Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Geometry and measures
Practise Area and perimeter. unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Area and perimeter of rectangles, triangles, parallelograms, trapezia, circles and shapes made from them. These questions are common on both tiers, often in a context such as fencing, flooring or costs, and on Higher sometimes with algebra.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
1
Find the perimeter by adding the sidesAdd the lengths of all the edges around the outside of the shape.
2
Find the area of rectangles and trianglesRectangle = length × width; triangle = ½ × base × perpendicular height.
3
Find the area of parallelograms and trapeziaParallelogram = base × perpendicular height; trapezium = ½(a + b)h.
4
Find the area and circumference of circlesArea = πr²; circumference = πd = 2πr.
4
Find areas and perimeters of compound shapesSplit into rectangles, triangles and parts of circles, finding any missing lengths first.
5
Solve area problems in context and backwardse.g. work out the cost of turf for a lawn, or the radius of a circle from its area.
Notes
Perimeter
Perimeter is the total distance around the outside of a shape, in units of length (cm, m).
In compound shapes, find any missing sides first; opposite sides of a rectangle are equal.
Circumference of a circle: \(C = \pi d = 2\pi r\).
Perimeter of a semicircle = \(\frac{1}{2}\pi d + d\): half the circumference plus the straight edge.
Area formulas
Rectangle: area = length × width.
Triangle: \(A = \frac{1}{2}bh\), where h is the perpendicular height (at right angles to the base).
Parallelogram: \(A = bh\), using the perpendicular height, not the sloping side.
Trapezium: \(A = \frac{1}{2}(a + b)h\), where a and b are the parallel sides and h is the distance between them.
Circle: \(A = \pi r^2\). Square the radius, not the diameter.
Compound shapes and problems
Split the shape into simple shapes and add their areas, or take a larger shape and subtract the part cut out.
Area units are squared: cm², m². 1 m² = 100 × 100 = 10 000 cm², and 1 cm² = 100 mm².
In context, read what is needed: if one tin of paint covers 12 m², always round the number of tins up.
Working backwards: if a circle has area 50 cm², then \(r^2 = 50 \div \pi\), so \(r = \sqrt{50 \div \pi} = 3.99\) cm.
Cheatsheet
Rectangle = length × width
Triangle = ½ × base × perpendicular height
Parallelogram = base × perpendicular height
Trapezium = ½(a + b)h
Circle: area = πr²; circumference = πd = 2πr
Semicircle: area = ½πr²; perimeter = ½πd + d
1 m² = 10 000 cm²; 1 cm² = 100 mm²
How to answer each type of question
Work out the area of a compound shape
3 marks4
Split the shape into simple parts and label them.
Find each area separately, finding any missing lengths first.
Add (or subtract) the areas and give squared units.
Example. A shape is made from a rectangle 12 cm by 8 cm and a semicircle. The diameter of the semicircle is one of the 8 cm sides of the rectangle. Work out the area of the shape. Give your answer correct to 1 decimal place.
Divide by the amount one pack covers and round up to a whole number of packs.
Multiply by the price and give the answer in pounds.
Example. A lawn is in the shape of a trapezium. Its parallel sides are 14 m and 9 m long, and the perpendicular distance between them is 6 m. Grass seed is sold in boxes. One box covers 20 m² and costs £8.50. Work out the cost of the boxes needed to seed the whole lawn.
Show the model answer
Area = ½ × (14 + 9) × 6 (M1) = 69 m² (A1) 69 ÷ 20 = 3.45, so 4 boxes are needed (M1) Cost = 4 × £8.50 = £34 (A1)
Circles: work backwards from the area or circumference
3 marks5
Write the formula with the value you are given, e.g. πr² = 30.
Rearrange to find r, keeping the unrounded value.
Use r to find what the question asks for.
Example. A circular pond has an area of 30 m². Work out the circumference of the pond. Give your answer correct to 3 significant figures.
Show the model answer
πr² = 30 (M1), so r = √(30 ÷ π) = 3.090… m (M1) Circumference = 2 × π × 3.090… = 19.4 m (A1)
Shortcuts and memory tricks
Trapezium: 'add the parallel sides, halve, times the height'.
Sense check an area by drawing a rectangle around the shape: the shape's area must be smaller.
Use the π button and round only at the end.
Changing area units: square the length conversion. 1 m = 100 cm, so 1 m² = 100² = 10 000 cm².
Where marks are lost
Using the sloping side instead of the perpendicular height for a triangle or parallelogram.
Forgetting the ½ in the triangle and trapezium formulas.
Putting the diameter into πr².
Leaving out the straight edge (the diameter) in the perimeter of a semicircle.
Dividing by 100 instead of 10 000 to change cm² to m².
Rounding the number of tins or boxes down.
Exam technique
Label each part of a compound shape and show its area; each part can earn a method mark.
Always give units, and square them for area.
Write money correctly, e.g. £34 or £34.00, never £34.0.
In 'show that' or decision questions, show every calculation and finish with a sentence that answers the question.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
The diagram shows a trapezium. Diagram NOT accurately drawn
[object Object]
Work out the area of the trapezium.[2]
Show the answer and mark scheme
Answer: \(204\) m²
M1 for \(\frac{1}{2}(12 + 22) \times 12\)
A1 for 204
Worked solution: Area \(= \frac{1}{2}(a + b)h = \frac{1}{2}(12 + 22) \times 12 = 204\) m²
Question 2Medium3 marks
The diagram shows a quarter circle with radius 10.4 cm. Diagram NOT accurately drawn
[object Object]
Work out the perimeter of the quarter circle. Give your answer correct to 1 decimal place.[3]
Show the answer and mark scheme
Answer: \(37.1\) cm
M1 for the curved edge \(\frac{1}{4} \times 2 \times \pi \times 10.4\) (= 16.33...)
M1 for adding the two radii, 20.8
A1 for 37.1 (37.13...)
Worked solution: Curved edge \(= \frac{1}{4} \times 2\pi \times 10.4 = 16.33\ldots\) cm Perimeter \(= 16.33\ldots + 20.8 = 37.13\ldots\) = 37.1 cm (1 d.p.)