Cambridge Lower Secondary Stage 7, Unit 8. Learn the parts of a circle, how to calculate circumference and area, work with sectors, test congruence, and find symmetry in polygons!
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Watch each part of the circle light up in turn!
Parts of a Circle
Radius (r)
Distance from the centre to any point on the circumference.
Diameter (d)
Distance across the circle through the centre. d = 2r
Circumference
The perimeter of the circle — the total distance around the outside.
Chord
A straight line joining two points on the circle (not through the centre).
Arc
A portion of the circumference between two points.
Sector
A "slice of pie" — bounded by two radii and an arc.
Tangent
A straight line that touches the circle at exactly one point (at 90° to the radius).
Segment
The region between a chord and the arc it cuts off.
Key Formulas
Circumference: C = πd = 2πr
Area: A = πr²
Arc Length: θ/360 × 2πr
Sector Area: θ/360 × πr²
Congruence Conditions
SSS
Three sides equal
SAS
Two sides and the included angle equal
ASA
Two angles and the included side equal
RHS
Right angle, hypotenuse, and one other side equal
Symmetry of Polygons
Lines of symmetry are mirror lines. Order of rotational symmetry is how many times the shape looks the same in one full 360° rotation.
Equilateral Triangle
3 lines · order 3
Square
4 lines · order 4
Rectangle
2 lines · order 2
Rhombus
2 lines · order 2
Regular Pentagon
5 lines · order 5
Regular Hexagon
6 lines · order 6
Isosceles Triangle
1 line · order 1
Parallelogram
0 lines · order 2
Circle Properties
All radii of a circle are equal in length.
Equal chords are the same distance from the centre.
The perpendicular from the centre to a chord bisects the chord (cuts it in half).
A tangent is perpendicular to the radius at the point of contact.
📝 Worked Examples
Study each example carefully before attempting the exercises. Use π = 3.14159... or leave answers in terms of π.
Example 1: Find the Circumference (given diameter)
Question: A circle has diameter 14 cm. Find its circumference. (Give your answer to 2 decimal places.)
Step 1 — Write the formula: C = πd
Step 2 — Substitute: C = π × 14
Step 3 — Calculate: C = 43.98 cm (2 d.p.)
Answer: C = 43.98 cm
Example 2: Find the Area (given radius)
Question: A circle has radius 6 cm. Find its area to 2 decimal places.
Step 1 — Write the formula: A = πr²
Step 2 — Substitute: A = π × 6² = π × 36
Step 3 — Calculate: A = 113.10 cm² (2 d.p.)
Answer: A = 113.10 cm²
Example 3: Find the Arc Length of a Sector
Question: A sector has radius 8 cm and angle 90°. Find the arc length to 2 decimal places.
Step 1 — Formula: Arc length = θ/360 × 2πr
Step 2 — Substitute: = 90/360 × 2 × π × 8
Step 3 — Simplify: = 1/4 × 16π = 4π
Step 4 — Calculate: = 12.57 cm (2 d.p.)
Answer: Arc length = 12.57 cm
Example 4: Find the Sector Area
Question: A sector has radius 10 cm and angle 72°. Find the sector area to 2 decimal places.
Step 1 — Formula: Sector area = θ/360 × πr²
Step 2 — Substitute: = 72/360 × π × 10²
Step 3 — Simplify: = 1/5 × 100π = 20π
Step 4 — Calculate: = 62.83 cm² (2 d.p.)
Answer: Sector area = 62.83 cm²
🔍 Circle Explorer — Visualizer
Radius
8 cm
Diameter
16 cm
Circumference
50.27 cm
Area
201.06 cm²
Drag the slider to change the radius and watch everything update live!
🏿 Ex 1: Circle Parts Labeller
Click a word from the word bank, then click the matching hotspot on the circle diagram to label it.
Word Bank — click a word, then click its hotspot:
Radius
Diameter
Circumference
Chord
Arc
Sector
Tangent
Segment
🔄 Ex 2: Circumference Calculator
Calculate the circumference. Give answers to 2 decimal places. Use C = πd or C = 2πr.
🍎 Ex 3: Area Calculator
Calculate the area of the circle. Give answers to 2 decimal places. Use A = πr².
🍕 Ex 4: Sector Challenge
Find arc length or sector area. Give answers to 2 decimal places. Hint: Arc = θ/360 × 2πr | Sector Area = θ/360 × πr²
△ Ex 5: Congruence Checker
For each pair of triangles, decide if they are CONGRUENT (and select the rule) or NOT CONGRUENT.
✨ Ex 6: Symmetry Sorter
For each shape, enter (a) the number of lines of symmetry and (b) the order of rotational symmetry.
📝 Practice Questions
Grab your paper and pencil! Give circumference and area answers to 2 d.p. unless told otherwise. Use π ≈ 3.14159.
Name the part of a circle that is the distance from the centre to the edge.
A circle has diameter 20 cm. What is its radius?
Find the circumference of a circle with diameter 7 cm. (2 d.p.)
Find the circumference of a circle with radius 9 cm. (2 d.p.)
Find the area of a circle with radius 5 cm. (2 d.p.)
Find the area of a circle with diameter 12 cm. (2 d.p.)
A sector has radius 6 cm and angle 60°. Find its arc length. (2 d.p.)
A sector has radius 10 cm and angle 120°. Find its arc length. (2 d.p.)
A sector has radius 8 cm and angle 45°. Find its area. (2 d.p.)
A sector has radius 5 cm and angle 90°. Find its area. (2 d.p.)
Two triangles have sides 5 cm, 7 cm, 9 cm each. Are they congruent? Which rule?
Triangle A has sides 6 cm and 8 cm with an included angle of 40°. Triangle B also has sides 6 cm and 8 cm with an included angle of 40°. Are they congruent? Which rule?
How many lines of symmetry does a regular hexagon have?
What is the order of rotational symmetry of a rectangle?
How many lines of symmetry does a parallelogram have?
A chord is drawn in a circle. The perpendicular from the centre meets the chord. What is true about the two halves of the chord?
What is the relationship between a tangent and the radius at the point of contact?
Find the circumference of a circle with radius 3.5 cm. (2 d.p.)
Find the area of a circle with radius 7 cm. (2 d.p.)
A sector has radius 12 cm and angle 150°. Find its area. (2 d.p.)
Show all your workings on paper! Use π ≈ 3.14159 or leave in terms of π.
A bicycle wheel has a diameter of 70 cm. How far does the bicycle travel when the wheel completes 50 full rotations? Give your answer in metres to 2 d.p.
A circular pizza has a circumference of 94.25 cm. Find the radius of the pizza to 2 d.p. (Use C = 2πr, rearrange to find r.)
A circular swimming pool has an area of 50.27 m². Find the radius of the pool to 2 d.p. (Rearrange A = πr².)
A garden has a circular lawn with radius 5 m. A path of width 1 m surrounds it. Find the area of the path only. Give your answer to 2 d.p.
A clock face has radius 15 cm. The minute hand sweeps through a sector with angle 120° in 20 minutes. Find (a) the arc length swept and (b) the area of the sector. (2 d.p.)
Two triangles: Triangle A has a right angle, hypotenuse 13 cm, and one leg 5 cm. Triangle B has a right angle, hypotenuse 13 cm, and one leg 5 cm. Are they congruent? State the rule and explain why.
A regular octagon has 8 equal sides and 8 equal angles. State (a) the number of lines of symmetry and (b) the order of rotational symmetry of a regular octagon. Explain your reasoning.
A sector of a circle has arc length 15.71 cm and radius 10 cm. Find the angle of the sector. (Use arc = θ/360 × 2πr.)
A semicircular rug has diameter 2.4 m. Find the total area of the rug to 2 d.p. (A semicircle is half a full circle.)
A composite shape is made by joining a rectangle (length 10 cm, width 6 cm) to a semicircle at one of its shorter ends. Find the total area of the composite shape to 2 d.p.
Circumference per rotation: C = π × 70 = 219.91 cm. Distance = 50 × 219.91 = 10995.57 cm = 109.96 m
C = 2πr ⇒ r = C/(2π) = 94.25 / (2π) = 94.25 / 6.2832 = 15.00 cm
A = πr² ⇒ r² = A/π = 50.27 / π = 16. So r = √16 = 4.00 m
Large circle (r=6): A = π × 36 = 113.10 m². Lawn (r=5): A = π × 25 = 78.54 m². Path = 113.10 − 78.54 = 34.56 m²
Yes, congruent by RHS (Right angle, Hypotenuse 13 cm, and one side 5 cm are all equal in both triangles).
(a) A regular octagon has 8 lines of symmetry (one through each pair of opposite vertices and one through midpoints of opposite sides). (b) Order of rotational symmetry is 8 (it looks the same every 45°).