GRE Circles
Circles are formula-driven, which is good news: the list is short and it does not change. Know the two formulas, treat arcs and sectors as fractions of the whole, and remember two angle facts, and circle questions become quick.
The two formulas
Everything builds on the radius r. The diameter is d = 2r, the circumference is C = 2(pi)r = (pi)d, and the area is A = (pi)r^2. The GRE usually wants answers in terms of pi, so do not rush to multiply it out. The most common slip is plugging the diameter into a formula that wants the radius, so label r before you compute.
Arcs and sectors: a fraction of 360
An arc is a piece of the circumference and a sector is a wedge of the area, and both are the same fraction of the whole circle: the central angle divided by 360. So arc length = (angle / 360) x 2(pi)r and sector area = (angle / 360) x (pi)r^2. A 90-degree slice is one quarter; a 60-degree slice is one sixth. Set up the fraction first, then the rest is plug-in.
Two angle facts worth points
- Semicircle makes a right angle: a triangle inscribed in a circle with the diameter as one side has a 90-degree angle opposite the diameter.
- Radius meets tangent at 90 degrees: a tangent line touches the circle at exactly one point and is perpendicular to the radius there.
Both facts turn a circle problem into a right-triangle problem, where the Pythagorean theorem takes over.
Circles with squares
A frequent setup is a circle inscribed in a square (the circle's diameter equals the square's side) or a square inscribed in a circle (the square's diagonal equals the circle's diameter). Identify which length is shared and the rest follows.
Frequently asked questions
What are the circle formulas I must know for the GRE?
Circumference C = 2(pi)r = (pi)d, and area A = (pi)r^2, where r is the radius and d = 2r is the diameter. Nearly every circle question comes down to these two plus the idea that an arc or sector is a fraction of the whole.
How do I find an arc length or a sector area?
Both are a fraction of the full circle, and the fraction is the central angle over 360. Arc length = (angle / 360) x 2(pi)r. Sector area = (angle / 360) x (pi)r^2. A 90-degree sector is one quarter of the circle.
What is the semicircle right-angle rule?
Any angle inscribed in a semicircle is a right angle. If a triangle is drawn inside a circle with one side as the diameter, the angle opposite that diameter is 90 degrees. This turns many circle problems into right-triangle problems.
Where do people lose points on circles?
Mixing up radius and diameter (using d where the formula needs r doubles or halves the answer), and confusing circumference with area. Read which one is given, and note GRE figures are not drawn to scale.
Drill circles with real GRE questions
Grezi turns circle geometry into interactive lessons and adaptive practice, with an AI tutor that walks you through the setup, not just the final number.
More GRE Quant: all topics, triangles, formula sheet.