Master GRE circle geometry — radius, diameter, circumference, area, arcs and sectors, inscribed and central angles, tangents, and circles in the coordinate plane.
Both use the radius. Halve the diameter first: $r = d/2$.
and
The central angle over $360^\circ$ is the fraction of the whole circle.
inscribed central (same arc)
An inscribed angle on a diameter is $90^\circ$.
radius tangent at the point of contact
Creates a right triangle for the Pythagorean relationship.
Center $(h, k)$; the signs flip. Complete the square from expanded form.
Note: Figure not drawn to scale.
A circle is the set of all points a fixed distance — the radius — from a center. Almost every quantity the GRE asks about is a short computation away from , so the first move on any circle problem is to pin down the radius. The diameter is , and the two most-used formulas both take the radius directly:
The single most common circle error on the GRE is feeding the diameter into a formula that wants the radius. If a circle has diameter , its area is , not — a factor-of-four blunder the answer choices are built to catch. Halve the diameter before you substitute, every time.
An arc is a slice of the circumference; a sector is the pie-shaped region between two radii. Both are just a fraction of the whole circle, and the fraction is the central angle over :
If a sector has central angle , it is exactly one quarter of the circle: one quarter of the circumference for the arc, one quarter of the area for the region. Master this one idea and arcs, sectors, and "slice of a pizza" word problems all collapse into the same two steps: find the fraction, then multiply.
Two kinds of angle sit inside a circle, and confusing them is a classic trap. A central angle has its vertex at the center; it equals the arc it cuts off. An inscribed angle has its vertex on the circle. The key theorem:
Two consequences the test loves. First, an inscribed angle that subtends a diameter is always (the diameter is a arc, and half of is ) — a right triangle appears for free. Second, all inscribed angles standing on the same arc are equal, no matter where their vertex sits on the far side.
A tangent line touches a circle at exactly one point, and the radius drawn to that point is perpendicular to the tangent. That right angle is the whole point: it lets you drop a right triangle onto the figure and reach for the Pythagorean relationship. If a tangent segment runs from an external point to the circle, the radius, the tangent, and the line to the center form a right triangle with the center-to-external-point distance as the hypotenuse.
| Feature | Fact you use |
|---|---|
| Radius to a tangent point | perpendicular to the tangent |
| Two tangents from one external point | equal length |
| Inscribed angle on a diameter | |
| Central vs. inscribed on same arc | central inscribed |
A circle centered at with radius has equation
Read the center off the signs — note they flip: is centered at with radius . When a circle is handed to you in expanded form such as , complete the square in and in to recover the center and radius. Completing the square is the recurring coordinate-circle skill the GRE rewards.
Find the radius first — halving a diameter if necessary. For arcs and sectors, write the central-angle fraction and multiply by the circumference or area. For inscribed-angle questions, remember the halving relationship and look for a diameter that forces a right angle. For tangents, draw the perpendicular radius and hunt for the right triangle. And in the coordinate plane, match the equation to , completing the square when the circle arrives expanded. Nearly every GRE circle question is one of these five moves.
A circle has area . What is its circumference, in terms of ?
A sector of a circle of radius has a central angle of . What is the area of the sector, in terms of ?
Points , , and lie on a circle, and is a diameter. If the inscribed angle at measures , what is the measure of the angle at ?
From an external point , a tangent touches circle at point . The radius and the distance . What is the length of the tangent segment ?
Note: Figure not drawn to scale.
The equation describes a circle in the -plane. What is its radius?
In a circle of radius , an arc has length . What is the measure of the central angle that subtends this arc?
Substituting the diameter where a formula wants the radius, making an area four times too large.
Halve any diameter before substituting: if , then and , never .
Treating an inscribed angle as equal to its arc, or equal to the corresponding central angle.
An inscribed angle is half the central angle on the same arc. If the central angle is , the inscribed angle is .
Reading the center of as and the radius as .
The signs flip: the center is . And the right side is , so , not .
Forgetting that the radius to a tangent point is perpendicular, and so missing the right triangle in tangent problems.
Always draw the radius to the point of contact — it meets the tangent at , giving a right triangle you can solve with the Pythagorean relationship.
A circle has area . What is its circumference, in terms of ?
A sector of a circle of radius has a central angle of . What is the length of the arc that bounds the sector, in terms of ?
A chord of a circle has length .
Quantity A: the length of the circle's radius
Quantity B:
From an external point , a line is tangent to circle at point . If the radius is and , what is the length of the tangent segment ?
The equation describes a circle in the -plane. What is the radius of the circle?
Quantity A: the area of a circle with radius
Quantity B: the area of a sector with central angle in a circle of radius
Master GRE angle relationships — complementary, supplementary, and vertical angles, the straight-line and full-turn sums, and parallel lines cut by a transversal — then chase unknown angles with confidence.
Command GRE triangle and polygon rules — the angle sum, exterior-angle theorem, triangle inequality, area, the Pythagorean special right triangles, similarity, and the interior-angle sum of any n-gon.
Master GRE solid geometry — volume and surface area of boxes, cubes, and cylinders, cone and sphere formulas, the space diagonal of a box, and combining 2D facts inside 3D figures.