Circle Basics

Parts of a circle and basic properties

Circle Basics

Definitions

Circle: The set of all points equidistant from a center point.

Radius: Distance from center to any point on the circle (symbol: $r$ )

Diameter: Distance across circle through center (symbol: $d$ ) $d = 2r$

Chord: Line segment connecting two points on the circle

Secant: A line that intersects the circle at two points

Tangent: A line that touches the circle at exactly one point

Key Properties

Tangent Property: A tangent line is perpendicular to the radius at the point of tangency.

Chord Property: A perpendicular from the center to a chord bisects the chord.

Equal Chords: Chords equidistant from the center are congruent.

Circumference

The distance around a circle: $C = 2\pi r = \pi d$

Area

$A = \pi r^2$

Arc Length

For a central angle of $\theta$ degrees: $\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

📚 Practice Problems

1Problem 1easy

❓ Question:

A circle has a radius of 5. Find the circumference and area.

💡 Show Solution

Circumference: $C = 2\pi r = 2\pi(5) = 10\pi$

Area: $A = \pi r^2 = \pi(5)^2 = 25\pi$

Answer: Circumference = $10\pi$ (or ≈ 31.4), Area = $25\pi$ (or ≈ 78.5)

2Problem 2medium

❓ Question:

A circle has diameter 16. Find the length of an arc with central angle $45°$ .

💡 Show Solution

Step 1: Find the radius $r = \frac{d}{2} = \frac{16}{2} = 8$

Step 2: Use arc length formula $\text{Arc length} = \frac{\theta}{360°} \times 2\pi r$

$= \frac{45}{360} \times 2\pi(8)$

$= \frac{1}{8} \times 16\pi$

$= 2\pi$

Answer: Arc length is $2\pi$ (or ≈ 6.28)

3Problem 3hard

❓ Question:

A chord is 8 cm from the center of a circle with radius 10 cm. Find the length of the chord.

💡 Show Solution

Draw a radius to the chord's endpoint and a perpendicular from center to chord.

This creates a right triangle:

Hypotenuse = radius = 10
One leg = distance from center = 8
Other leg = half the chord length

Use Pythagorean Theorem: $8^2 + \left(\frac{c}{2}\right)^2 = 10^2$

$64 + \frac{c^2}{4} = 100$

$\frac{c^2}{4} = 36$

$c^2 = 144$

$c = 12$

Answer: The chord length is 12 cm

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