45-45-90 Triangle Calculator
A 45-45-90 triangle is a right isosceles triangle whose legs are equal and whose hypotenuse is exactly the leg multiplied by the square root of 2. Enter any one known value (a leg, the hypotenuse, the altitude, the area, or the perimeter) and this calculator instantly returns every side, the altitude, the inradius, the circumradius, the perimeter, and the area, with full show-your-work steps.
Formula
Worked example
Leg = 5 cm: hypotenuse = 5 * sqrt(2) approx 7.0711 cm, altitude = 5 / sqrt(2) approx 3.5355 cm, perimeter approx 17.0711 cm, area = 12.5 cm^2, inradius approx 1.4645 cm, circumradius approx 3.5355 cm.
What Is a 45-45-90 Triangle?
A 45-45-90 triangle is the only right triangle that is also isosceles. Its angles are always 45 degrees, 45 degrees, and 90 degrees, which forces both legs to be exactly the same length. Because the angles are fixed, the three sides always keep the ratio 1 : 1 : sqrt(2), where sqrt(2) is approximately 1.41421. This constant ratio means that if you know any single measurement, whether a leg, the hypotenuse, the altitude, the area, or the perimeter, you can reconstruct the whole triangle. The shape appears in architecture, woodworking, and tiling anywhere a perfect 45-degree miter cut is needed.
Solving from Any Known Value
Most calculators only let you enter a leg or the hypotenuse. This calculator also accepts the altitude to the hypotenuse, the area, or the perimeter as starting points. From a leg, multiply by sqrt(2) to get the hypotenuse and divide by sqrt(2) to get the altitude. From the hypotenuse, divide by sqrt(2) to recover the leg. From the altitude h, the leg equals h times sqrt(2), because the altitude to the hypotenuse in a 45-45-90 triangle is always leg divided by sqrt(2). From the area A, the leg is sqrt(2 times A). From the perimeter P, the leg is P divided by (2 plus sqrt(2)).
Altitude, Inradius, and Circumradius
Three additional measurements are less commonly taught but often needed in technical work. The altitude from the right-angle vertex perpendicular to the hypotenuse equals the leg divided by sqrt(2), which is the same as half the hypotenuse. The inradius, the radius of the largest circle that fits inside the triangle, equals leg times (2 minus sqrt(2)) divided by 2, approximately 0.2929 times the leg. The circumradius, the radius of the circle that passes through all three vertices, equals half the hypotenuse for any right triangle, and in this case is the same as the altitude. Knowing the circumradius lets you find the triangle inside a given circle, while the inradius is useful for drafting the inscribed circle in technical drawings.
Unit Switching: Metric and Imperial
Choose any unit from the dropdown next to the value field: metres, centimetres, millimetres, feet, or inches. All outputs are automatically returned in the same unit. The area output is in that unit squared, so if you switch from centimetres to inches, both the side lengths and the area update immediately. This avoids the common error of mixing a side measured in inches with an area assumed to be in square feet.
Real-World Applications
Cutting a square along its diagonal always produces two 45-45-90 triangles, so the hypotenuse of each triangle equals the square side length times sqrt(2). This is why a 4-foot-wide square panel has a diagonal of about 5.657 feet. The same rule governs the diagonal of any square floor tile, the run of a 45-degree staircase ramp, and the distance between opposite corners of a square window opening. In the unit circle, the point at 45 degrees has coordinates (sqrt(2)/2, sqrt(2)/2), each equal to half the hypotenuse of a 45-45-90 triangle inscribed in the circle.
45-45-90 Triangle: All Properties as Multiples of the Leg
| Property | Exact expression | Decimal multiple of leg |
|---|---|---|
| Leg a (or b) | a | 1.00000 |
| Hypotenuse c | a * sqrt(2) | 1.41421 |
| Altitude h | a / sqrt(2) | 0.70711 |
| Perimeter P | a * (2 + sqrt(2)) | 3.41421 |
| Area A | a^2 / 2 | 0.50000 * a |
| Inradius r | a * (2 - sqrt(2)) / 2 | 0.29289 |
| Circumradius R | a * sqrt(2) / 2 | 0.70711 |
Every measurement scales linearly with the leg length a. Multiply the decimal multiple by your leg value to get the result.
Frequently asked questions
How do you find the hypotenuse of a 45-45-90 triangle?
Multiply either leg by sqrt(2), which is approximately 1.41421. For example, a leg of 6 cm gives a hypotenuse of 6 times sqrt(2), which is about 8.485 cm. Both legs are equal, so it does not matter which one you measure.
How do you find the leg if you only know the hypotenuse?
Divide the hypotenuse by sqrt(2), which is the same as multiplying by sqrt(2) and then dividing by 2. A hypotenuse of 10 inches gives a leg of 10 divided by sqrt(2), which is about 7.071 inches. Both legs will have that same length.
What is the area of a 45-45-90 triangle?
Because the two equal legs meet at the right angle, they act as the base and the height of the triangle, so the area is one half of a leg squared. With a leg of 8 m, the area is 8 squared divided by 2, which is 32 square metres. If you know the hypotenuse c instead, the area equals c squared divided by 4.
What is the altitude of a 45-45-90 triangle?
The altitude drawn from the right-angle vertex perpendicular to the hypotenuse equals the leg divided by sqrt(2), which is the same as the circumradius. For a leg of 5 cm, the altitude is about 3.536 cm. This altitude also bisects the right angle into two 45-degree angles.
What are the inradius and circumradius of a 45-45-90 triangle?
The circumradius of any right triangle equals half the hypotenuse, so for a 45-45-90 triangle with leg a, the circumradius is a times sqrt(2) divided by 2. The inradius equals a times (2 minus sqrt(2)) divided by 2, or about 0.2929 times the leg. For a leg of 10 cm, the inradius is about 2.929 cm and the circumradius is about 7.071 cm.
Can I solve a 45-45-90 triangle if I only know the perimeter?
Yes. The perimeter is always the leg times (2 plus sqrt(2)), so divide the perimeter by (2 plus sqrt(2)) to get the leg, then use the leg to find everything else. For a perimeter of 20 cm, the leg is 20 divided by (2 plus sqrt(2)), which is about 5.858 cm.