Rule
The central angle subtended by an arc is twice any inscribed angle subtended by the same arc.
Angles in the same segment, standing on the same arc, are equal.
An angle inscribed in a semicircle is a right angle ().
Example 1 — central vs inscribed angle
If (inscribed), find (central, same arc ).
- Central angle inscribed angle .
Answer:
Rule
Opposite angles of a cyclic quadrilateral (all four vertices on one circle) add up to .
Rule
A tangent touches a circle at exactly one point, and is always perpendicular to the radius at that point.
From an external point, exactly two tangents can be drawn to a circle, and they are equal in length.
From the centre of a circle, no tangent can be drawn (the centre is inside).
Arc length ; sector area ( in radians)
Example 2 — finding the radius from the arc length
If cm and , find .
- Convert to radians: .
- .
Answer: cm
Example 3 — sector from perimeter and area
A sector has perimeter cm and radius cm. Find its arc length and area.
- Perimeter cm.
- rad.
- Area .
Answer: cm, area cm
Quick Recap
- ●Central angle inscribed angle on the same arc; angles in the same segment are equal
- ●Angle in a semicircle
- ●Cyclic quadrilateral: opposite angles sum to
- ●Tangent radius at the point of contact; two equal tangents from an external point
- ●; sector area
