# Circle theorems: angles, chords, tangents and arcs

Canonical: https://duckyhelper.com/learn/geometry/circle-theorems/
Updated: 2026-10-01

A central angle, with its vertex at the center, equals its arc. An inscribed angle, with its vertex on the circle, is half its arc, so an angle inscribed in a semicircle is 90°. A tangent line is perpendicular to the radius where it touches the circle. Arc length is \(\frac{\theta}{360} \cdot 2\pi r\) and sector area is \(\frac{\theta}{360} \cdot \pi r^2\), where \(\theta\) is the central angle in degrees.

## The key idea

Most circle problems come down to a short list of facts. For angles, the first thing to check is where the vertex sits: at the center, or on the circle.

**The circle facts you will use most**

| Fact | What it says |
| --- | --- |
| Central angle (vertex at the center) | Equals its intercepted arc |
| Inscribed angle (vertex on the circle) | Half of its intercepted arc |
| Inscribed angles on the same arc | Equal to each other |
| Angle inscribed in a semicircle | Always 90° |
| Quadrilateral inscribed in a circle | Opposite angles add to 180° |
| Tangent line | Perpendicular to the radius at the point where it touches |
| Two tangent segments from one outside point | Equal in length |
| Radius perpendicular to a chord | Cuts the chord in half |

$$
\text{central angle} = \text{arc} \qquad \text{inscribed angle} = \frac{1}{2} \cdot \text{arc}
$$

### Arc length and sector area

An arc is a fraction of the whole circle, and that fraction is the central angle over 360°. Take that fraction of the circumference for arc length, or of the area for a sector (a pizza slice).

$$
\text{arc length} = \frac{\theta}{360} \cdot 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \cdot \pi r^2
$$

## Worked examples

**Example 1: inscribed and central angles**

Problem: Points A, B and C lie on a circle with center O. Inscribed angle ABC intercepts arc AC, which measures 110°. Find angle ABC and central angle AOC.

1. Angle ABC has its vertex on the circle, so it is inscribed. An inscribed angle is half its arc.

   $$
   \frac{1}{2} \cdot 110 = 55
   $$
2. Angle AOC has its vertex at the center, so it equals the arc: 110°.
3. Notice the inscribed angle is half the central angle on the same arc.

Answer: Angle ABC is \(55^\circ\) and angle AOC is \(110^\circ\).

**Example 2: a tangent and a radius**

Problem: Line PT is tangent to a circle at T. The circle has center O and radius 9, and \(PT = 12\). How far is P from the center? How far is P from the nearest point on the circle?

1. A tangent is perpendicular to the radius at the point where it touches, so angle OTP is 90°. Triangle OTP is a right triangle, and its hypotenuse is OP. Call OP d.
2. Use the Pythagorean theorem.

   $$
   9^2 + 12^2 = d^2
   $$
3. Simplify.

   $$
   225 = d^2
   $$
4. Take the positive root.

   $$
   d = 15
   $$
5. The nearest point on the circle lies on segment OP, one radius from O.

   $$
   15 - 9 = 6
   $$

Answer: P is 15 units from the center and 6 units from the circle.

**Example 3: arc length and sector area**

Problem: A circle has radius 6. A central angle of 150° cuts off an arc and a sector. Find the arc length and the sector area in terms of \(\pi\).

1. Find the fraction of the circle.

   $$
   \frac{150}{360} = \frac{5}{12}
   $$
2. Arc length: that fraction of the circumference, \(2\pi(6) = 12\pi\).

   $$
   \frac{5}{12} \cdot 12\pi = 5\pi
   $$
3. Sector area: that fraction of the area, \(\pi(6)^2 = 36\pi\).

   $$
   \frac{5}{12} \cdot 36\pi = 15\pi
   $$

Answer: Arc length \(5\pi\) (about 15.71). Sector area \(15\pi\) (about 47.12).

**Example 4 (test-hard): a chord and its distance from the center**

Problem: A chord 24 cm long is 5 cm from the center of a circle. Find the radius and the circumference.

1. Draw a segment from the center that meets the chord at a right angle. It is 5 cm long, and it cuts the chord in half, into two 12 cm pieces.
2. Draw a radius r to one end of the chord. Now there is a right triangle with legs 5 and 12, and r is its hypotenuse.

   $$
   5^2 + 12^2 = r^2
   $$
3. So \(r^2 = 169\). Take the positive root.

   $$
   r = 13
   $$
4. Circumference is \(2\pi r\).

   $$
   2\pi \cdot 13 = 26\pi
   $$

Answer: The radius is 13 cm. The circumference is \(26\pi\) cm, about 81.68 cm.

## Common mistakes

- **Forgetting to halve.** An inscribed angle is half its arc. If your inscribed angle equals its arc, check where the vertex is.
- **Halving a central angle.** Central angles are not halved. Vertex at the center means angle equals arc.
- **Using the diameter as r.** Arc length uses \(2\pi r\) and sector area uses \(\pi r^2\). If you are given the diameter, halve it first.
- **Mixing up arc length and sector area.** Arc length is a distance, in units. Sector area is a region, in square units. Check which one the question asks for.
- **Missing a hidden right angle.** A tangent meets a radius at 90°, and an angle inscribed in a semicircle is 90°. These are often the key to a Pythagorean step.

## Quick methods

> **Tip: Draw the radius**
>
> When you are stuck, draw a radius to every labeled point on the circle, especially tangent points and the ends of chords. All radii are equal, which gives you isosceles triangles and right angles to work with. This one habit opens up most textbook circle problems.

## Practice

**5 practice questions**

1. A central angle measures 80°. An inscribed angle intercepts the same arc. What does the inscribed angle measure?
   A. \(40^\circ\)
   B. \(80^\circ\)
   C. \(100^\circ\)
   D. \(160^\circ\)

   Answer: \(40^\circ\). The arc is 80°, the same as the central angle, and the inscribed angle is half the arc, 40°. 80° forgets to halve, 160° doubles instead of halving, and 100° is \(180 - 80\).

2. Quadrilateral ABCD is inscribed in a circle. Angle A measures \((3x + 10)^\circ\) and angle C measures \((2x + 20)^\circ\). What is the measure of angle C?
   A. \(30^\circ\)
   B. \(40^\circ\)
   C. \(80^\circ\)
   D. \(100^\circ\)

   Answer: \(80^\circ\). Opposite angles of an inscribed quadrilateral add to 180°: \(5x + 30 = 180\), so \(x = 30\) and angle C is \(2(30) + 20 = 80\). 30 is x, 100° is angle A, and 40° comes from setting the two angles equal.

3. From a point P outside a circle, two tangent segments touch the circle at A and B. \(PA = 2x + 5\) and \(PB = 4x - 7\). What is PA?
   A. \(6\)
   B. \(12\)
   C. \(17\)
   D. \(34\)

   Answer: \(17\). Tangent segments from the same point are equal, so \(2x + 5 = 4x - 7\) and \(x = 6\). Then \(PA = 2(6) + 5 = 17\). 6 is x, 12 forgets the + 5, and 34 adds PA and PB together.

4. A circle has radius 10. How long is the arc cut off by a central angle of 72°?
   A. \(2\pi\)
   B. \(4\pi\)
   C. \(8\pi\)
   D. \(20\pi\)

   Answer: \(4\pi\). \(\frac{72}{360} = \frac{1}{5}\), and the circumference is \(20\pi\), so the arc is \(4\pi\). \(20\pi\) is the whole circumference (and, by coincidence, also the sector area here). \(8\pi\) uses 180 instead of 360, and \(2\pi\) uses \(\pi r\) instead of \(2\pi r\).

5. Triangle ABC is inscribed in a circle, and side AB is a diameter. Angle A measures 34°. What is the measure of angle B, in degrees?

   Answer: 56. Angle C is inscribed in a semicircle, so it is 90°. Then angle B is \(180 - 90 - 34 = 56\).

## Frequently asked questions

### What is the difference between a central angle and an inscribed angle?

Look at the vertex. A central angle has its vertex at the center of the circle, and it equals its arc. An inscribed angle has its vertex on the circle, and it is half its arc. When both open onto the same arc, the inscribed angle is half the central angle.

### Why is an angle inscribed in a semicircle always 90°?

A semicircle's arc is 180°, and an inscribed angle is half its arc, so the angle is 90°. This is often called Thales' theorem. It means any triangle drawn inside a circle with a diameter as one side is a right triangle.

### How do I find arc length if the angle is in radians?

In radians the formulas are shorter: arc length is \(r\theta\) and sector area is \(\frac{1}{2}r^2\theta\). They give the same answers as the degree formulas on this page, because 360° is the same as \(2\pi\) radians.

### How can I tell if a line is tangent to a circle?

A tangent touches the circle at exactly one point and is perpendicular to the radius there. If you know the three sides of the triangle formed by the center, the touch point and a point on the line, check whether \(a^2 + b^2 = c^2\). If it holds, the line is tangent.

## Related

- [The Pythagorean theorem](https://duckyhelper.com/learn/geometry/pythagorean-theorem/)
- [Angles, parallel lines and polygon angle sums](https://duckyhelper.com/learn/geometry/angles-and-parallel-lines/)
- [Circles on the SAT](https://duckyhelper.com/learn/sat-math/circles/)
- [Geometry study guides](https://duckyhelper.com/learn/geometry/)

## Try asking Ducky

- "Is this angle inscribed or central? The vertex looks like it's on the circle."
- "I got 160 degrees for the inscribed angle. Should I have halved it instead?"
- "Explain why a tangent is perpendicular to the radius."

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