters with the numbers you know.76 = ½ × (5 + b) × 8
- Double the area, then divide by the height. This undoes the ½ and the × height, and leaves the bracket (a + b).76 × 2 = 152
152 ÷ 8 = 19
so 5 + b = 19 - Subtract the known side. Undo the “+ a”.b = 19 − 5 = 14
- Write the answer. Give the unit.b = 14 cm
Check. ½ × (5 + 14) × 8 = 76. ✓
Watch out. After doubling and dividing you have a + b, not b. Do not forget to subtract a.
Method. Write the area using the formula, put it equal to 55, and solve.
- Write the formula with the expressions. Add the parallel sides first.Area = ½ × (x + (x + 6)) × 5 = ½ × (2x + 6) × 5
- Set equal to 55 and clear the half. Multiply both sides by 2.(2x + 6) × 5 = 110
- Divide by 5. Undo the multiplication by 5.2x + 6 = 22
- Solve. Subtract 6, then divide by 2.2x = 16
x = 8
Check. Parallel sides 8 cm and 14 cm: ½ × 22 × 5 = 55. ✓
Watch out. Collect x + (x + 6) as 2x + 6 before you continue.
Exercise 18C Trapezia
Non-calculatorKey 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 = 9 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 = 5 + 9 = 14
- Multiply by the height. This gives the area of the double (parallelogram).14 × 4 = 56
- Halve it. The trapezium is half of the parallelogram.56 ÷ 2 = 28
- Write the units. Area is in square cm.Area = 28 cm²
Check. The average of the parallel sides is 14 ÷ 2 = 7; 7 × 4 = 28. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
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. Ignore the slant lengths.a = 8 cm, b = 22 cm, h = 12 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 = 8 + 22 = 30
- Multiply by the height. This gives the area of the double (parallelogram).30 × 12 = 360
- Halve it. The trapezium is half of the parallelogram.360 ÷ 2 = 180
- Write the units. Area is in square cm.Area = 180 cm²
Check. The average of the parallel sides is 30 ÷ 2 = 15; 15 × 12 = 180. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. Perimeter is the distance all the way round: add the four sides. The dotted height is not on the edge, so leave it out.
- List the four sides. Two parallel sides and two slanting sides.8, 22, 13, 15
- Add them. The perimeter is the total of the four sides.8 + 22 + 13 + 15 = 58
- Write the units. Perimeter is a length.Perimeter = 58 cm
Check. Add in pairs: (8 + 22) + (13 + 15) = 30 + 28 = 58. ✓
Watch out. Include both slanting sides. The dotted height is not part of the boundary.
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 = 7 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 = 7 + 10 = 17
- Multiply by the height. This gives the area of the double (parallelogram).17 × 4 = 68
- Halve it. The trapezium is half of the parallelogram.68 ÷ 2 = 34
- Write the units. Area is in square cm.Area = 34 cm²
Check. The average of the parallel sides is 17 ÷ 2 = 8.5; 8.5 × 4 = 34. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. Perimeter is the distance all the way round: add the four sides. The dotted height is not on the edge, so leave it out.
- List the four sides. Two parallel sides and two slanting sides.7, 10, 4, 5
- Add them. The perimeter is the total of the four sides.7 + 10 + 4 + 5 = 26
- Write the units. Perimeter is a length.Perimeter = 26 cm
Check. Add in pairs: (7 + 10) + (4 + 5) = 17 + 9 = 26. ✓
Watch out. Include both slanting sides. The dotted height is not part of the boundary.
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 = 3.5 m, b = 6.5 m, h = 4 m
- 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 = 3.5 + 6.5 = 10
- Multiply by the height. This gives the area of the double (parallelogram).10 × 4 = 40
- Halve it. The trapezium is half of the parallelogram.40 ÷ 2 = 20
- Write the units. Area is in square m.Area = 20 m²
Check. The average of the parallel sides is 10 ÷ 2 = 5; 5 × 4 = 20. ✓
Watch out. Add the two parallel sides before multiplying. Do not use a slanting side as the height.
Method. Put the area and the parallel sides into the trapezium formula, then undo each operation to find the height.
- Write the formula and substitute. Replace the letters with the numbers you know.90 = ½ × (7 + 11) × h
90 = ½ × 18 × h - Double the area. Undo the ½.90 × 2 = 180, so 18 × h = 180
- Divide by the sum of the parallel sides. Undo the multiplication by 18.h = 180 ÷ 18 = 10
- Write the answer. Give the unit.Height = 10 cm
Check. ½ × 18 × 10 = 90 cm². ✓
Watch out. Double the area first, then divide by (a + b), not by one of the sides.
Method. Put the area, one parallel side and the height into the formula. First find a + b, then take away the side you know.
- Write the formula and substitute. Replace the letters with the numbers you know.76 = ½ × (5 + b) × 8
- Double the area, then divide by the height. This undoes the ½ and the × height, and leaves the bracket (a + b).76 × 2 = 152
152 ÷ 8 = 19
so 5 + b = 19 - Subtract the known side. Undo the “+ a”.b = 19 − 5 = 14
- Write the answer. Give the unit.b = 14 cm
Check. ½ × (5 + 14) × 8 = 76. ✓
Watch out. After doubling and dividing you have a + b, not b. Do not forget to subtract a.
Method. Find the area of the patio, then multiply by the cost per square metre.
- Find the area. Use ½ × (a + b) × h with the parallel sides 4.5 and 7.5 and the distance between them, 3.a + b = 4.5 + 7.5 = 12
12 × 3 = 36
36 ÷ 2 = 18 m² - Multiply by the cost. Each square metre costs £35.18 × 35 = £630
Check. The average width is 6 m and 6 × 3 = 18 m²; 18 × 35 = 630. ✓
Watch out. The 3 m is the height between the parallel sides, so it goes in the formula as h.
Method. The base of the parallelogram is the two parallel sides placed end to end.
- Find the two pieces along the bottom. The bottom edge is the longer parallel side of one trapezium (12 cm) followed by the shorter parallel side of the other (8 cm).pieces: 12 cm and 8 cm
- Add the parallel sides. The base is the two pieces together.12 + 8 = 20 cm
Check. Look at the top edge: 8 + 12 = 20 cm as well. ✓
Method. Area of a parallelogram = base × perpendicular height.
- Write the base and the height. The base is the answer to part (a); the height is the dotted line, 5 cm.base = 20 cm, height = 5 cm
- Multiply. No half for a parallelogram.20 × 5 = 100 cm²
Check. 20 × 5 = 5 × 20 = 100. ✓
Method. The two trapeziums are identical, so each is half of the parallelogram.
- Say how many equal pieces there are. The parallelogram is made of two identical trapeziums.2 equal pieces
- Halve the area. One trapezium is one of the two equal pieces.100 ÷ 2 = 50 cm²
Check. Formula: ½ × (8 + 12) × 5 = 50. ✓
Method. Check which length Tom used for the height.
- Spot the mistake. The 5 cm side slants. The height is the side at a right angle to the parallel sides, which is 4 cm.Tom used 5 cm instead of 4 cm.
- Use the correct height. Area = ½ × (a + b) × h.½ × (7 + 10) × 4 = ½ × 17 × 4 = 34 cm²
Check. 34 ÷ 4 = 8.5 = the average of 7 and 10. ✓
Watch out. The height is perpendicular (at 90°) to the parallel sides.
Method. Find the area of the trapezium, then find the width of the rectangle that has the same area.
- Area of the trapezium. ½ × (a + b) × h.½ × (6 + 10) × 5 = ½ × 80 = 40 cm²
- Width of the rectangle. Rectangle area = length × width, so width = area ÷ length.40 ÷ 8 = 5 cm
Check. 8 × 5 = 40 cm². ✓
Watch out. Find the area first; it is the link between the two shapes.
Method. Find each part from the diagram, add the areas, then use the formula and compare.
- Find the base of each triangle. The bottom is 14 cm. The rectangle in the middle has width 8 cm (the same as the top). The rest is shared between the two triangles.14 − 8 = 6 cm, split as 2 cm on the left and 4 cm on the right
- Area of the rectangle. length × width.8 × 6 = 48 cm²
- Area of the left triangle. ½ × base × height.½ × 2 × 6 = 6 cm²
- Area of the right triangle. ½ × base × height.½ × 4 × 6 = 12 cm²
- Add. The trapezium is the three pieces together.48 + 6 + 12 = 66 cm²
- Use the formula. Compare with ½(a + b)h.½ × (8 + 14) × 6 = ½ × 22 × 6 = 66 cm²
- Conclude. State what you have shown.Both methods give 66 cm², so they agree.
Check. The 2 cm and 4 cm triangle bases add to 6 cm, the difference of the parallel sides. ✓
Watch out. End a “show that” question with a sentence saying the two answers match.
Method. Use the area formula with the algebraic sides and solve the equation.
- Write the formula. Add the parallel sides first.½ × (2x + 6) × 5 = 55
- Multiply both sides by 2. Undo the half.(2x + 6) × 5 = 110
- Divide by 5. Undo the multiplication.2x + 6 = 22
- Solve. Subtract 6 then divide by 2.2x = 16, so x = 8
Check. ½ × (8 + 14) × 5 = 55. ✓
Watch out. Combine x + (x + 6) into 2x + 6 first.
Method. Write the area formula, simplify the bracket, and solve.
- Write the formula. Add the two parallel sides: (2x + 1) + (x + 5) = 3x + 6.½ × (3x + 6) × 4 = 54
- Simplify. ½ × 4 = 2, so multiply the bracket by 2.2(3x + 6) = 54
6x + 12 = 54 - Solve. Subtract 12, then divide by 6.6x = 42
x = 7 - Find the sides. Substitute x = 7.2 × 7 + 1 = 15 cm and 7 + 5 = 12 cm
Check. ½ × (15 + 12) × 4 = ½ × 27 × 4 = 54. ✓
Watch out. Add the two parallel sides before you multiply by the height.
Method. Find the height from the trapezium, then use it in the parallelogram formula.
- Height of the trapezium. Put the numbers in ½(a + b)h and undo.96 = ½ × (9 + 15) × h = 12h
h = 96 ÷ 12 = 8 cm - Base of the parallelogram. Same height 8 cm, same area 96 cm², so base = area ÷ height.96 ÷ 8 = 12 cm
- Notice. Compare 12 with the two parallel sides.12 is exactly halfway between 9 and 15: it is (9 + 15) ÷ 2.
Check. ½ × (9 + 15) × 8 = 96 and 12 × 8 = 96. ✓
Watch out. This shows why the trapezium formula uses the average of the parallel sides.