Circles Geometry: Definitions, Formulas, Properties, and Examples

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Updated: Dec 15, 2024

Lesson

Circles are fundamental geometric shapes with wide-ranging applications. Their properties are essential in fields like architecture, physics, and engineering.

A circle's constant ratio of circumference to diameter (π) is crucial for calculations involving arcs, areas, and volumes. The principles of circle geometry help in various applications like designing of structures, analysis of circular motion, and optimization of manufacturing processes.

Introduction to Circles in Geometry

A circle is formed by all the points in a plane that are the same distance away from a single, central point. This central point is called the center of the circle.

The distance from the center to any point on the circle is called the radius. The radius determines the size of the circle: a larger radius creates a larger circle.

Parts of a Circle

Circles have different parts with special names that help us understand and measure them.

Radius (r): The distance from the center to any point on the circle.

Diameter (d): A line segment passing through the center with endpoints on the circle. The diameter is twice the radius (d = 2r).

Center: The fixed point equidistant from all points on the circle.

Chord: A line segment with endpoints on the circle. The diameter is the longest chord.

Secant: A line that intersects the circle at two points.Tangent: A line that intersects the circle at exactly one point.

Arc: A portion of the circle's circumference.

Sector: A region bounded by two radii and an arc. Think of it like a slice of pie.

Segment: A region bounded by a chord and an arc.

Annulus: The region between two concentric circles (circles that share the same center).

Take this quiz -

Identify the different parts of a circle

Circles Formulas

Circles have special formulas that help us measure their size and different parts.

Property	Formula	Explanation
Circumference (C)	C = 2πr or C = πd	The distance around the circle.
Area (A)	A = πr²	The space enclosed within the circle.
Diameter (d)	d = 2r	A line segment through the center with endpoints on the circle.
Arc Length (s)	s = (θ/360°) × 2πr	The length of a portion of the circumference, where θ is the central angle in degrees.
Area of a Sector	Area of Sector = (θ/360°) × πr²	The area of a "slice" of the circle, where θ is the central angle in degrees.
Length of a Chord (c)	c = 2r sin(θ/2)	The distance between two points on the circle, where θ is the central angle in degrees.
Area of a Segment	Area of Segment = (θ/360°) × πr² - (1/2)r²sinθ	The area bounded by a chord and an arc, where θ is the central angle in degrees.

Take this quiz -

Circle-identify radius and diameter.

Properties of Circles

Circles possess unique properties that govern their angles, chords, and relationships with other shapes. These properties are essential for solving geometric problems and applying circle concepts in various contexts.

Property	Description
Angle in a Semicircle	An angle inscribed in a semicircle is always a right angle (90 degrees).
Cyclic Quadrilateral	If a quadrilateral can be inscribed in a circle (all vertices on the circle), the opposite angles are supplementary (add up to 180 degrees).
Tangent-Chord Angle	The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.
Intersecting Chords	When two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.
Equal Chords	Chords that are equidistant from the center of a circle are equal in length.
Concentric Circles	Circles with the same center but different radii are called concentric circles.

Take this quiz -

Circumference and Area of Circle Quiz! Trivia

Examples of Circle Calculations

1. A bicycle wheel has a radius of 30 cm. What is the circumference of the wheel?

Solution:

We are given the radius (r) = 30 cm.
Recall the formula for circumference: C = 2πr
Substitute the values into the formula: C = 2 * π * 30 cm
Calculate the result: C ≈ 2 * 3.14159 * 30 cm ≈ 188.5 cm

Therefore, the circumference of the bicycle wheel is approximately 188.5 cm

2. A circular garden has a diameter of 10 meters. Calculate the area of the garden.

Solution:

We are given the diameter (d) = 10 m.
Calculate the radius: r = d/2 = 10 m / 2 = 5 m
Recall the formula for the area of a circle: A = πr²
Substitute the values into the formula: A = π * (5 m)²
Calculate the result: A ≈ 3.14159 * 25 m² ≈ 78.54 m²

Therefore, the area of the circular garden is approximately 78.54 square meters.

3. The circumference of a circular pond is 88 meters. Find the radius of the pond.

Solution:

We are given the circumference (C) = 88 m.
Recall the formula for circumference: C = 2πr
Rearrange the formula to solve for the radius: r = C / (2π)
Substitute the values into the formula: r = 88 m / (2 * π)
Calculate the result: r ≈ 88 m / (2 * 3.14159) ≈ 14 m

Therefore, the radius of the circular pond is approximately 14 meters.

4. A circular mirror has an area of 154 square centimeters. What is the diameter of the mirror?

Solution:

We are given the area (A) = 154 cm².
Recall the formula for the area of a circle: A = πr²
Rearrange the formula to solve for the radius: r² = A / π => r = √(A / π)
Substitute the values into the formula: r = √(154 cm² / π)
Calculate the result: r ≈ √(154 cm² / 3.14159) ≈ 7 cm
Calculate the diameter: d = 2r = 2 * 7 cm = 14 cm

Therefore, the diameter of the circular mirror is approximately 14 centimeters.

5. A sector of a circle with a radius of 12 cm has a central angle of 60 degrees. Find the length of the arc.

Solution:

We are given the radius (r) = 12 cm and the central angle (θ) = 60°.
Recall the formula for arc length: s = (θ/360°) × 2πr
Substitute the values into the formula: s = (60°/360°) × 2 * π * 12 cm
Calculate the result: s ≈ (1/6) × 2 * 3.14159 * 12 cm ≈ 12.57 cm

Therefore, the length of the arc is approximately 12.57 cm.5. A sector of a circle with a radius of 12 cm has a central angle of 60 degrees. Find the length of the arc.

Solution: