Circles Cheat Sheet

A printable reference covering circle circumference, area, arcs, sectors, chords, tangents, secants, and circle equations for grades 9-12.

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Geometry: Circles covers the measurements, angles, lines, and equations connected to circles. Students need this cheat sheet because circle problems often combine diagrams, algebra, and angle relationships in one question. It is useful for reviewing formulas, identifying which theorem applies, and checking work on homework, quizzes, and tests. The most important ideas are radius, diameter, circumference, area, arc length, sector area, and the relationships among chords, tangents, and secants. Central angles, inscribed angles, and intercepted arcs are closely connected. Coordinate geometry adds the standard circle equation $(x - h)^2 + (y - k)^2 = r^2$ , where $(h,k)$ is the center and $r$ is the radius.

Key Facts

The diameter is twice the radius, so $d = 2r$ and $r = \frac{d}{2}$ .
The circumference of a circle is $C = 2\pi r$ or $C = \pi d$ .
The area of a circle is $A = \pi r^2$ .
Arc length is $s = \frac{\theta}{360^\circ} \cdot 2\pi r$ when $\theta$ is measured in degrees.
Sector area is $A = \frac{\theta}{360^\circ} \cdot \pi r^2$ when $\theta$ is measured in degrees.
An inscribed angle equals half its intercepted arc, so $m\angle ABC = \frac{1}{2}m\widehat{AC}$ .
A tangent line is perpendicular to the radius at the point of tangency, so $\text{radius} \perp \text{tangent}$ .
The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$ .

Vocabulary

Radius: A radius is a segment from the center of a circle to any point on the circle.
Diameter: A diameter is a chord that passes through the center of the circle and has length $d = 2r$ .
Chord: A chord is a segment whose endpoints both lie on the circle.
Tangent: A tangent is a line that touches a circle at exactly one point and is perpendicular to the radius at that point.
Secant: A secant is a line that intersects a circle at two points.
Central Angle: A central angle is an angle whose vertex is the center of the circle and whose sides are radii.

Common Mistakes to Avoid

Using diameter instead of radius in area formulas is wrong because $A = \pi r^2$ requires the radius, not the diameter.
Forgetting to square the radius in $A = \pi r^2$ is wrong because area is measured in square units and grows with $r^2$ .
Treating an inscribed angle as equal to its intercepted arc is wrong because an inscribed angle is half the measure of its intercepted arc.
Using $360^\circ$ formulas with radians is wrong because $s = \frac{\theta}{360^\circ} \cdot 2\pi r$ assumes the angle is measured in degrees.
Writing the circle equation with the wrong signs is wrong because $(x - h)^2 + (y - k)^2 = r^2$ has center $(h,k)$ , so $(x + 3)^2$ means the center has $x$ -coordinate $-3$ .

Practice Questions

1 A circle has radius $6\text{ cm}$ . Find its circumference and area in terms of $\pi$ .
2 A sector has radius $10\text{ m}$ and central angle $72^\circ$ . Find the arc length and sector area.
3 Find the center and radius of the circle $(x - 4)^2 + (y + 2)^2 = 49$ .
4 Explain why a tangent line must be perpendicular to the radius drawn to the point of tangency.

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