GRE Circles, Quadrilaterals and Polygons

Circles, quadrilaterals and polygons make up the flat-shape half of GRE Geometry. GRE Quantitative Reasoning covers both topics inside its Geometry content area, alongside lines, angles, triangles, 3-D figures and the Pythagorean theorem. The formula list is short and fixed, so most lost points come from a bad setup rather than unfamiliar math.

This guide covers circle area, circumference, arc length and sector area, inscribed and central angles, quadrilateral area formulas and special-case properties, and polygon interior and exterior angle sums. Worked examples run through every section. The last section handles the two setup traps costing points most often: mixing up radius and diameter, and forcing one formula onto an irregular shape. Practice both topics with our drills at three difficulty levels.

Where Circles and Polygons Sit in GRE Geometry

GRE Quantitative Reasoning organizes content into four areas: arithmetic, algebra, geometry and data analysis. Read the full breakdown in our GRE Quantitative Reasoning overview. The Geometry content area covers circles, quadrilaterals and other polygons, plus triangles, parallel and perpendicular lines, congruent and similar figures, 3-D figures, area, perimeter, volume, the Pythagorean theorem and angle measurement in degrees.

Two related guides round out GRE Geometry. Our guide to coordinate geometry, lines, angles and triangles covers the point-and-line half of the content area, and our guide to solids and volume covers the 3-D half. Proof construction is not tested on the GRE, and trigonometry and calculus are excluded too, so the formulas below cover the full extent of what a question asks you to apply.

Circle Formulas: Area, Circumference, Arc Length and Sector Area

Every circle formula builds from the radius, the distance from the center to the edge. The diameter, twice the radius, shows up directly in the circumference formula and in problems giving you the width of a circle instead of its radius. Read each problem for which of the two it states before you write anything down.

Area and Circumference

A circle's area equals pi times the radius squared: A = pi r squared. The circumference, the distance around the circle, equals 2 times pi times the radius, or pi times the diameter: C = 2 pi r = pi d. Take a circle with a radius of 5. Its area is pi times 5 squared, or 25 pi, and its circumference is 2 pi times 5, or 10 pi.

Arc Length and Sector Area

An arc is a piece of the circle's edge, and a sector is the pie-slice piece of the circle's area between two radii. Both scale by the same fraction: the central angle divided by 360 degrees. A 90-degree central angle cuts off a quarter of the circumference and a quarter of the area. The two rules below apply the fraction to the full-circle results.

• Arc length equals the central angle divided by 360, times the circumference.
• Sector area equals the central angle divided by 360, times the area.

Take a circle with a radius of 6 and a central angle of 60 degrees. The full circumference is 2 pi times 6, or 12 pi, so the arc length is 60 divided by 360, times 12 pi, or 2 pi. The full area is pi times 6 squared, or 36 pi, so the sector area is 60 divided by 360, times 36 pi, or 6 pi. The fraction 60 over 360 reduces to one sixth, and both answers land at one sixth of the full-circle values.

The table below collects the four circle formulas in one place. Each row assumes you already have the radius. Halve the diameter first when a problem states one instead.

Inscribed and Central Angles

A central angle has its vertex at the circle's center, with two radii as its sides, and its measure always equals the measure of the arc it cuts off. An inscribed angle has its vertex on the circle itself, with two chords as its sides, and it measures half the central angle cutting the same arc, so it also measures half the arc. Take a central angle of 80 degrees. It cuts off an arc measuring 80 degrees, and any inscribed angle with its vertex on the far arc and its sides passing through the same two endpoints measures half of 80, or 40 degrees.

One special case is worth memorizing. An inscribed angle cutting off a semicircle, with its two endpoints sitting on a diameter, always measures 90 degrees, no matter where the vertex sits on the circle. A diameter splits the circle into two 180-degree arcs, and half of 180 is 90. Spotting a diameter as one side of an inscribed triangle turns the figure into a right triangle, which brings the Pythagorean theorem into play, covered in our guide to coordinate geometry, lines, angles and triangles.

Quadrilateral Area Formulas and Special-Case Properties

Four quadrilaterals appear in the official GRE Math Review geometry chapter: the rectangle, the square, the parallelogram and the trapezoid. Add the rhombus, a parallelogram with four equal sides, and you have the full set worth memorizing. Each shape carries its own area formula plus its own rules for sides, angles and diagonals.

Area Formulas by Shape

Every area formula below reduces to a base times a height, once you know where the height sits. The height runs perpendicular to the base, never along the slanted side of a parallelogram or a trapezoid. The table below pairs each shape with its formula.

The rhombus row carries a caveat. The official Math Review lists area formulas for the rectangle, the square, the parallelogram and the trapezoid, and a rhombus is a parallelogram, so base times height already covers it. The half-product of the diagonals is a correct shortcut for the case where both diagonals are known, not a separate ETS formula. Reach for it when a problem hands you the diagonals, and fall back to base times height otherwise.

Special-Case Properties

Beyond area, each quadrilateral carries fixed rules about its sides, angles and diagonals. Questions built on these rules ask for a missing angle or a diagonal length with no area calculation at all. The table below lists the defining properties of each shape.

These categories nest rather than sitting side by side. A square satisfies every rule listed for the rectangle and the rhombus, and both of those are parallelograms, so a square is a parallelogram too. The ETS definition asks for at least one pair of parallel sides in a trapezoid, so every parallelogram, rectangle, rhombus and square qualifies as a trapezoid as well. On a quantitative comparison question, a figure called a trapezoid rules nothing out: its legs need not be unequal, its angles need not differ, and the second pair of sides is free to be parallel. Treat the extra properties as unknown, and expect answer D unless the problem pins the figure down.

Polygon Interior and Exterior Angle Sums

Every polygon's interior angles follow one formula, based only on the number of sides. A polygon with n sides splits into (n - 2) triangles, and each triangle contributes 180 degrees, so the interior angles sum to (n - 2) times 180 degrees. A regular polygon spreads the sum evenly across its vertices, so each interior angle measures (n - 2) times 180, divided by n. A pentagon has an interior sum of 3 times 180, or 540 degrees, and each angle of a regular pentagon measures 540 divided by 5, or 108 degrees.

The table below lists the interior angle sum and the size of each interior angle in a regular version of six common polygons. The sum rises by 180 degrees with every side you add. Regular means every side and every angle is equal.

Exterior angles follow a simpler rule. The exterior angles of any convex polygon, one at each vertex, always sum to 360 degrees, no matter how many sides it has. In a regular polygon, each exterior angle measures 360 divided by n, and it pairs with its interior angle to fill a straight line of 180 degrees. A regular octagon has 8 sides, so each exterior angle measures 360 divided by 8, or 45 degrees, and each interior angle measures 180 minus 45, or 135 degrees, matching the table above.

Common Traps on Circle and Polygon Questions

A small set of setup errors accounts for most missed points across circle and polygon questions. Two show up more than the rest: swapping the radius for the diameter, and reaching for a single formula on a shape with none. Both land before the arithmetic starts, so rechecking your calculation never catches them.

Radius vs Diameter

Problems state a circle's diameter as often as its radius, and every area formula above needs the radius. Plugging a diameter straight into the area formula, without halving it first, produces an answer four times too large. Write the radius on its own line before you touch a formula. The three checks below keep the two measurements apart.

• Halve a stated diameter before using the area formula, A = pi r squared.
• Circumference accepts either measurement directly, since C = 2 pi r = pi d, so confirm which one a problem states before substituting.
• A radius of 8 and a diameter of 8 produce different areas: 64 pi versus 16 pi.

Irregular Polygon Area

Irregular polygons, shapes with no single area formula, turn up as L-shapes, arrow shapes and combined figures. Split one into rectangles, triangles or trapezoids, shapes with a known formula, then add the pieces for a combined region. The other route subtracts a missing piece from a larger complete shape, which is faster when the figure looks like a full rectangle with a bite taken out of it.

Take an L-shaped figure made from a 10 by 8 rectangle with a 4 by 3 rectangle cut from one corner. The full rectangle's area is 10 times 8, or 80, and the missing corner is 4 times 3, or 12, so the L-shape's area is 80 minus 12, or 68. Splitting the same figure into a 10 by 5 rectangle and a 6 by 3 rectangle gives 50 plus 18, or 68 again. Pick whichever route needs fewer measurements from the figure.

We offer three difficulty levels of drills for Circles and for Quadrilaterals & Polygons on our GRE practice page, along with full-length practice tests and Quant section practice. Every drill set is free, with no signup required. Start at the easy level to lock the formulas in, then move up once your setup errors drop to zero.

Sources

• ETS: GRE General Test, Quantitative Reasoning content preparation
• ETS: GRE Math Review (PDF)