Circles
Circles are fundamental shapes in geometry, appearing everywhere from the wheels of a car to the orbits of planets. Understanding their properties is crucial for many areas of mathematics and science.
This guide covers key definitions, circle properties, angle theorems, tangent rules, arc length, sector area, equations of circles, worked examples, and a practice quiz.
1Introduction
A circle is the set of all points in a plane that are equidistant from a fixed central point. Circles are everywhere in the real world and form the basis for understanding arcs, angles, and many geometric relationships.
Mastering circles means knowing their vocabulary (radius, diameter, chord, arc, sector), core formulas (circumference, area), angle theorems (central, inscribed), tangent properties, and the algebraic equation of a circle in the coordinate plane.
A pizza has a 14-inch diameter. What is the length of crust on one slice if the pizza is cut into 8 equal pieces? That is an arc-length problem, and by the end of this guide you will solve it in seconds.
Real-World Uses
Engineering & Design
Gears, wheels, pipes, and bearings all depend on circle geometry for precision.
Navigation & GPS
Satellites use circles and arcs to triangulate positions on Earth.
Architecture
Arches, domes, and rotundas use circular geometry for structural strength.
Sports & Recreation
Running tracks, Ferris wheels, and dartboards are all based on circles.
2Key Definitions
Parts of a Circle
Circle
The set of all points in a plane equidistant from a fixed center point.
Center (C)
The fixed central point from which all points on the circle are equidistant.
Radius (r)
A segment from the center to any point on the circle. All radii are equal.
Diameter (d)
A segment through the center connecting two points on the circle. d = 2r.
Chord
A segment connecting any two points on the circle. The diameter is the longest chord.
Secant
A line that intersects a circle at two distinct points. It contains a chord.
Tangent
A line that touches the circle at exactly one point (the point of tangency).
Arc
A continuous portion of the circumference. Minor arc < 180°, major arc > 180°.
Sector
A "pizza slice" region bounded by two radii and the arc between them.
Segment
A region bounded by a chord and the arc it subtends.
3Properties of Circles
- All radii of a circle are congruent.
- A diameter divides a circle into two semicircles.
- Perpendicular Chord Theorem: If a diameter is perpendicular to a chord, it bisects the chord and its intercepted arc.
- Congruent Chords: In the same circle, congruent chords are equidistant from the center and subtend congruent arcs.
Core Formulas
Circumference
C = 2πr = πd
Distance around the circle. Linear units (cm, m).
Area
A = πr²
Space enclosed by the circle. Square units (cm², m²).
Key Relationship
d = 2r
The diameter is always twice the radius.
4Angles in Circles
Angles formed by chords, tangents, and secants have specific relationships with the arcs they intercept.
Central Angle vs Inscribed Angle
A. Central Angles
A central angle has its vertex at the center and its sides are radii. Its measure equals the intercepted arc.
Central angle = 60°
Intercepted arc = 60°
B. Inscribed Angles
An inscribed angle has its vertex on the circle and its sides are chords. Its measure is half the intercepted arc.
Intercepted arc = 100°
Inscribed angle = ½ × 100° = 50°
Special Cases
Angle in a Semicircle
An inscribed angle that intercepts a semicircle (180° arc) is always 90°.
Same Arc Angles
Inscribed angles intercepting the same arc are congruent.
Cyclic Quadrilateral
Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
C. Tangent-Chord Angle
An angle formed by a tangent and a chord at the point of tangency equals half the intercepted arc — the same rule as inscribed angles.
5Tangent Properties
Tangent-Radius Theorem
A tangent is perpendicular (90°) to the radius at the point of tangency. This often creates right triangles for Pythagorean theorem problems.
Two-Tangent Theorem
Two tangent segments drawn from the same external point are congruent. If PA and PB are tangents from point P, then PA = PB.
Tangent-Radius Right Angle
Whenever you see a tangent and a radius meeting, you have a right angle. This means you can use the Pythagorean theorem to find missing lengths. It is one of the most frequently tested relationships in circle geometry.
6Arc Length & Sector Area
Both arc length and sector area are fractions of the full circle, determined by the central angle.
Arc Length (degrees)
L = (θ/360) × 2πr
In radians: L = rθ
Result is a linear measurement (cm, m).
Sector Area (degrees)
A = (θ/360) × πr²
In radians: A = ½r²θ
Result is a square measurement (cm², m²).
The fraction θ/360 tells you "what portion of the full circle am I dealing with?" Then you multiply that fraction by the full circumference (for arc length) or full area (for sector area).
7Equations of Circles
A circle can be represented algebraically in the coordinate plane.
Standard Form (Center-Radius Form)
(x − h)² + (y − k)² = r²
where (h, k) is the center and r is the radius
At the Origin
x² + y² = r²
Special case when center is (0, 0).
General Form
Ax² + Ay² + Dx + Ey + F = 0
Use completing the square to convert to standard form.
Reading the Equation
Given: (x + 1)² + (y − 5)² = 81
x − h = x + 1 → h = −1
y − k = y − 5 → k = 5
r² = 81 → r = 9
Center: (−1, 5), Radius: 9
Write the equation for center (−2, 3) and radius 4:
(x − (−2))² + (y − 3)² = 4²
(x + 2)² + (y − 3)² = 16
8Worked Examples
Example 1: Circumference from Diameter
A circle has a diameter of 14 cm. Find its radius and circumference.
r = d / 2 = 14 / 2 = 7 cm
C = πd = π(14)
C = 14π ≈ 43.98 cm
Example 2: Central & Inscribed Angles
Arc AB = 80°. Find the central angle AOB and inscribed angle ACB.
Central angle AOB = arc AB = 80°
Inscribed angle ACB = ½ × arc AB
ACB = ½ × 80° = 40°
Example 3: Tangent Length (Pythagorean Theorem)
Tangent AB from external point A to circle with center O at point B. OB = 5, AO = 13. Find AB.
OB ⊥ AB (tangent-radius theorem)
OB² + AB² = AO² (Pythagorean theorem)
5² + AB² = 13²
25 + AB² = 169
AB² = 144
AB = 12 units
Example 4: Arc Length & Sector Area
Circle with r = 6 cm, central angle = 120°. Find arc length and sector area.
L = (120/360) × 2π(6)
L = (1/3) × 12π
L = 4π ≈ 12.57 cm
A = (120/360) × π(6)²
A = (1/3) × 36π
A = 12π ≈ 37.70 cm²
Example 5: Equation of a Circle
Write the equation of a circle with center (3, −4) and radius 6.
(x − h)² + (y − k)² = r²
(x − 3)² + (y − (−4))² = 6²
(x − 3)² + (y + 4)² = 36
9Common Mistakes
Confusing central and inscribed angles
A central angle equals its arc; an inscribed angle is half its arc. Always check where the vertex is.
Mixing up radius and diameter
Most formulas use the radius. If the problem gives you the diameter, divide by 2 first.
Wrong units on answers
Arc length is in linear units (cm), but sector area is in square units (cm²). Circumference is linear, area is square.
Forgetting the Pythagorean theorem for tangent problems
A tangent perpendicular to a radius creates a right triangle. Use a² + b² = c² to find missing sides.
Sign errors in circle equations
In (x − h)² + (y − k)² = r², note the minus signs. If you see (x + 2)², that means h = −2, not 2.
Quick Revision Summary
- ✓Diameter = 2 × Radius — always convert before using formulas.
- ✓Circumference: C = 2πr = πd (distance around).
- ✓Area: A = πr² (space inside).
- ✓Central angle = intercepted arc. Inscribed angle = ½ intercepted arc.
- ✓Angle in a semicircle is always 90°.
- ✓Tangent ⊥ Radius at the point of tangency — use Pythagorean theorem.
- ✓Two tangents from the same external point are congruent.
- ✓Arc length: L = (θ/360) × 2πr. Sector area: A = (θ/360) × πr².
- ✓Standard equation: (x − h)² + (y − k)² = r², center (h, k), radius r.
- ✓Cyclic quadrilateral: opposite angles are supplementary (sum to 180°).
Frequently Asked Questions
- What's the difference between a chord and a secant?
- A chord is a line segment whose endpoints are on the circle. A secant is a line that passes through the circle, intersecting it at two points. So a chord is part of a secant line.
- When do I use 2πr and when do I use πr²?
- Use 2πr (or πd) to find the circumference (the distance around the circle). Use πr² to find the area (the space inside the circle). Remember: circumference is in linear units (cm, m) and area is in square units (cm², m²).
- How do I know if an angle is central or inscribed?
- Check the vertex. If the vertex of the angle is at the center of the circle, it's a central angle. If the vertex is on the circle (and its sides are chords), it's an inscribed angle.
- What is π (pi)?
- Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It is an irrational number, approximately 3.14159, and is used in almost all circle calculations.
- What if I'm given the general form of a circle's equation?
- If you have something like x² + y² − 4x + 6y − 3 = 0, you need to use completing the square for both x and y terms to convert it into standard form (x − h)² + (y − k)² = r². This lets you identify the center (h, k) and radius r.
Practice Quiz
Test your understanding of circles — select the correct answer for each question.
1.Radius is half of:
2.A line that touches a circle at exactly one point is called a:
3.Find the area of a circle with radius 4.
4.Find the circumference of a circle with radius 3.
5.Which formula gives the arc length (in degrees)?
6.By the inscribed angle theorem, an inscribed angle equals:
7.What is the equation of a circle with center (0, 0) and radius 6?
8.A tangent line is always perpendicular to the:
9.Find the sector area when r = 6 and the central angle is 60°.
10.A chord that passes through the center of a circle is called a:
Final Study Advice
- 1.Always check whether you are given the radius or diameter before plugging into a formula.
- 2.Draw a diagram for every problem — label the center, radii, chords, and angles.
- 3.When you see a tangent, immediately think right angle and Pythagorean theorem.
- 4.For inscribed angle problems, always find the intercepted arc first, then halve it.
- 5.Watch the signs when reading circle equations: (x + 2) means h = −2.