Circles
Circle questions reward knowing which angle or segment relationship applies. Central angles equal their intercepted arcs. An inscribed angle equals half its intercepted arc, and inscribed angles that intercept the same arc are congruent. An angle inscribed in a semicircle is a right angle, and opposite angles of a cyclic quadrilateral are supplementary. For angles formed by chords, secants, and tangents: an angle with its vertex inside the circle equals half the sum of the intercepted arcs; an angle with its vertex outside the circle equals half the difference of the intercepted arcs; a tangent and a chord form an angle equal to half the intercepted arc.
For segments, remember: a tangent is perpendicular to the radius at the point of tangency; two tangents from an external point are congruent; when two chords intersect, the products of the segments are equal (a times b = c times d); when two secants or a secant and a tangent meet outside, the product of the external part and the whole secant is equal for both.
Circle measurement: circumference = 2 times pi times r; area = pi times r squared. Arc length = (central angle / 360) times circumference; sector area = (central angle / 360) times area. The equation of a circle centered at (h, k) with radius r is (x - h)^2 + (y - k)^2 = r^2.
Traps: using the arc rather than half the arc for an inscribed angle, mixing up arc length with sector area, and forgetting to complete the square when a circle equation is given in expanded form.
Worked example. A sector has a central angle of 90 degrees in a circle of radius 6. Arc length = (90/360)(2 pi times 6) = (1/4)(12 pi) = 3 pi. Sector area = (1/4)(36 pi) = 9 pi.