Area and perimeter
Section: Mensuration | Syllabus: Cambridge IGCSE Mathematics (0580)
Perimeter and Area of Rectangles
The perimeter and area of a rectangle are found using its length and width, and questions often work backwards from a given perimeter or area to find an unknown side.
- Perimeter of a rectangle =2×(length+width)
- Area of a rectangle =length
- Given the perimeter (or area) and one side, the other side can be found by rearranging the formula
Worked Example: Finding an Unknown Side from the Perimeter
- Question: A rectangle has a perimeter of 34 cm. Its length is 11 cm. Find its width.
- Step 1: 2×(11+w)=34
- Step 2: 11+w=17
- Step 3: w=17-11=6
- Answer: width = 6 cm
Common Mistakes
MistakeForgetting to divide the total perimeter by 2 first, before subtracting the known side
Fixalways divide the total perimeter by 2 first - this gives length + width, before either side can be isolated
MistakeConfusing the area formula with the perimeter formula when a question doesn't clearly specify which is meant
Fixcheck the units given in the question - a plain length (e.g. cm) suggests perimeter, while a squared unit (e.g. cm²) confirms area
Area of Triangles and Parallelograms
The area of a triangle or a parallelogram is found using its base and its perpendicular height - the height must always be measured at a right angle to the chosen base, not along a slanted side.
- Area of a triangle =(1/2)
- Area of a parallelogram =base
- The height used must always be the perpendicular height (measured at 90° to the base), not the length of a slanted side
Worked Example: Finding the Area of a Triangle
- Question: A triangle has a base of 14 cm and a perpendicular height of 9 cm. Find its area.
- Step 1: (1/2)149
- Step 2: =63
- Answer: 63 cm²
Worked Example: Finding the Area of a Parallelogram
- Question: A parallelogram has a base of 12 cm and a perpendicular height of 7.5 cm. Find its area.
- Step 1: 127.5
- Step 2: =90
- Answer: 90 cm²
Common Mistakes
MistakeForgetting to halve the result when calculating the area of a triangle
Fixthe triangle area formula always includes the ½ - a triangle is exactly half of the equivalent parallelogram
MistakeUsing a slanted side length instead of the perpendicular height
Fixthe height must be measured straight up from the base at a right angle - a slanted side is usually longer than the true perpendicular height
Area and Perimeter of Trapeziums
A trapezium has exactly one pair of parallel sides. Its area uses both parallel sides, while its perimeter sometimes requires Pythagoras' theorem to find a missing slanted side.
- Area of a trapezium =(a+b/2)× h, where a and b are the two parallel sides and h is the perpendicular height between them
- To find the perimeter, all four sides must be known or calculated - a missing slanted side can often be found using Pythagoras' theorem, by forming a right-angled triangle with the horizontal and vertical differences between the parallel sides
- Always check which sides are actually parallel before applying the trapezium area formula
Worked Example: Finding the Area of a Trapezium
- Question: A trapezium has parallel sides of 7 m and 15 m, and a perpendicular height of 8 m. Find its area.
- Step 1: (7+15/2)8
- Step 2: =118=88
- Answer: 88 m²
Worked Example: Finding a Missing Side Using Pythagoras
- Question: A right-angled trapezium has a horizontal base of 6 cm. Its two parallel vertical sides are 18 cm and 12 cm. Find the length of the fourth (slanted) side.
- Step 1: The horizontal offset between the slanted side's ends is 6 cm; the vertical difference between the parallel sides is 18-12=6 cm
- Step 2: Use Pythagoras: √(6^2+6^2)=√(72)
- Step 3: √(72)=6√(2)8.49
- Answer: 6√2 cm (≈ 8.49 cm)
Common Mistakes
MistakeUsing a slanted side length in the area formula instead of the perpendicular height
Fixthe trapezium area formula needs the height measured straight between the two parallel sides, not a slanted side
MistakeForgetting to find a missing slanted side using Pythagoras before adding up all four sides for the perimeter
Fixcheck whether every side length is already given - if not, look for a right-angled triangle formed by the horizontal and vertical differences to calculate it first
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