Circle Geometry
Explore the parts of a circle and calculate circumference, arc length, and sector area using key formulas.
Parts of a Circle
Understanding the terminology of circles is essential before applying formulas. Each part has a specific name and mathematical definition.
Key parts of a circle labelled.
Radius (r)
Distance from the centre to any point on the circle. The diameter = 2r.
Diameter (d)
A chord passing through the centre. It is the longest chord: d = 2r.
Arc
A portion of the circumference (the curved line) between two points on the circle.
Sector
A "pizza slice" region bounded by two radii and an arc. Like a slice of pie.
Circumference
The circumference is the perimeter of the circle — the total length of the boundary. It is calculated using the constant π (pi ≈ 3.14159).
Using radius
C = 2πr
Using diameter
C = πd
Tip: Use the π button on your calculator for accuracy. Only round your final answer, not intermediate steps.
Arc Length and Sector Area
An arc and a sector are both fractions of the full circle. If the sector angle is θ degrees, then the fraction of the circle is θ/360.
Arc Length
l = (θ/360) × 2πr
Sector Area
A = (θ/360) × πr²
Quick example: sector with r = 6 cm, θ = 90°
Arc length = (90/360) × 2π × 6 = (1/4) × 12π = 3π ≈ 9.42 cm
Sector area = (90/360) × π × 36 = (1/4) × 36π = 9π ≈ 28.27 cm²
Key Vocabulary
| Term | Definition |
|---|---|
| Circumference | The perimeter of a circle. C = 2πr or C = πd. |
| Arc | A portion of the circumference between two points on the circle. |
| Sector | The region bounded by two radii and an arc; a "pizza slice" of the circle. |
| Chord | A straight line segment joining two points on the circle. The diameter is the longest chord. |
Worked Examples
Circumference from Radius
A circle has a radius of 7 cm. Find the circumference. (Use π ≈ 3.14)
Formula: C = 2πr
Substitute: C = 2 × 3.14 × 7 = 43.96 cm
Answer: C ≈ 43.96 cm
Arc Length
Find the arc length of a sector with radius 10 cm and angle 120°.
Formula: l = (θ/360) × 2πr
Substitute: l = (120/360) × 2 × π × 10
Calculate: l = (1/3) × 20π = 20π/3 ≈ 20.94 cm
Sector Area
A sector has radius 5 m and angle 72°. Find its area.
Formula: A = (θ/360) × πr²
Substitute: A = (72/360) × π × 25
Calculate: A = (1/5) × 25π = 5π ≈ 15.71 m²
Knowledge Check
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Key Concepts Summary
- ●Circumference: C = 2πr = πd. The perimeter of the circle.
- ●Area of circle: A = πr².
- ●Arc length: l = (θ/360) × 2πr. A fraction of the circumference.
- ●Sector area: A = (θ/360) × πr². A fraction of the full circle area.
- ●Diameter = 2 × radius. Always use the π key on your calculator for accurate answers.