Geometry

Circle Theorems

Circles have a beautiful set of angle and length relationships. In this lesson, we’ll explore central angles, inscribed angles, arc length, and sector area — all interactively.

Central Angle vs. Inscribed Angle

A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle itself. The stunning fact:

An inscribed angle is always half the central angle that subtends the same arc.

Central angle (degrees)80

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\text{Central angle} = 80° \quad \Rightarrow \quad \text{Inscribed angle} = \frac{80°}{2}

The graph below shows both angles on the unit circle. The purple rays form the central angle from the center (0, 0) to two points on the circle. The red rays form the inscribed angle from a point on the circle (-1, 0) to the same two arc endpoints. Notice the inscribed angle is always half the central angle.

Try This

Try this: Set the central angle to 180 degrees. The inscribed angle becomes 90 degrees — that’s Thales’ theorem: any angle inscribed in a semicircle is a right angle!

Arc Length

An arc is a portion of the circle’s circumference. The arc length depends on the radius and the central angle:

s = r \cdot \theta \quad \text{(theta in radians)}

Radius (r)3

Central angle (degrees)90

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r = 3, \quad \theta = 90° = 90 \cdot \frac{\pi}{180} \text{ rad}

s = 3 \cdot 90 \cdot \frac{\pi}{180}

The graph below shows the circle. The red curve is the arc from 0° up to the chosen angle. The cyan vertical line marks where the arc ends on the circle. As you increase the angle, the arc sweeps counterclockwise.

Try This

Try this: Set the angle to 180° — the arc is exactly a semicircle and s = pi * r. Set it to 360° — the arc wraps the full circle and s = 2 * pi * r. For r = 3, the full circumference is about 18.85.

Sector Area

A sector is the “pie slice” region between two radii and an arc. Its area is:

A = \frac{1}{2} r^2 \theta \quad \text{(theta in radians)}

Or equivalently:

A = \frac{\theta}{360°} \cdot \pi r^2

Radius (r)3

Central angle (degrees)90

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A = \frac{90}{360} \cdot \pi \cdot 3^2

The graph shows the sector arc in yellow on the circle. The green vertical line marks where the sector ends. The “pie slice” is the region between the arc, the x-axis radius, and the angled radius.

Connection

Connection: When theta = 360 degrees, the sector area equals the full circle area: pi * r^2. The sector area formula is just a fraction of the circle’s total area.

Tangent Lines

A tangent line touches a circle at exactly one point and is perpendicular to the radius at that point.

Point of tangency (degrees)45

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The graph shows the unit circle with a tangent line at the chosen point. Notice how the tangent is always perpendicular to the radius.

Try This

Try this: Set the tangent point to 90 degrees. The radius points straight up, and the tangent line is horizontal — clearly perpendicular! Try different angles to see how the tangent rotates.

Challenge

Challenge: A circle has radius 10 cm. A central angle of 72 degrees cuts off an arc. Find (a) the arc length and (b) the area of the sector. Express your answers in terms of pi.

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