GRADE 8 Mathematics MEASUREMENTS – Circles Notes
MEASUREMENTS — Circles
Subject: Mathematics | Target age: 13 (Kenya)
1. What is a circle?
A circle is the set of all points in a plane that are the same distance from a fixed point called the centre. The distance from the centre to any point on the circle is the radius.
2. Parts of a circle (simple names)
- Centre (O)
- Radius (r): from centre to the circle
- Diameter (d): passes through centre; d = 2r
- Circumference: the distance around the circle
- Chord: a straight line joining two points on the circle
- Arc: part of the circumference between two points
- Sector: area between two radii and the arc
- Tangent: a line that touches the circle at exactly one point
3. Useful constants
π (pi) is used in circle formulae. For school work use either π ≈ 3.14 or π = 22/7 (choose the one the question asks for).
4. Important formulae
- Circumference: C = 2πr = πd
- Area of circle: A = πr²
- Arc length (for angle θ in degrees): arc = (θ/360) × 2πr
- Area of sector (angle θ in degrees): sector area = (θ/360) × πr²
- Diameter and radius: d = 2r, r = d/2
5. Visual — labels on a circle
6. Worked examples
If r = 7 cm, find the circumference and area. Use π = 22/7.
Circumference C = 2πr = 2 × (22/7) × 7 = 44 cm.
Area A = πr² = (22/7) × 7 × 7 = 154 cm².
If d = 10 cm, find circumference. Use π ≈ 3.14.
C = πd ≈ 3.14 × 10 = 31.4 cm.
For a circle with r = 14 cm and central angle θ = 60°:
Arc length = (θ/360) × 2πr = (60/360) × 2π × 14 = (1/6) × 28π = 28π/6 ≈ 14.67 cm (using π = 22/7 gives 14 2/3 cm).
Sector area = (θ/360) × πr² = (60/360) × π × 14² = (1/6) × π × 196 ≈ 32.55 × π? (use π value); using π = 22/7 gives (1/6) × (22/7) × 196 = (1/6) × 616 = 102.67 cm² approx.
7. Helpful tips
- Always check whether the question wants π = 22/7, π = 3.14, or an exact answer in terms of π.
- Keep units with your answers (cm, m, cm², m²).
- Convert degrees to a fraction of 360 when finding arcs or sectors.
- Use d = 2r to switch between diameter and radius quickly.
8. Practice questions (try these)
- Find the circumference and area of a circle with radius 5 cm (use π = 3.14).
- A wheel has diameter 0.8 m. Find the distance it moves in one full turn (circumference). (Give answer to 2 dp.)
- Find the length of an arc of a circle with radius 21 cm and angle 30°. Use π = 22/7.
- Find the area of a sector with radius 10 cm and central angle 120°. Use π = 3.14.
- Given area of a circle is 154 cm² with π = 22/7. Find the radius.
9. Answers to practice
- C = 2πr = 2 × 3.14 × 5 = 31.4 cm. A = πr² = 3.14 × 25 = 78.5 cm².
- d = 0.8 m so C = πd ≈ 3.14 × 0.8 = 2.512 m → 2.51 m (to 2 d.p.).
- Arc = (30/360) × 2πr = (1/12) × 2 × (22/7) × 21 = (1/12) × 924/7? Simpler: (1/12)×2×22×3 = (1/12)×132 = 11 cm.
- Sector area = (120/360) × π × 10² = (1/3) × 3.14 × 100 = 104.67 cm² (approx).
- A = πr² = 154 and π = 22/7. So r² = 154 ÷ (22/7) = 154 × 7 / 22 = 49 ⇒ r = 7 cm.
End of notes — practise more sector and arc questions to get confident. If you want printable notes or more worked examples, tell me which area you find hardest.