Circles
Circles show up in geometry questions worth real points — and they almost always come down to a handful of formulas plus one clever trick involving right triangles. Learn those, and these questions become free.
The radius runs from the center to the edge; circumference wraps around, area fills inside.
A circle in the plane: center (h, k) and radius r define its equation (x-h)² + (y-k)² = r².
The chord trick: radius is the hypotenuse, half the chord and the center-distance are the legs.
Less often, you'll see arcs (part of the circumference) and sectors (a pizza-slice region). For these, set up a proportion using the central angle out of 360°:
- Arc length
= (angle/360) × 2πr - Sector area
= (angle/360) × πr²
Master the four core formulas, the standard-form equation, and the chord-to-right-triangle move, and you've covered nearly every circle question on the test.
Check your understanding with a question from this topic:
In a circle with radius 10, a chord is 16 units long. What is the distance from the center of the circle to the chord?
Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).
Worked examples
A circle has a diameter of 14. What is the area of the circle?
A circle has center (3, -2) and radius 5. Which of the following is the equation of the circle?
In a circle with radius 10, a chord is 16 units long. What is the distance from the center of the circle to the chord?
Common pitfalls
Area is πr², so a given diameter must be halved before squaring. Squaring the diameter first inflates your answer by a factor of 4 — and that wrong value is usually one of the choices.
In (x-h)² + (y-k)² = r², the number on the right is the radius squared. If it says = 36, the radius is 6, not 36. Reverse this when going from radius to equation.
The formula subtracts h and k. A center of (-4, 5) gives (x + 4)² + (y - 5)². Always flip the sign of each coordinate when writing the equation.
The right-triangle leg is half the chord, not the whole chord, because the perpendicular from the center bisects it. Using the full chord length wrecks the Pythagorean setup.
Key takeaways
Memorize the four basics: d = 2r, C = 2πr, A = πr². Find the radius first, always.
Circle equation: (x - h)² + (y - k)² = r², where (h, k) is the center and the right side is r² (signs of h and k flip).
For chords, draw the radius and a perpendicular from the center to form a right triangle: radius² = (half chord)² + (distance)².
Arc length and sector area both use the fraction angle/360 of the full circle.
Watch & learn
Curated Khan Academy walkthroughs on Circles. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.
Try it yourself
5 practice questions on Circles, drawn from the question bank. The tutor is one click away if you get stuck.