Calculate a missing right-triangle side with Thales’ theorem from two side lengths, and get the circumradius, area, and perimeter.

Precision

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).

c² = a² + b²; 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 right-triangle value to solve for, or choose Check three sides. This calculator implements the circle/right-triangle case of Thales’ theorem.
  2. Choose one common length unit and enter the displayed side lengths. The shared unit relabels all measurements without converting their numbers.
  3. Press Calculate to solve the missing side and display the circumradius (r = c/2), triangle area, and perimeter.
  4. To check a triangle, enter all three sides. Hypotenuse c must exceed each leg. The selected precision also sets the relative squared-side tolerance: 10 to the power of minus the selected decimal places.
  5. A consistency result is approximate within that tolerance, not proof of an exact right angle. The displayed area and circumradius then use the right-triangle approximation. Intercept-theorem and coordinate checks are not provided here.

Example Problem : 

Choose Hypotenuse c and use the following values. The result is c = 5 units, circumradius 2.5 units, area 6 square units, and perimeter 12 units.

Leg a = 3 units

Leg b = 4 units