Thales' Theorem — Definition, Formula & Examples

Thales' Theorem states that if you draw a triangle inside a circle where one side is the diameter, the angle opposite that diameter is always 90°. In other words, any angle inscribed in a semicircle is a right angle.

Let $AB$ be a diameter of a circle with center $O$ , and let $C$ be any point on the circle distinct from $A$ and $B$ . Then $\angle ACB = 90°$ . This is a special case of the inscribed angle theorem, where the intercepted arc is a semicircle of measure $180°$ .

Key Formula

\angle ACB = \frac{1}{2} \times 180° = 90°

Where:

$A, B$ = Endpoints of a diameter of the circle
$C$ = Any other point on the circle
$\angle ACB$ = The inscribed angle opposite the diameter

How It Works

Pick any diameter

AB

of a circle. Choose any point

C

on the circle (not at

A

B

) and connect

C

to both

A

and

B

. The resulting triangle

ABC

always has a right angle at

C

. This works because the inscribed angle

\angle ACB

intercepts an arc of

180°

, and an inscribed angle is half its intercepted arc, giving

\frac{180°}{2} = 90°

. The converse also holds: if

\angle ACB = 90°

, then

AB

must be a diameter of the circle passing through

A

B

, and

C

Worked Example

Problem: A circle has center O and diameter AB = 10 cm. Point C lies on the circle such that AC = 6 cm. Find the length of BC.

Apply Thales' Theorem: Since AB is a diameter and C is on the circle, Thales' Theorem tells us that angle ACB is 90°.

\angle ACB = 90°

Use the Pythagorean Theorem: Triangle ACB is right-angled at C, so AB is the hypotenuse.

AB^2 = AC^2 + BC^2

Solve for BC: Substitute the known values and solve.

10^2 = 6^2 + BC^2 \implies 100 = 36 + BC^2 \implies BC^2 = 64 \implies BC = 8 \text{ cm}

Answer: BC = 8 cm

Why It Matters

Thales' Theorem gives you a reliable way to construct a right angle using only a compass and straightedge, which is fundamental in geometric constructions. It appears frequently in high school geometry proofs involving cyclic quadrilaterals and circle theorems, and it is used in engineering and surveying to verify perpendicularity.

Common Mistakes

Mistake: Assuming any chord (not just a diameter) creates a 90° inscribed angle.

Correction: The inscribed angle is 90° only when the opposite side is a diameter. For other chords, the inscribed angle equals half the intercepted arc, which will not be 90° unless that arc is exactly 180°.