Perimeter, Area and Circles

Calculating perimeter and area is a fundamental skill for Edexcel GCSE Mathematics. This lesson covers the formulae for rectangles, triangles, parallelograms, trapeziums and circles, then extends to sectors and composite shapes. Pay careful attention to which formulae are given on the Edexcel formula sheet and which you must memorise.

---

Key Vocabulary

Term	Meaning
Perimeter	The total distance around the outside of a 2D shape
Area	The amount of 2D space inside a shape, measured in square units (cm², m², etc.)
Circumference	The perimeter of a circle
Sector	A "pizza slice" of a circle (bounded by two radii and an arc)
Composite shape	A shape made up of two or more simpler shapes

---

Formulae You Must Memorise

These are NOT given on the Edexcel formula sheet:

Shape	Area formula	Perimeter
Rectangle	$A = l \times w$A=l×w	P = 2(l + w)
Triangle	$A = \frac{1}{2} \times b \times h$A=21​×b×h	Sum of 3 sides
Parallelogram	$A = b \times h$A=b×h	Sum of 4 sides
Circle	A = $\pi r^2$πr2	C = $\pi d$πd = $2\pi r$2πr

Given on the Edexcel Formula Sheet: The area of a trapezium ($A = \frac{1}{2}(a + b) \times h$A=21​(a+b)×h) IS provided on the Edexcel formula sheet, so you do not need to memorise it — but you should be confident using it.

---

Rectangles

Worked Example 1

A rectangle has length 12 cm and width 5 cm. Find its perimeter and area.

Perimeter = 2(12 + 5) = 2 x 17 = 34 cm Area = 12 x 5 = 60 cm²

---

Triangles

The height must be perpendicular to the base.

Worked Example 2

A triangle has base 8 cm and perpendicular height 6 cm. Find its area.

Area = 1/2 x 8 x 6 = 24 cm²

---

Parallelograms

Worked Example 3

A parallelogram has base 10 cm and perpendicular height 7 cm. Find its area.

Area = 10 x 7 = 70 cm²

Common Mistake: Do not use the slant height — always use the perpendicular height.

---

Trapeziums

A trapezium has two parallel sides (a and b) and a perpendicular height (h).

Area = 1/2 (a + b) x h

Worked Example 4

A trapezium has parallel sides of 6 cm and 10 cm, and a perpendicular height of 4 cm.

Area = 1/2 (6 + 10) x 4 = 1/2 x 16 x 4 = 32 cm²

---

Circles

Circumference

C = $\pi d$πd = $2\pi r$2πr

Worked Example 5

A circle has radius 5 cm. Find its circumference. Give your answer to 1 decimal place.

C = 2 x pi x 5 = 10pi = 31.4 cm (1 d.p.)

Area

A = $\pi r^2$πr2

Worked Example 6

A circle has diameter 14 cm. Find its area. Give your answer to 1 decimal place.

Radius = 14 / 2 = 7 cm A = pi x 7² = 49pi = 153.9 cm² (1 d.p.)

Common Mistake: Forgetting to halve the diameter to find the radius before using $A = \pi r^2$A=πr2.

Worked Example 7

The circumference of a circle is 40 cm. Find its radius. Give your answer to 2 decimal places.

C = $2\pi r$2πr 40 = $2\pi r$2πr r = 40 / (2 x pi) = 20 / pi = 6.37 cm (2 d.p.)

---

Sectors

A sector is a fraction of a circle. The fraction is determined by the angle at the centre (theta).

Arc length = (theta / 360) x $2\pi r$2πr

Sector area = (theta / 360) x $\pi r^2$πr2

Edexcel Formula Sheet: These sector formulae are NOT explicitly given. You need to understand how to use them.

Worked Example 8

A sector has radius 9 cm and angle 80°. Find (a) the arc length and (b) the area. Give answers to 1 d.p.

(a) Arc length = (80/360) x 2 x pi x 9 = (2/9) x 18pi = 4pi = 12.6 cm (1 d.p.)

(b) Sector area = (80/360) x pi x 9² = (80/360) x 81pi = (2/9) x 81pi = 18pi = 56.5 cm² (1 d.p.)

Worked Example 9

Find the perimeter of the sector in Example 8.

Perimeter = arc length + 2 x radius = 12.566... + 9 + 9 = 30.6 cm (1 d.p.)

---

Composite Shapes

Break the shape into simpler parts (rectangles, triangles, semicircles, etc.), then add or subtract areas as needed.

Worked Example 10

An L-shaped room can be split into two rectangles: one 8 m by 3 m and one 5 m by 4 m.

Total area = (8 x 3) + (5 x 4) = 24 + 20 = 44 m²

Worked Example 11

A running track consists of a rectangle 100 m by 60 m with a semicircle on each short end.

Area of rectangle = 100 x 60 = 6000 m² Each semicircle has diameter 60 m, so radius = 30 m. Area of each semicircle = 1/2 x pi x 30² = 450pi Total area = 6000 + 2 x 450pi = 6000 + 900pi = 8827.4 m² (1 d.p.)

Perimeter = 2 x 100 + 2 x (1/2 x pi x 60) = 200 + 60pi = 388.5 m (1 d.p.)

---

Common Mistakes and Misconceptions

• Using slant height instead of perpendicular height for parallelograms and triangles.
• Forgetting to halve the diameter to get the radius for circle calculations.
• Confusing circumference and area formulas. Remember: circumference is linear (units like cm); area is square (units like cm²).
• Forgetting to add the two straight edges when finding the perimeter of a sector.
• Not giving units — always include cm, cm², m, m², etc.

---

Edexcel Exam Tips