Work with radius, diameter, circumference, and area, plus the angle relationships formed inside a circle — central angles, inscribed angles, and arcs all connect back to the same 360° whole. Every question comes with a written walkthrough of the relationship used.
C = 2πr and A = πr² both need the radius specifically — plugging a given diameter directly into either formula without halving it first doubles or quadruples the result.
An inscribed angle is half its intercepted arc, not equal to it — that's the rule for a central angle, a completely different angle sharing the same arc.
A sector's arc length and area both scale with its own central angle out of 360°, not with any other angle that happens to appear in the same figure.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
A circle has a diameter of 18 cm. Find its circumference. (Use π ≈ 3.14)
A — 56.52 cm. Circumference is π times the diameter: 3.14 × 18 = 56.52 cm. 28.26 cm (B) comes from halving the diameter to get the radius and then using C = πr instead of C = 2πr — dropping the necessary factor of 2.
A sector of a circle with radius 10 cm has a central angle of 72°. Find the arc length. (Use π ≈ 3.14)
A — 12.56 cm. The arc's fraction of the full circle is 72°/360° = 1/5. Multiply that by the full circumference: (1/5) × 2π(10) = (1/5)(62.8) = 12.56 cm. 62.8 cm (B) is the full circumference — the result of forgetting to apply the 72°/360° fraction at all.
In a circle, an inscribed angle intercepts an arc of 84°. A central angle in the same circle intercepts an arc that is twice as large as the inscribed angle's arc. Find the measure of the central angle.
A — 168°. A central angle equals its intercepted arc directly. The inscribed angle's arc is 84°, so the central angle's arc — twice as large — is 168°, and the central angle itself measures 168°. 84° (B) comes from applying the inscribed-angle rule (half the arc) to the central angle by mistake, halving 168° instead of leaving it as the direct arc measure.
Circle problems are mostly about tracking which measurement — radius, diameter, arc, or angle — a given formula actually wants, since one wrong substitution changes the whole answer.
Label every given measurement as either a radius or a diameter before touching a formula. Untimed practice is where that labeling habit sticks.
Move to timed sessions once radius/diameter tracking and the central/inscribed angle rules are automatic.
Pair circles with area and perimeter in a mock — composite figures frequently combine both.
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