A central angle has its vertex at the centre. Its measure equals the arc it cuts off.
Step 1: Let's Learn
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Inscribed angles
An inscribed angle has its vertex on the circle. It measures exactly half its arc.
A striking consequence
Every inscribed angle on the same arc is equal, no matter where on the circle its vertex sits.
The angle in a semicircle
An angle inscribed in a semicircle is always 90°, since the arc is 180° and the angle is half of it.
Tangents
A tangent touches the circle at exactly one point and is perpendicular to the radius drawn to that point.
Two tangents from a point
Two tangents drawn from the same external point are equal in length.
Where the vertex sits changes the measure
A central angle equals its arc; an inscribed angle is half of it. The position of the vertex is what determines the relationship, which is why the cases are named separately.
Angles on the same arc are equal
Every inscribed angle standing on a given arc has the same measure, wherever on the circle its vertex sits. That is the consequence which makes the theorem so widely useful.
The angle in a semicircle is a right angle
An inscribed angle on a diameter subtends a 180° arc, so it measures 90°. Known since Thales, it is one of the oldest theorems in geometry and appears constantly.
Draw the radii
All radii are equal, so drawing them creates isosceles triangles with equal base angles. That single move is the first step in a large proportion of circle proofs.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Diego Use the Inscribed Angle
An arc measures 80°. Diego needs the inscribed angle standing on it.
- Step 1
A central angle on that arc would measure 80°, matching the arc exactly.
Step 4: Your Turn
Practice makes it stick.
The Inscribed Angle
Problem 1 of 2
An arc measures 80°. What is the inscribed angle on it, in degrees?
degrees
The Semicircle
Problem 2 of 2
An angle inscribed in a semicircle. What is its measure, in degrees?
degrees
Angles and Arcs
1 of 8
An arc of 100°. Inscribed angle on it, in degrees?
2 of 8
An arc of 100°. Central angle on it, in degrees?
3 of 8
An inscribed angle of 35°. What is its arc, in degrees?
4 of 8
The angle between a tangent and the radius at the point of contact, in degrees?
5 of 8
One tangent from a point is 12 long. How long is the other?
6 of 8
How many points does a tangent share with the circle?
7 of 8
Match each angle type with its relationship to its arc.
Tap a card on the left to start.
8 of 8
An arc of 140°. Inscribed angle, in degrees?
Step 5: Quick Check
Show what you know.
Question 1 of 2
An arc of 120°. What is the inscribed angle on it, in degrees?
Question 2 of 2
What decides whether an angle equals its arc or halves it?
What You Learned
- A central angle equals its arc; an inscribed angle is half of it.
- All inscribed angles on the same arc are equal.
- A tangent is perpendicular to the radius at the point of contact.