GRE · Geometry · TOPIC LESSON
Translate radii, turns and shaded regions into exact lengths and areas.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GRE General Test study; it does not reproduce official questions, certify a writing score or predict test content.
LEARN THROUGH ORIGINAL PRACTICE
All teaching passages, writing prompts, examples and exercises in this lesson are original. No official ETS questions, prompts or explanations are reproduced. Current structure and observed public sample metadata are separate from this original teaching; no official score or question probability is implied.
01 / UNDERSTAND & APPLY
All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.
A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.
A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.
02 / UNDERSTAND & APPLY
A circle is the set of points in a plane at a fixed distance r from its center. That distance is the radius. A diameter passes through the center and joins two points on the circle, so its length is d = 2r. A chord joins any two points on the circle; a chord that does not pass through the center is not a diameter.
An arc is part of the circumference. A sector is the region enclosed by two radii and an intervening arc, like a slice reaching the center. A segment is enclosed by a chord and an arc. A sector area formula does not directly give a segment area; a triangle must usually be subtracted.
Ordinary circle figures are not necessarily to scale. A point that looks like the center or a line that appears tangent is not enough; rely on labels and conditions. Once a center is specified, every segment from it to the circle is a radius even if the sketch draws those segments at different apparent lengths.
03 / UNDERSTAND & APPLY
Circumference is the full boundary length C = 2πr = πd. The enclosed area is A = πr². Doubling the radius doubles the circumference but multiplies the area by four. This difference is dimensional: one formula contains one power of length and the other contains two.
If C is known, first solve r = C/(2π), then use that radius for area. If A is known, solve r = √(A/π), taking the positive root. Keep π symbolic until approximation is requested. A calculator is useful for a final rounded measurement but should not replace the radius–diameter decision.
ORIGINAL WORKED EXAMPLE
A circular patio has diameter 14 m. Find its circumference and area.
Answer Circumference 14π m; area 49π m².
ORIGINAL WORKED EXAMPLE
A circle has circumference 18π cm. Find its area.
Answer 81π cm².
04 / UNDERSTAND & APPLY
A central angle has its vertex at the circle’s center. For a specified arc with central angle θ degrees, the fraction of a full turn is θ/360, and its length is (θ/360)2πr. This applies to a minor arc with θ < 180, a semicircle with θ = 180, or a major arc with θ > 180, as long as θ refers to the chosen arc.
Between two endpoints there are generally two arcs. If one has central angle 80°, the other has 280°, and their lengths add to the circumference. An arc length is not its chord length: a straight chord cuts across the interior, while an arc follows the curve.
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ORIGINAL WORKED EXAMPLE
A circle has radius 12 cm. Find the length of an arc whose central angle is 75°.
Answer 5π cm.
05 / UNDERSTAND & APPLY
The sector determined by a central angle θ has area (θ/360)πr². The same turn fraction appears in arc length and sector area, but one multiplies circumference and the other multiplies full-circle area. Sector perimeter includes the curved arc and the two bounding radii; its value is arc length + 2r for a proper sector with two radius edges.
If the fraction of the circle is given directly, such as three eighths, use that fraction instead of first converting to degrees. Conversely, divide the sector area by πr² to recover the circle fraction and multiply by 360 to recover its central angle.
ORIGINAL WORKED EXAMPLE
A sector of radius 10 cm has central angle 72°. Find its area and perimeter.
Answer Area 20π cm²; perimeter 20 + 4π cm.
06 / UNDERSTAND & APPLY
A radius to a point of tangency is perpendicular to the tangent line. This can expose a right triangle with a point outside the circle. Do not assume a line is tangent just because the sketch touches the curve in one visible place; tangency must be given or established.
An inscribed angle has its vertex on the circle and sides along chords. It measures half the central angle subtending the same intercepted arc, or equivalently half that arc’s degree measure. In particular, an inscribed angle intercepting a diameter is 90° because its intercepted arc is a semicircle. Keep track of which arc is intercepted; it is the arc opposite the inscribed vertex.
Equal chords in the same circle cut off equal minor arcs. A perpendicular from the center to a chord bisects the chord, so half the chord, its distance from the center, and a radius form a right triangle. These are geometric relationships, not reasons to measure a chord on the drawing.
ORIGINAL WORKED EXAMPLE
O is the center of a circle with radius 5 cm. Point P lies outside the circle, OP = 13 cm, and PT is tangent at T. Find PT.
Answer 12 cm.
07 / UNDERSTAND & APPLY
An annulus is the region between concentric circles. If outer radius R and inner radius r satisfy R > r ≥ 0, its area is π(R² − r²). The radial width is R − r, but π(R − r)² is not the area of the ring. If total boundary length is requested, include both circumferences: 2πR + 2πr.
For a circle inscribed in a square, the square’s side equals the circle’s diameter. For a square inscribed in a circle, the square’s diagonal equals the diameter. Those are different configurations and produce different relations. “Fits inside” alone does not guarantee tangency to all sides.
For a circular segment, subtract the area of the triangle formed by the two radii and chord from the appropriate sector. A 90° minor segment of radius r has area πr²/4 − r²/2 because the radius triangle is right.
ORIGINAL WORKED EXAMPLE
Two concentric circles have radii 8 m and 5 m. Find the area between them.
Answer 39π m².
08 / UNDERSTAND & APPLY
For two circles, a radius ratio of a:b is also the diameter and circumference ratio a:b. Their area ratio is a²:b². If an area becomes 49/25 times as large, the positive radius factor is √(49/25) = 7/5. A 40% radius increase thus produces a 96% area increase, because 1.4² = 1.96.
Convert length units before using a square. Since 1 m = 100 cm, 1 m² = 10,000 cm². A radius of 30 cm is 0.3 m, whose square is 0.09 m². Do not divide an area in cm² by only 100 to convert it to m².
Numeric entry may ask for the coefficient of π or give a specific approximation such as π ≈ 3.14. Enter exactly what is requested. If rounding is specified, retain extra digits during intermediate calculations and round only at the end.
09 / UNDERSTAND & APPLY
A circle’s diameter is its longest chord. All other chords are shorter, but their lengths vary with their distance from the center. A radius alone does not determine an arbitrary chord or an arbitrary sector. For QC, ask which feature can move while the radius remains fixed.
A fixed circumference does determine both radius and area. Similarly, two circles with equal areas have equal positive radii. Unknown numerical values do not imply that their relationship is unknown.
ORIGINAL WORKED EXAMPLE
QC. A circle has radius 5. Quantity A: the length of a chord. Quantity B: 8.
Answer D: the relationship cannot be determined.
10 / UNDERSTAND & APPLY
Use the QC key from the first section. Do not substitute a decimal for π unless the item asks for one.
ORIGINAL PRACTICE · 1
81
The radius is 9, so area is 81π. The requested coefficient is 81.
ORIGINAL PRACTICE · 2
B. 20π
r² = 100 gives r = 10, so C = 2πr = 20π.
ORIGINAL PRACTICE · 3
6
The arc is one third of the circumference 18π, giving 6π.
ORIGINAL PRACTICE · 4
8
The fraction is 45/360 = 1/8, and the full area is 64π. The sector area is 8π.
ORIGINAL PRACTICE · 5
A, C and D
Diameter, circumference, and fixed-angle arc length scale linearly with radius; area scales with its square. Thus B confuses the two scale powers.
11 / UNDERSTAND & APPLY
Use the wording to identify central angles, tangent lines, and inscribed shapes before applying their special relationships.
ORIGINAL PRACTICE · 1
6
The curved half-circumference is 6π and the straight diameter is 12. The coefficient k is 6.
ORIGINAL PRACTICE · 2
C
Equal values of 2πr give equal radii, which give equal values of πr².
ORIGINAL PRACTICE · 3
D
A 45° sector has area 2π, below 4π; a 180° sector has area 8π, above it. Both satisfy the given radius.
ORIGINAL PRACTICE · 4
B. 100 − 25π
The square’s side is the diameter, so r = 5. Subtract disk area 25π from square area 100.
ORIGINAL PRACTICE · 5
67
The inscribed angle is half its intercepted arc measure: 134/2 = 67°.
ORIGINAL PRACTICE · 6
0.785
Area is 3.14×(0.5)² = 3.14×0.25 = 0.785 m². Squaring the diameter would give four times too much.
KEEP THE BIG IDEAS
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