Circle Angles, Arcs, Sectors, and Arc Length Explained

Central and inscribed angles measure arcs differently

A central angle has its vertex at the center of a circle, so its degree measure equals the measure of its intercepted arc. An inscribed angle has its vertex on the circle and measures half its intercepted arc:

$$m\angle ABC=\frac12m\widehat{AC}.$$

Thus an inscribed angle intercepting a $146^\circ$ arc measures $73^\circ$. Conversely, a $41^\circ$ inscribed angle intercepts an $82^\circ$ arc.

Chords, secants, and tangents locate the vertex

When two chords intersect inside a circle, the angle equals half the sum of the intercepted arcs. When secants or tangents meet outside, the angle equals half the difference of the larger and smaller intercepted arcs. A tangent-chord angle with its vertex on the circle equals half its intercepted arc.

The vertex location—center, circle, inside, or outside—selects the theorem. This decision should come before substituting any numbers.

Cyclic quadrilaterals connect opposite angles

All four vertices of a cyclic quadrilateral lie on one circle. Each pair of opposite angles is supplementary because their intercepted arcs together make a full circle. If one angle is $118^\circ$, the opposite angle is $62^\circ$.

Arc length and sector area use the same fraction

An angle of $\theta$ degrees selects the fraction $\theta/360^\circ$ of a full circle. Multiply that fraction by circumference for arc length and by circle area for sector area:

$$L=\frac{\theta}{360^\circ}(2\pi r), \qquad A=\frac{\theta}{360^\circ}(\pi r^2).$$

For a $72^\circ$ sector with radius 10, the fraction is $1/5$. The arc length is $4\pi$, while the area is $20\pi$. Arc length uses linear units; sector area uses square units.

Keep pi exact until the final line

Leaving answers in terms of $\pi$ preserves exactness and makes formulas easier to audit. If a decimal is required, round only once at the end and state the precision. Differences between a calculator value and a classroom key that uses $3.14$ are rounding differences, not evidence that the circle theorem changed.