Circles – The Perfect Shape
The circle is one of the most perfect shapes in mathematics and nature. From wheels to planets, circles appear everywhere. Knowing the parts of a circle and how to calculate its measurements is essential geometry.
Parts of a Circle
| Part | Definition |
|---|---|
| Centre | The fixed point at the middle, equidistant from every point on the circle |
| Radius (r) | Distance from the centre to any point on the circle |
| Diameter (d) | A chord through the centre; d = 2r |
| Chord | A line segment joining any two points on the circle |
| Arc | A portion of the circumference (boundary curve) |
| Sector | A pie-slice region bounded by two radii and an arc |
| Segment | The region between a chord and the arc it cuts off |
| Tangent | A line that touches the circle at exactly one point |
Circumference and Area
Circumference (perimeter) = 2πr = πd. Area = πr². Use π ≈ 3.14159 or leave answers in terms of π.
A circle has radius 7 cm. Find its circumference and area.
Circumference = 2 × π × 7 = 14π ≈ 43.98 cm. Area = π × 7² = 49π ≈ 153.94 cm².
A circle has diameter 12 cm. Find its circumference.
r = 6 cm. Circumference = 2 × π × 6 = 12π ≈ 37.70 cm.
Arc Length and Sector Area
| Measurement | Formula |
|---|---|
| Arc length | (θ / 360) × 2πr, where θ is the angle in degrees |
| Sector area | (θ / 360) × πr² |
Find the arc length and sector area for a 90° sector of a circle with radius 10 cm.
Arc length = (90/360) × 2π(10) = ¼ × 20π = 5π ≈ 15.71 cm. Sector area = (90/360) × π(100) = 25π ≈ 78.54 cm².
Circle Theorems (Key Rules)
- A tangent meets a radius at exactly 90°.
- The angle at the centre is twice the angle at the circumference (for the same arc).
- Angles in a semicircle are always 90°.
- Opposite angles in a cyclic quadrilateral sum to 180°.
Key Takeaways
- Circumference = 2πr; Area = πr².
- A sector is a slice of a circle; its area and arc length use the angle fraction (θ/360).
- A tangent to a circle is always perpendicular to the radius at the point of contact.
- Angles in a semicircle = 90° (a fundamental circle theorem).
Practice Questions
- Find the circumference and area of a circle with radius 5 cm. Leave answers in terms of π.
- A circular track has diameter 100 m. How far is one lap?
- Find the arc length of a 120° sector with radius 9 cm.
- Find the area of a 45° sector with radius 8 cm.
- A circle has circumference 31.4 cm. Find its radius (use π = 3.14).