An inscribed angle is exactly half the central angle standing on the same arc.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The position does not matter
Every inscribed angle on the same arc is equal, wherever on the circle its vertex sits.
The angle in a semicircle
A diameter subtends a central angle of 180°, so the inscribed angle is 90°. Always.
Opposite angles
In a cyclic quadrilateral, opposite angles add to 180°.
Half the central angle
An inscribed angle is half the central angle subtending the same arc. The proof uses the isosceles triangles formed by radii, which is why the isosceles result earlier in the course was worth proving carefully.
Angles on the same arc are equal
Every inscribed angle standing on the same arc has the same measure, wherever on the circle its vertex sits. This is the consequence that makes the theorem so useful in problems and proofs.
The angle in a semicircle is a right angle
An inscribed angle on a diameter subtends a 180° arc, so it is 90°. This special case, known since Thales, is one of the oldest theorems in geometry and turns up constantly.
Opposite angles of a cyclic quadrilateral
A quadrilateral with all four vertices on a circle has opposite angles summing to 180°. It follows from the inscribed angle theorem applied to the two arcs, and its converse proves four points lie on a circle.
Step 2: Try It Yourself
Tap and try it out.
- Diameter10 cm
- Circumference31.42 cm
- Area78.54 cm²
- Circumference ÷ diameter3.14
Circumference is a length, measured in cm. Area is a coverage, measured in cm². The units tell them apart.
Step 3: Watch an Example
One step at a time.
Watch Kofi Halve a Central Angle
A central angle of 100° stands on an arc.
- Step 1
The inscribed angle on that same arc is half of it.
Step 4: Your Turn
Practice makes it stick.
Halving
Problem 1 of 2
Central angle 80°. Inscribed angle on the same arc, in degrees?
The Semicircle
Problem 2 of 2
An angle inscribed in a semicircle, in degrees?
Angles in a Circle
1 of 8
Central 140°. Inscribed on the same arc?
2 of 8
Inscribed 35°. Central on the same arc?
3 of 8
Cyclic quadrilateral, one angle 85°. Its opposite, in degrees?
4 of 8
Two inscribed angles on the same arc, one is 25°. The other?
5 of 8
Angle in a semicircle, in degrees?
6 of 8
Select every true statement about inscribed angles.
7 of 8
Central 200°. Inscribed on the same arc?
8 of 8
Cyclic quadrilateral, one angle 110°. Its opposite?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Central 90°. Inscribed on the same arc?
What You Learned
- An inscribed angle is half the central angle on the same arc.
- Its size does not depend on where the vertex sits.
- An angle inscribed in a semicircle is always a right angle.