Circle Essentials

Explore the geometry of circles: radius, diameter, circumference, area, sectors, arcs, and tangent lines.

Geometry

Beginner

~45 min

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1. What is a Circle?

A circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the center).

Distance from center to any point on the circle.

Distance across the circle through the center. $d = 2r$

A line segment connecting two points on the circle.

A portion of the circumference (curved part of the circle).

A "pizza slice" region bounded by two radii and an arc.

Region between a chord and its arc.

2. Circumference

C = 2\pi r = \pi d

The circumference is the perimeter of a circle.

Pi is the ratio of circumference to diameter for any circle: $\pi = \frac{C}{d} \approx 3.14159...$

Common approximations: 3.14, 22/7, or use the π button on your calculator.

3. Area

A = \pi r^2

The area enclosed by a circle.

If $r = 5$:

A = \pi(5)^2 = 25\pi \approx 78.54

A_{semi} = \frac{1}{2}\pi r^2

4. Sectors and Arcs

s = r\theta

where θ is in radians

s = \frac{\theta°}{360°} \times 2\pi r

where θ is in degrees

A = \frac{1}{2}r^2\theta

where θ is in radians

A = \frac{\theta°}{360°} \times \pi r^2

where θ is in degrees

5. Tangent Lines

A tangent line touches the circle at exactly one point.

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

radius ⊥ tangent at the point of contact

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