Arcs, sectors and circles
A circle is a whole, and every arc and sector question is asking for a piece of that whole. The fraction comes from the central angle. Find the fraction, apply it to the circumference or the area, and the formula you were trying to remember stops mattering.
The two foundations
The circumference is 2πr and the area is πr². The diameter is twice the radius, and a question that gives you the diameter is testing whether you halve it before squaring. Everything else in the topic is built from these two measurements, including the volume of a cylinder, which is the area of a circle times a height.
An arc is a fraction of the circumference
The central angle tells you how much of the circle you have. A 90 degree angle is 90/360, which is one quarter, so the arc is one quarter of the circumference. In general, arc length is the central angle over 360 times 2πr. The same fraction applied to the area gives the sector: central angle over 360 times πr². Write the fraction first, then multiply, because the fraction is usually a friendly number.
Radians are the same fraction
A full circle is 360 degrees, which is 2π radians. Converting is a proportion question, and the reference panel gives you the equality. A question that gives an angle in radians and asks for an arc length still works through the same fraction: the angle over 2π replaces the angle over 360.
The equation of a circle
A circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². The signs invert, which is the single most common slip: (x + 2)² means the center x is -2, and (y − 5)² means the center y is 5. The right side is the radius squared, so a 25 on the right is a radius of 5, not 25.
When the equation is not in that form
If the squared terms are expanded, complete the square for x and for y, and add the same constants to both sides. The equation x² + y² - 6x + 4y = 12 becomes (x - 3)² + (y + 2)² = 25 after adding 9 and 4, so the center is (3, -2) and the radius is 5. A question that gives a point on the circle and asks for the radius is a distance calculation from the center, nothing more.
Angles inside the circle
An inscribed angle is half the central angle that opens onto the same arc. An angle inscribed in a semicircle is a right angle, because the arc is 180 degrees. Both facts turn an angle question into a fraction of the circle, which is the same move as everything else here.
How do I find an arc length?
Take the central angle over 360, then multiply by the circumference. For a 60 degree angle on a circle of radius 6, the fraction is one sixth and the arc is one sixth of 12π, which is 2π.
What is the center of (x + 2)² + (y − 5)² = 16?
The center is (-2, 5) and the radius is 4. The signs inside the brackets flip, and the right side is the square of the radius.
How do I complete the square for a circle equation?
Group the x terms and the y terms, add the square of half of each coefficient to both sides, then write each group as a squared bracket. The constants you added appear on the right, which becomes the radius squared.
Is a chord length proportional to its arc?
No. Equal arcs have equal chords, but a chord is not a fraction of anything, so it cannot be scaled the way an arc length can. Use the triangle formed with the radii instead.
Related guides
Updated 2026-10-25. Free to use, no account needed. SAT is a trademark of College Board. Not affiliated with or endorsed by College Board.