1MA1 · G16–G17Geometry & MeasuresFoundation & HigherGrades 1–5

Perimeter & Area

Perimeter is the distance around the edge of a shape; area is the amount of flat surface it covers. A handful of formulas — for rectangles, triangles, parallelograms and trapezia — combine to handle almost any shape, including the L-shapes, gardens and floor plans that fill exam papers.

40 min Mensuration

The big picture

Every area formula grows out of the rectangle. A parallelogram rearranges into a rectangle, so its area is base × perpendicular height. A triangle is half a parallelogram. A trapezium averages its two parallel sides. Understanding these links means you can reconstruct any formula you forget.

Edexcel mensuration questions are often multi-step problems in context — the cost of turf for a garden, how many tiles fit a floor — where area is only one stage, and unit conversions like cm² to m² catch many candidates.

What you'll learn

Calculate perimeters of shapes, including compound shapes
Find areas of rectangles, triangles, parallelograms and trapezia
Find areas of compound shapes by adding and subtracting
Solve problems involving area in context, including unit conversions

Area formulas

Rectangle: length × width. Parallelogram: base × perpendicular height. Triangle: $\tfrac12$ × base × perpendicular height. Trapezium: $\tfrac12(a + b)h$ , where $a$ and $b$ are the parallel sides.

The height must be $\textbf{perpendicular}$ to the base — not a slanting side.

A = \tfrac12 bh

Triangle.

A = \tfrac12(a + b)h

Trapezium.

Example 1Find the area of a trapezium with parallel sides 5 cm and 11 cm, and height 6 cm.

A = \tfrac12(5 + 11) \times 6

= 8 \times 6

Answer48 cm²

Your turnSame method, new numbers — try it before moving on.

Find the area of a triangle with base 9 cm and perpendicular height 7 cm, in cm².

Your answer

Show worked solution

Solution

1 $\frac{1}{2} \times 9 \times 7 = 31.5$ cm².

Answer: $31.5$ cm²

Why this works

Two copies of a trapezium, one flipped, fit together to make a parallelogram with base $a + b$ and height $h$ . Half of that is the trapezium formula.

Perimeter

Perimeter is the total length of all the edges. For compound shapes, find any missing side lengths first using the lengths you know.

Perimeter is a length, measured in cm or m — not cm².

Example 1An L-shaped room is a 10 m by 8 m rectangle with a 4 m by 3 m rectangle removed from one corner. Find its perimeter and area.

1Removing a corner rectangle does not change the perimeter: the missing edges are replaced by equal inside edges.

2Perimeter

= 2(10 + 8) = 36

3Area

= 80 - 12 = 68

m².

AnswerPerimeter 36 m, area 68 m²

Your turnSame method, new numbers — try it before moving on.

A square has perimeter 28 cm. Find its area, in cm².

Your answer

Show worked solution

Solution

1Side $= 28 \div 4 = 7$ cm.
2Area $= 49$ cm².

Answer: $49$ cm²

Tip — For a compound shape, mark every side length on the diagram before adding. Missing sides are found by adding or subtracting known sides.

Compound shapes

Split a compound shape into rectangles, triangles and trapezia, find each area, then add.

Alternatively, find the area of a larger simple shape and subtract the parts that are missing — often quicker.

Example 1A path 1 m wide runs around the outside of a 12 m by 8 m lawn. Find the area of the path.

1Outer rectangle:

14 \times 10 = 140

m².

2Lawn:

12 \times 8 = 96

m².

3Path

= 140 - 96

Answer44 m²

Your turnSame method, new numbers — try it before moving on.

A path 2 m wide runs around the outside of a 10 m by 6 m lawn. Find the area of the path, in m².

Your answer

Show hint

The outer rectangle is 14 m by 10 m.

Show worked solution

Solution

1Outer rectangle: $14 \times 10 = 140$ m².
2Path $= 140 - 60 = 80$ m².

Answer: $80$ m²

Units and problems

Area units are squared: $1$ m² $= 100 \times 100 = 10\,000$ cm², and $1$ cm² $= 100$ mm².

In cost or quantity problems, find the area, convert to the unit being priced, then round up to whole packs where needed.

Example 1Turf costs £4.50 per m² and is sold in whole square metres. A rectangular lawn measures 7.4 m by 5.2 m. How much does the turf cost?

1Area

= 38.48

m².

2Buy 39 m² (round up).

3Cost

= 39 \times 4.50

Answer£175.50

Tip — In context, decide whether to round up (you need enough turf) or down (how many whole items fit). Explain your choice.

Think like an examiner

Write the formula, substitute, then calculate.
Include units: cm² for area, cm for perimeter.
Show the split of compound shapes clearly.

Area

A = lw

Rectangle.

A = bh

Parallelogram.

A = \tfrac12 bh

Triangle.

A = \tfrac12(a + b)h

Trapezium.

Watch out for these

Using the slant side as the height of a parallelogram.

Use the perpendicular height.

Forgetting the

\tfrac12

for a triangle.

Triangle area is half base × height.

Converting 3 m² to 300 cm².

3 \times 10\,000 = 30\,000

cm².

Giving perimeter in cm².

Perimeter is a length, in cm.

Stretch yourself

A rectangle has area 48 cm² and perimeter 28 cm. Find its length and width.

Hint — Let the sides be $l$ and $w$ . Write two equations and form a quadratic.

Reveal solution

SolutionFrom the perimeter,

2(l + w) = 28

, so

l + w = 14

and

w = 14 - l

. From the area,

l(14 - l) = 48

, so

l^{2} - 14l + 48 = 0

, which factorises as

(l - 6)(l - 8) = 0

. The sides are 8 cm and 6 cm. Check:

8 \times 6 = 48

✓ and

2(8 + 6) = 28

✓.

\blacksquare

Questions students ask

What is the difference between area and perimeter?

Perimeter measures the boundary (a length); area measures the surface enclosed (a squared unit).

Do I need to learn the trapezium formula?

Yes — it is not given on the Edexcel formula sheet.

How do I convert cm² to m²?

Divide by 10 000, because 1 m² is 100 cm by 100 cm.

Key takeaways

Perimeter: total length of the edges.
Rectangle $lw$ ; parallelogram $bh$ ; triangle $\tfrac12 bh$ ; trapezium $\tfrac12(a + b)h$ .
Heights must be perpendicular.
Compound shapes: split and add, or subtract from a larger shape.
Area units scale by the square of length conversions.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 21 mark

A parallelogram has base 12 cm and perpendicular height 7 cm. Work out its area.[1]
Show mark scheme
- B1 $84$ cm²
Answer: $84$ cm²

Question 2Grade 42 marks

A trapezium has area 60 cm² and parallel sides of 8 cm and 12 cm. Work out its height.[2]
Show mark scheme
- M1 $\frac{1}{2}(8 + 12)h = 60$
- A1 $6$ cm
Answer: $6$ cm

Question 3Grade 53 marks

A room is 4.2 m by 3.5 m. Carpet costs £18.50 per m² and is sold in whole square metres. Work out the cost of carpeting the room.[3]
Show mark scheme
- P1Area $= 14.7$ m²
- P1Rounds up to 15 m²
- A1£277.50
Answer: £277.50

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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