📐 Right Isosceles Triangle Calculator
Calculate sides and area of a right isosceles triangle.
What is a Right Isosceles Triangle?
A triangle with two equal sides and a 90° angle between them. The other two angles are each 45°. The ratio of sides is 1 : 1 : √2.
Given base (a)
Given height (h)
Given hypotenuse (b)
Given area (S)
30°-60°-90° Triangle
There is also the 30°-60°-90° special right triangle, where the ratio of sides is 1 : √3 : 2.
Worked examples
Legs of 6 give a hypotenuse of 6√2 ≈ 8.4853, an area of 6² / 2 = 18 and a perimeter of 12 + 8.4853 ≈ 20.4853.
Legs of 10 give a hypotenuse of 14.1421, an area of 50 and a perimeter of about 34.1421; the angles are always 45°, 45° and 90°.
What the formulas assume
The right angle sits between the two equal legs, so the hypotenuse is the only unequal side. The hypotenuse is irrational for every whole-number leg, hence the rounding. No input changes the angles: the ratio 1 : 1 : √2 is fixed, and the area comes out in squared units of the leg.
Frequently asked questions
- Why is the hypotenuse always √2 times a leg?
- Pythagoras gives a² + a² = 2a², so the hypotenuse is a√2. Since √2 is irrational, no whole-number leg ever produces a whole-number hypotenuse.
- How do I get the area from the hypotenuse alone?
- Square it and divide by four: area = h² / 4. A hypotenuse of 8.4853 gives 72 / 4 = 18, the same as legs of 6.
- How does this triangle relate to a square?
- The diagonal of a square splits it into two 45-45-90 triangles, so the diagonal is the side times √2 and each triangle holds half the square's area.