GCSE Maths: Area of triangles and parallelograms (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 triangles and parallelograms
Rectangles are easy because you can see the rows of squares. To find the area of a triangle or a parallelogram we turn it into a rectangle, so that we can see why the formula works.
Triangles
Choose one side of the triangle to be the base. The perpendicular height is the distance from the base straight up (at a right angle, 90°) to the opposite corner. The perpendicular height is usually shown as a dotted line with a small square at the bottom. It is not the same as a slanting side.
Now draw a rectangle round the triangle, with the same base and the same height. The triangle always fills exactly half of that rectangle: the two leftover pieces at the sides together make a copy of the triangle. So
area of a triangle = ½ × base × perpendicular height
For some triangles the height line falls outside the triangle. This happens when one angle is bigger than 90°. You extend the base with a dotted line and measure the height down to it. The formula still works.
Parallelograms
A parallelogram has two pairs of parallel sides. Cut a right-angled triangle off one end and slide it to the other end: the parallelogram becomes a rectangle with the same base and the same perpendicular height. So the area of a parallelogram is
area = base × perpendicular height
(no half this time, because nothing was lost). The slanting sides are used for the perimeter, but never for the area.
Key ideas
- The perpendicular height is measured at 90° to the base. It is not a slanting side.
- Triangle: area = ½ × base × perpendicular height.
- Parallelogram: area = base × perpendicular height.
- Perimeter = add all the outside sides (the slanting sides count; the dotted height does not).
- To find a missing base or height, work backwards: double the area first for a triangle, then divide.
Watch out
Never use a slanting side as the height, and never forget the half for a triangle. In a triangle with sides 13 cm, 14 cm and 15 cm and a height of 12 cm, the area is ½ × 14 × 12 = 84 cm², not ½ × 14 × 13.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. The slanting sides are not used for the area.base = 14 cm, height = 12 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.14 × 12 = 168
- Halve it. The triangle is half of that rectangle.168 ÷ 2 = 84
- Write the units. Area is in square cm.Area = 84 cm²
Check. Use half the base instead: 7 × 12 = 84. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
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, so add the three sides. Use the sloping sides, not the dotted height.
- Identify the three sides of the triangle. The dotted perpendicular height is not part of the boundary, so it is left out.13 cm, 14 cm, 15 cm
- Add them. Perimeter is the total length of the boundary.13 + 14 + 15 = 42
- Write the units. A perimeter is a length.Perimeter = 42 cm
Check. Add in a different order: 15 + 14 + 13 = 42. ✓
Watch out. The dotted height is a helper line. Do not include it in the perimeter.
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 = 12 cm, height = 6 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.12 × 6 = 72
- Write the units. Area is in square cm.Area = 72 cm²
Check. Swap the order: 6 × 12 = 72. ✓
Watch out. Use the perpendicular height (dotted line), not the slanting side. There is no half in the parallelogram formula.
Method. A parallelogram has two pairs of equal sides. Add all four sides; the slanting sides count, the dotted height does not.
- List the four sides. Opposite sides are equal.12, 10, 12, 10
- Add. The perimeter is the total of the four sides.12 + 10 + 12 + 10 = 44
- Write the units. Perimeter is a length.Perimeter = 44 cm
Check. 2 × (12 + 10) = 2 × 22 = 44. ✓
Watch out. The perimeter uses the slanting sides, not the perpendicular height.
Method. Use Area = ½ × base × height. Put in the area and the base, then undo the operations to find the height.
- Write the formula and substitute. Replace the letters with the numbers you know.60 = ½ × 15 × ?
- Undo the half by doubling. Multiplying by ½ halves the number; the opposite is to multiply by 2.60 × 2 = 120
so 15 × ? = 120 - Divide by the base. The opposite of multiplying by 15 is dividing by 15.? = 120 ÷ 15 = 8
- Write the answer. Give the unit.The height is 8 cm
Check. ½ × 15 × 8 = 60 cm². ✓
Watch out. Remember to double the area first. A common mistake is to divide the area by the other length without doubling.
Method. First find the base from the area, then make an equation and solve it for x.
- Find the base. Use the triangle method: double the area, then divide by the height.54 × 2 = 108
108 ÷ 9 = 12 cm - Make an equation. The base is 2x + 2, and it equals 12.2x + 2 = 12
- Solve. Subtract 2 from both sides, then divide by 2.2x = 10
x = 5
Check. Base = 2 × 5 + 2 = 12 and ½ × 12 × 9 = 54. ✓
Watch out. Do not forget to double the area first.
Exercise 18B Triangles and parallelograms
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.base = 8 cm, height = 5 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.8 × 5 = 40
- Halve it. The triangle is half of that rectangle.40 ÷ 2 = 20
- Write the units. Area is in square cm.Area = 20 cm²
Check. Use half the base instead: 4 × 5 = 20. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
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. The slanting sides are not used for the area.base = 14 cm, height = 12 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.14 × 12 = 168
- Halve it. The triangle is half of that rectangle.168 ÷ 2 = 84
- Write the units. Area is in square cm.Area = 84 cm²
Check. Use half the base instead: 7 × 12 = 84. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
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, so add the three sides. Use the sloping sides, not the dotted height.
- Identify the three sides of the triangle. The dotted perpendicular height is not part of the boundary, so it is left out.13 cm, 14 cm, 15 cm
- Add them. Perimeter is the total length of the boundary.13 + 14 + 15 = 42
- Write the units. A perimeter is a length.Perimeter = 42 cm
Check. Add in a different order: 15 + 14 + 13 = 42. ✓
Watch out. The dotted height is a helper line. Do not include it in the perimeter.
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 = 7.5 m, height = 4 m
- 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.7.5 × 4
Ignore the decimal points: 75 × 4 = 300
Together there are 1 decimal place, so 300 becomes 30 - Halve it. The triangle is half of that rectangle.30 ÷ 2 = 15
- Write the units. Area is in square m.Area = 15 m²
Check. Use half the base instead: 3.75 × 4 = 15. ✓
Watch out. Use the perpendicular (dotted) height, never the slanting side, and do not forget to halve.
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 = 6 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 × 6 = 60
- Halve it. The triangle is half of that rectangle.60 ÷ 2 = 30
- Write the units. Area is in square cm.Area = 30 cm²
Check. Use half the base instead: 5 × 6 = 30. ✓
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 = 12 cm, height = 6 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.12 × 6 = 72
- Write the units. Area is in square cm.Area = 72 cm²
Check. Swap the order: 6 × 12 = 72. ✓
Watch out. Use the perpendicular height (dotted line), not the slanting side. There is no half in the parallelogram formula.
Method. A parallelogram has two pairs of equal sides. Add all four sides; the slanting sides count, the dotted height does not.
- List the four sides. Opposite sides are equal.12, 10, 12, 10
- Add. The perimeter is the total of the four sides.12 + 10 + 12 + 10 = 44
- Write the units. Perimeter is a length.Perimeter = 44 cm
Check. 2 × (12 + 10) = 2 × 22 = 44. ✓
Watch out. The perimeter uses the slanting sides, not the perpendicular height.
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 = 9.5 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.9.5 × 4
Ignore the decimal points: 95 × 4 = 380
Together there are 1 decimal place, so 380 becomes 38 - Write the units. Area is in square cm.Area = 38 cm²
Check. Swap the order: 4 × 9.5 = 38. ✓
Watch out. Use the perpendicular height (dotted line), not the slanting side. There is no half in the parallelogram formula.
Method. A parallelogram has two pairs of equal sides. Add all four sides; the slanting sides count, the dotted height does not.
- List the four sides. Opposite sides are equal.9.5, 5, 9.5, 5
- Add. The perimeter is the total of the four sides.9.5 + 5 + 9.5 + 5 = 29
- Write the units. Perimeter is a length.Perimeter = 29 cm
Check. 2 × (9.5 + 5) = 2 × 14.5 = 29. ✓
Watch out. The perimeter uses the slanting sides, not the perpendicular height.
Method. Use Area = ½ × base × height. Put in the area and the base, then undo the operations to find the height.
- Write the formula and substitute. Replace the letters with the numbers you know.60 = ½ × 15 × ?
- Undo the half by doubling. Multiplying by ½ halves the number; the opposite is to multiply by 2.60 × 2 = 120
so 15 × ? = 120 - Divide by the base. The opposite of multiplying by 15 is dividing by 15.? = 120 ÷ 15 = 8
- Write the answer. Give the unit.The height is 8 cm
Check. ½ × 15 × 8 = 60 cm². ✓
Watch out. Remember to double the area first. A common mistake is to divide the area by the other length without doubling.
Method. Use Area = ½ × base × height. Put in the area and the height, then undo the operations to find the base.
- Write the formula and substitute. Replace the letters with the numbers you know.42 = ½ × 7 × ?
- Undo the half by doubling. Multiplying by ½ halves the number; the opposite is to multiply by 2.42 × 2 = 84
so 7 × ? = 84 - Divide by the height. The opposite of multiplying by 7 is dividing by 7.? = 84 ÷ 7 = 12
- Write the answer. Give the unit.The base is 12 cm
Check. ½ × 7 × 12 = 42 cm². ✓
Watch out. Remember to double the area first. A common mistake is to divide the area by the other length without doubling.
Method. Area = base × height. We know the area and the base, so divide to find the perpendicular height.
- Write the formula and substitute. Use a question mark for the height.84 = 12 × ?
- Undo the multiplication. Divide by 12.? = 84 ÷ 12 = 7
- Write the answer. Give the unit.Perpendicular height = 7 cm
Check. 12 × 7 = 84. ✓
Watch out. Divide the area by the base. Do not double the area (that is for triangles).
Method. Find the area of the sail, then multiply by the cost of each square metre.
- Area of the triangle. half × base × height.½ × 2.4 × 3.5
2.4 × 3.5 = 8.4 (24 × 35 = 840, one decimal place in each, so two altogether: 8.40)
8.4 ÷ 2 = 4.2 m² - Multiply by the cost per square metre. Each square metre costs £18.4.2 × 18 = £75.60
Check. Estimate: 4 m² × £18 = £72, close to £75.60. ✓
Watch out. Cost uses the area (m²). The perimeter is not needed here.
Method. Find the area of the bed, then divide by the area one bag covers. Round up, because you cannot buy part of a bag.
- Area of the parallelogram. base × perpendicular height. The slanting side is not needed.9 × 4 = 36 m²
- Divide by the area one bag covers. This tells us how many bags are needed.36 ÷ 5 = 7.2
- Round up. 7 bags would not cover it all, so we need the next whole number.8 bags
Check. 8 bags cover 8 × 5 = 40 m², which is at least 36 m²; 7 bags cover only 35 m². ✓
Watch out. When a number of items must be whole, round up (not to the nearest) if you need to cover everything.
Method. Check which length Ravi used as the height, then correct it.
- Spot the mistake. The 13 cm side is a slanting side. The height must be at a right angle to the base, which is the dotted line of 12 cm.Ravi used 13 cm instead of 12 cm.
- Use the perpendicular height. Area = ½ × base × perpendicular height.½ × 10 × 12 = 120 ÷ 2 = 60 cm²
Check. Half of the 10 cm by 12 cm rectangle: 120 ÷ 2 = 60. ✓
Watch out. The height is the dotted line at right angles to the base, never a slanting side.
Method. Use the same formula for each triangle: only the base and the perpendicular height matter.
- Area of A. Half base × perpendicular height.½ × 10 × 6 = 30 cm²
- Area of B. The same base and the same perpendicular height.½ × 10 × 6 = 30 cm²
- Conclude. State your answer.Yes, Zara is correct: both areas are 30 cm², even though the triangles look different.
Check. The apex of each triangle is 6 cm above the base, so each fills half of a 10 cm by 6 cm rectangle. ✓
Watch out. Where the top corner is along the line does not change the area, only the height does.
Method. Area = base × height. We know the area and the base, so divide to find the perpendicular height.
- Write the formula and substitute. Use a question mark for the height.72 = 12 × ?
- Undo the multiplication. Divide by 12.? = 72 ÷ 12 = 6
- Write the answer. Give the unit.Perpendicular height = 6 cm
Check. 12 × 6 = 72. ✓
Watch out. Divide the area by the base. Do not double the area (that is for triangles).
Method. The perimeter is two lengths plus two widths. Take away the two lengths you know, then share what is left between the two widths.
- Take away the two lengths. A rectangle has two sides of length 12 cm. Whatever is left of the perimeter is the two widths.2 × 12 = 24
44 − 24 = 20 - Share between the two widths. There are two equal widths, so divide by 2.20 ÷ 2 = 10
- Write the answer. Give the unit.Width = 10 cm
Check. Add the four sides: 12 + 10 + 12 + 10 = 44 cm. ✓
Watch out. Half the perimeter is length + width, not just the width. Take away the length too.
Method. Work out both areas with their own formulas, then compare.
- Area of the triangle. half × base × height.½ × 12 × 5 = 60 ÷ 2 = 30 cm²
- Area of the parallelogram. base × height (no half).12 × 5 = 60 cm²
- Compare. Divide the larger by the smaller.60 ÷ 30 = 2, so the parallelogram is double the triangle.
Check. A diagonal cuts a parallelogram into two triangles of equal area, and each triangle has the same base and height. ✓
Watch out. The only difference in the formulas is the half for the triangle.
Method. Find the base from the area, then solve for x.
- Find the base. Double the area, then divide by the height.54 × 2 = 108
108 ÷ 9 = 12 cm - Make an equation. The base is 2x + 2.2x + 2 = 12
- Solve. Subtract 2, then divide by 2.2x = 10
x = 5
Check. ½ × (2 × 5 + 2) × 9 = ½ × 12 × 9 = 54. ✓
Watch out. Double the area first.
Method. Write the area as an expression and put it equal to 40.
- Write the area. Area = base × height.(x + 3) × x = x2 + 3x
- Set equal to 40. The area is 40 cm².x2 + 3x = 40
- Rearrange to equal 0. Subtract 40 from both sides.x2 + 3x − 40 = 0
Check. Substitute x = 5: 25 + 15 − 40 = 0. ✓
Method. Factorise the quadratic and solve. A length cannot be negative.
- Find two numbers. They multiply to −40 and add to +3.8 and −5
- Factorise. Write as two brackets.(x + 8)(x − 5) = 0
- Solve. Each bracket can equal 0.x = −8 or x = 5
- Choose the sensible answer. A length must be positive.x = 5
Check. Base = 8 cm, height = 5 cm, area = 8 × 5 = 40 cm². ✓
Watch out. Reject the negative solution: a length cannot be negative.