Use the calculator below to apply Thales’ theorem for a right triangle in a circle: if a triangle is inscribed in a circle and one side is the diameter, then the opposite angle is 90°. In the “Right Triangle (Circle Thales)” tab you can enter any two of the sides (a, b, c) to solve the third side and get the circumradius, area, and perimeter.

Thales Theorem Calculator

Use the right-triangle version of Thales’ theorem. Enter any 2 side lengths and leave 1 blank to solve it. If you enter all 3, the calculator will check whether they form a valid right triangle.

Enter two side lengths, then press Calculate.

Thales Theorem Formula

The following formulas are used in the calculator for the right-triangle (Thales) case: a right triangle satisfies the Pythagorean theorem, and its circumradius equals half the hypotenuse (because the hypotenuse is the diameter of the circumcircle).

c^2 = a^2 + b^2
r = c/2

Variables:

  • a and b are the legs of the right triangle (units)
  • c is the hypotenuse (units); in the Thales setup, c is also the diameter of the circumcircle
  • r is the circumradius (units)

To solve for a missing side, rearrange the Pythagorean theorem. Once c is known, the circumradius is r = c/2, the area is (1/2)ab, and the perimeter is a + b + c.

What is Thales Theorem?

Thales’ theorem states that if A, B, and C are points on a circle and AB is a diameter of the circle, then the angle ACB is a right angle (90 degrees). Equivalently, if a triangle is right-angled, then its circumcenter is the midpoint of the hypotenuse, and the hypotenuse equals the diameter of its circumcircle.

How to Use Thales Theorem Calculator

The following steps outline how to use the calculator for common Thales-theorem tasks.

  1. Select the appropriate tab for your problem (Right Triangle, Intercept Theorem, or Coordinate Check).
  2. For the Right Triangle tab, choose a length unit and enter any two of a, b, and c (leave exactly one blank).
  3. Press Calculate to solve the missing side and display the circumradius (r = c/2), triangle area, and perimeter.
  4. For the Intercept Theorem tab, enter five of the six values (a1, a2, b1, b2, c1, c2) to solve the missing one and see the scale factor and consistency check.
  5. For the Coordinate Check tab, enter coordinates for A, B, and C (with AB treated as the diameter) to compute ∠ACB and test whether C lies on the circle with AB as diameter.

Example Problem : 

Use the following values as an example to test the Right Triangle (Circle Thales) tab.

Leg a = 3 units

Leg b = 4 units