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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Central angle=80°⇒Inscribed angle=80°0\text{Central angle} = 80° \quad \Rightarrow \quad \text{Inscribed angle} = \frac{80°}0

The graph below shows both angles. We use the unit circle (radius = 1). The central angle opens from the center, and the inscribed angle opens from a point on the circle.

-2-1.5-1-0.50.511.52-2-1.5-1-0.50.511.52Upper semicircleLower semicircleCentral angle (upper ray)Central angle (lower ray)Inscribed angle (half)
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⋅θ(theta in radians)s = r \cdot \theta \quad \text{(theta in radians)}
Radius (r)3
15
Central angle (degrees)90
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r=3,θ=90°=90⋅π0 radr = 3, \quad \theta = 90° = 90 \cdot \frac{\pi}0 \text{ rad}
s=3⋅90⋅π0s = 3 \cdot 90 \cdot \frac{\pi}0

Below, the x-axis represents the angle in degrees (0 to 360). The curves show arc length as a function of angle for the current radius and for comparison radii.

4590135180225270315360Arc length (current r)Arc length (r = 1)Current arc length
Try This

Try this: Set the angle to 360 degrees. The arc length equals the full circumference: s = 2 * pi * r. For r = 3, that’s about 18.85.

Sector Area

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

A=00r2θ(theta in radians)A = \frac00 r^2 \theta \quad \text{(theta in radians)}

Or equivalently:

A=θ360°⋅πr2A = \frac{\theta}{360°} \cdot \pi r^2
Radius (r)3
15
Central angle (degrees)90
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A=900⋅π⋅32A = \frac{90}0 \cdot \pi \cdot 3^2

Below, the x-axis represents the central angle. The curves show how sector area grows with angle for different radii.

459013518022527031536045Sector area (current r)Sector area (r = 1)Current sector area
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.

-3-2-1123-3-2-1123Upper semicircleLower semicircleRadius to tangent pointTangent line
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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