Circles, arcs and sectors.
Use a central-angle fraction of the circle, then count the complete boundary when required.
Calculate circumference and circle area · find minor and major arcs and sectors · give exact π answers
Use radius, diameter and π correctly
A circle’s radius runs from its centre to the boundary. Its diameter crosses the whole circle through the centre and is twice the radius. Circumference names the distance around the circular boundary.
The constant π links circumference to diameter for every circle. Area instead measures the enclosed surface and depends on the radius squared. Decide which part of the circle is being measured before choosing a formula, and convert a given diameter to a radius where needed.
The radius is half the diameter. Keep π in an exact answer when requested; otherwise retain calculator precision until the final rounding step.
A circle has diameter 14 cm. Find its circumference and area in terms of π.
Radius r = 7 cm.
Area = π × 7² = 49π cm²
Pause and explainA circle has diameter 10 cm. What radius belongs in its area formula?
5 cm. The radius is half the diameter.
A sector takes the same fraction of the whole circle
A sector is a slice of a circle bounded by two radii and an arc. If the central angle is 90°, the slice is one quarter of a full 360° turn.
That same fraction applies to both the circumference’s arc and the circle’s area, because the slice takes the same fraction of the complete turn. First calculate the fraction of the circle, then apply it to the relevant whole-circle measure. The two results have different units.
Sector area = θ360° × πr²
The angle θ is at the centre and is measured in degrees here. These formulas are supplied in the List of formulas.
Arc length: 60°360° × 12π = 2π cm
Sector area: 60°360° × 36π = 6π cm²
Sector perimeter: 12 + 2π cm
Find the arc length and sector area for r = 9 cm and θ = 80°.
Area = 80360 × 81π = 18π cm²
Pause and explainWhy is a 180° sector half the circle’s area?
Its angle is half of 360°, so it contains half the equal-angle slices that make up the full circle.
Major sectors and complete boundaries
A minor sector uses the smaller central angle, while the major sector uses the rest of the full turn. The perimeter of either wedge includes straight radii as well as the curved arc.
Sketch the selected region and trace its entire boundary with a pencil. This prevents answering an arc-length question when a perimeter is requested. If the given angle names the minor sector but the question asks for the major one, subtract it from 360° before calculating.
A radius-6 cm circle has a minor sector of 60°. Find the major sector’s area and perimeter.
Major angle = 360° − 60° = 300°.
Major arc = 300360 × 12π = 10π cm
Major-sector perimeter = 10π + 6 + 6 = 10π + 12 cm
Find the perimeter of a semicircle of radius 5 cm.
The curved part alone is 5π cm, but the complete perimeter also includes the 10 cm straight edge.
Pause and explainDoes the perimeter of a sector equal just its arc length?
No. Add the two radii as well, unless the shape’s boundary is described differently.
Your turn. Decide which boundary is required.
Label radius, diameter and the required central angle on a sketch. Decide whether the answer needs circumference, area, arc length or sector perimeter.
Keep π exact if requested, and delay decimal rounding until the final result. Trace the boundary when finding perimeter so that straight edges are not forgotten.
Find the circumference and area of a radius-4 cm circle in terms of π.
Reveal worked answer
A sector has radius 10 cm and angle 72°. Find its arc length and area.
Reveal worked answer
Find the perimeter of the sector in the previous question.
Reveal worked answer
A circle of radius 3 cm has a minor angle of 120°. Find the area of the corresponding major sector.