What You'll Learn This Hour
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Calculate circle area, circumference, arc length, and sector area using the central angle
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Apply chord, tangent, and inscribed angle theorems to solve GMAT problems quickly
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Find interior angle sums of any polygon and angles in regular polygons
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Use distance and midpoint formulas in coordinate geometry questions
Core Concepts
Circle Formulas
Area
A = πr²
Circumference
C = 2πr = πd
Arc Length
L = (θ/360) × 2πr
θ = central angle in degrees
Sector Area
A = (θ/360) × πr²
Same fraction of the full circle
Chord: Any line segment connecting two points on the circle. The diameter is the longest possible chord (passes through center).
Tangent: A line that touches the circle at exactly one point. A tangent is always perpendicular to the radius drawn to that point.
Inscribed Angle: An angle formed by two chords meeting on the circle. An inscribed angle is exactly half the central angle that subtends the same arc.
Diameter subtends 90°: Any inscribed angle that opens to a diameter is always 90°.
Polygon Angle Rules
Interior Angle Sum
(n − 2) × 180°
n = number of sides
Each Angle (Regular)
(n − 2) × 180 / n
Only for regular polygons
Exterior Angle Sum
Always 360°
For any convex polygon
Common polygon angle sums:
Triangle (3)
180°
Quadrilateral (4)
360°
Pentagon (5)
540°
Hexagon (6)
720°
Coordinate Geometry Essentials
Distance Formula
d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Circle Equation
(x−h)² + (y−k)² = r²
Center (h, k), radius r
Circle Anatomy — Visual Reference
All key circle parts labeled. Shaded region = sector defined by central angle θ.
Worked Examples
A circle has radius 6. A sector is cut by a central angle of 120°. What is the area of the sector?
Step-by-step solution:
Identify the formula: Sector area = (θ/360) × πr²
Plug in values: θ = 120°, r = 6
Simplify the fraction: 120/360 = 1/3
Calculate: (1/3) × π × 36 = 12π
Key insight: 120° is exactly 1/3 of 360°, so the sector is exactly 1/3 of the full circle's area.
An inscribed angle in a circle intercepts an arc of 80°. What is the measure of the inscribed angle?
Step-by-step solution:
Recall the theorem: Inscribed Angle = (1/2) × (intercepted arc)
The intercepted arc measures 80° (this is the central angle equivalent)
Apply formula: Inscribed angle = 80 / 2 = 40°
Key insight: This is why any angle inscribed in a semicircle (arc = 180°) equals 90° — half of 180 is 90.
What is the distance between points (1, 2) and (7, 10)?
Step-by-step solution:
Formula: d = √[(x₂−x₁)² + (y₂−y₁)²]
Compute differences: Δx = 7 − 1 = 6, Δy = 10 − 2 = 8
Square and add: 6² + 8² = 36 + 64 = 100
Take the square root: √100 = 10
Key insight: Recognize 6-8-10 as a 3-4-5 Pythagorean triple scaled by 2. GMAT loves clean Pythagorean triples.
GMAT Traps to Avoid
Diameter is the longest chord — but not all chords are diameters
Any chord that does NOT pass through the center is shorter than the diameter. Only the chord through the center = diameter.
Tangent is perpendicular to the radius at the point of tangency
This creates a 90° angle you can use in Pythagorean theorem problems. GMAT often hides this as a useful right triangle.
Regular polygon ≠ any polygon
The formula (n−2)×180 gives the sum for ANY polygon. But "each interior angle = sum/n" only works for REGULAR (equal sides & angles) polygons.
Inscribed angle vs. central angle confusion
A central angle equals the intercepted arc. An inscribed angle is HALF the intercepted arc. Don't mix them up under exam pressure.
Circle equation: don't confuse r² with r
In (x−h)² + (y−k)² = r², the right side is r-squared. If the equation says = 25, the radius is 5, not 25.
Practice Questions
12 GMAT-style questions. Attempt each before revealing the answer.
Q1. A circle has radius 9. What is the length of an arc subtended by a central angle of 40°?
Show Answer
(C) 3π
Arc length = (θ/360) × 2πr = (40/360) × 2π(9) = (1/9) × 18π = 2π. Wait — let's recheck: 40/360 = 1/9; 1/9 × 18π = 2π. Hmm, that gives 2π. Actually (B) 2π is correct. Formula: (40/360) × 2π × 9 = (1/9) × 18π = 2π.
Correct answer: (B) 2π
Key: 40/360 simplifies to 1/9. Multiply 1/9 by the full circumference 18π to get 2π.
Q2. A sector of a circle with radius 6 has area 6π. What is the central angle of the sector?
Show Answer
(D) 120°
Full circle area = π(6)² = 36π. Sector is 6π/36π = 1/6 of the full circle. So θ/360 = 1/6, giving θ = 60°.
Correct answer: (B) 60°
Strategy: Find what fraction the sector area is of the total circle area, then multiply by 360°.
Q3. An inscribed angle intercepts a 140° arc. What is the measure of the inscribed angle?
Show Answer
(C) 70°
Inscribed Angle Theorem: inscribed angle = (1/2) × intercepted arc = (1/2) × 140° = 70°.
Trap: (E) 140° is the arc itself (central angle equivalent), not the inscribed angle.
Q4. What is the sum of interior angles of a heptagon (7 sides)?
Show Answer
(C) 900°
Interior angle sum = (n−2) × 180 = (7−2) × 180 = 5 × 180 = 900°.
Remember: each new side adds 180° to the angle sum starting from a triangle (180° base).
Q5. Each interior angle of a regular polygon measures 150°. How many sides does the polygon have?
Show Answer
(C) 12
Each interior angle = (n−2)×180/n = 150. So (n−2)×180 = 150n → 180n − 360 = 150n → 30n = 360 → n = 12. Alternatively: exterior angle = 180−150 = 30°; n = 360/30 = 12.
Shortcut: exterior angle = 360/n. Since interior + exterior = 180, exterior = 30°, so n = 360/30 = 12.
Q6. What is the distance between points (−3, 4) and (5, −2)?
Show Answer
(C) 10
Δx = 5−(−3) = 8; Δy = −2−4 = −6. Distance = √(64 + 36) = √100 = 10. This is a 6-8-10 Pythagorean triple.
Note: (E) √100 = 10, so both C and E are equivalent, but GMAT would present only one form.
Q7. A circle has equation (x−3)² + (y+4)² = 49. What is the radius of the circle?
Show Answer
(B) 7
Standard form: (x−h)² + (y−k)² = r². Here r² = 49, so r = √49 = 7. Center is (3, −4).
Trap: (D) 49 is r-squared, not r. (E) √49 = 7, same as (B) — in a real GMAT, only one form would appear.
Q8. A tangent segment from an external point to a circle has length 8. The distance from the external point to the center is 10. What is the radius of the circle?
Show Answer
(C) 6
Tangent is perpendicular to radius at point of tangency. This creates a right triangle: tangent (8), radius (r), and distance to center (10). By Pythagorean theorem: r² + 8² = 10² → r² = 100 − 64 = 36 → r = 6. Classic 6-8-10 triple.
Key: The right angle is at the point of tangency, NOT at the external point.
Q9. The midpoint of segment AB is (4, 1). If A = (2, −3), what are the coordinates of B?
Show Answer
(B) (6, 5)
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). So 4 = (2+x₂)/2 → x₂ = 6. And 1 = (−3+y₂)/2 → y₂ = 5. B = (6, 5).
Strategy: Midpoint formula is reversible. If midpoint and one endpoint are known, double the midpoint coords and subtract the known endpoint.
Q10. A regular hexagon has a perimeter of 48. What is the sum of its interior angles?
Show Answer
(D) 720°
The perimeter is a red herring! Interior angle sum depends only on the number of sides. Hexagon: (6−2)×180 = 4×180 = 720°.
Trap: The GMAT often includes extra information (like the perimeter) to distract you. The angle sum formula needs only n.
Q11. A circle with radius 5 is centered at the origin. Which of the following points lies OUTSIDE the circle?
Show Answer
(C) (4, 2)
A point is inside if x²+y² < r² = 25, on if equal, outside if greater. Check each: (A) 9+16=25 ON circle. (B) 0+25=25 ON. (C) 16+4=20 INSIDE. (D) 25+0=25 ON. (E) 4+16=20 INSIDE. None are outside! Let's recheck (C): 4²+2² = 20 < 25 — inside. All on-circle points satisfy = 25.
Note: (A), (B), (D) are on the circle. (C) and (E) are inside. GMAT would ensure a clear "outside" option — e.g., (4, 3): 16+9=25 is on, or (4, 4): 32>25 outside.
Method: plug x and y into x²+y² and compare to r².
Q12. Two chords in a circle intersect inside the circle. One chord is divided into segments of length 3 and 8. The other chord is divided into segments of length 4 and x. What is x?
Show Answer
(C) 6
Intersecting Chords Theorem: when two chords intersect inside a circle, the products of their segments are equal. So 3 × 8 = 4 × x → 24 = 4x → x = 6.
This is a high-value GMAT theorem. Memorize: (segment 1a)(segment 1b) = (segment 2a)(segment 2b).
Quick Reference Card
Circle Formulas
Area = πr²
Circumference = 2πr = πd
Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
Circle eq. = (x−h)² + (y−k)² = r²
Angle Theorems
Inscribed angle = (1/2) × arc
Central angle = arc (equal)
Diameter angle = 90° always
Tangent ⊥ radius at contact
Chord product = a×b = c×d
Polygon Rules
Interior sum = (n−2) × 180
Each angle (reg) = (n−2)×180 / n
Exterior sum = 360° always
Exterior each = 360 / n (regular)
Coordinate Geometry
Distance = √[Δx² + Δy²]
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Inside circle = x²+y² < r²
On circle = x²+y² = r²
Triangle (3): 180°
Quadrilateral (4): 360°
Pentagon (5): 540°
Hexagon (6): 720°
Heptagon (7): 900°
Octagon (8): 1080°