Triangle 45 45 90 Calculator - Sides, Area, and Perimeter

A triangle 45 45 90 calculator is a right-triangle tool that solves the isosceles right triangle from any one side, using the fixed 1 : 1 : sqrt(2) side ratio. Pick which side you know, type the value, and get the equal legs, hypotenuse, perimeter, area, inradius, circumradius, and the three fixed interior angles.

• • Geometry and trigonometry students: Confirm the 1 : 1 : sqrt(2) side ratio and see how one known side recovers the rest of the triangle.
• • Carpenters and cabinet makers: Cut a 45 degree miter on a baseboard or square frame where one side is the known measurement.
• • Roof framers and stair builders: Compute the diagonal of a square stair tread or roof pitch where the run and rise meet at a right angle.
• • Drafters and 3D modelers: Resolve the missing sides for a 45-45-90 facet in a frame, panel, or surface model.

A 45-45-90 right triangle is the second of the two special right triangles. Its two legs are equal; the hypotenuse is the leg times the square root of 2.

Most classroom problems state one side and ask for the rest, and the same tool works for shop layouts and square-diagonal design problems.

When the same triangle is described by its shape rather than its angle set, the isosceles right triangle calculator solves the same 1:1:sqrt(2) family and adds inradius and circumradius outputs.

The 45-45-90 calculator does not solve a system of equations. It maps your known side to the leg a, then multiplies a by sqrt(2) to recover the second leg, the hypotenuse, the area, the inradius, and the circumradius.

• a (leg): One of the two equal legs. Equal to half the hypotenuse times sqrt(2).
• sqrt(2): Factor that scales a leg into the hypotenuse (about 1.4142135623730951).
• hypotenuse (a * sqrt(2)): Longest side, opposite the 90 degree angle. Equal to a times the square root of 2.

The only number the calculator needs is one side in linear units. The leg a is recovered by dividing the supplied side by sqrt(2) (hypotenuse input) or by using the value directly (leg input).

From a single leg, the rest follows: the second leg equals a, the hypotenuse is a times sqrt(2), and the area is a squared divided by 2.

According to Wikipedia: Special right triangle, a 45-45-90 right triangle has sides in the ratio 1 : 1 : sqrt(2), with the two legs each of length a and the hypotenuse of length a * sqrt(2).

When the right triangle is from the 30-60-90 family instead of the 45-45-90 family, the triangle 30 60 90 calculator applies the 1 : sqrt(3) : 2 ratio from any one side.

These four ideas cover why the 45-45-90 family is a clean, closed-form shape, and why a single side is enough to recover the whole triangle.

Those four ideas explain why a 45-45-90 right triangle is one of the easiest right triangles to solve by hand: the angles are fixed, the ratio family is fixed, and every numeric answer is a single radical.

Because the two legs are equal, the calculator reports them as both 'leg a (first)' and 'leg a (second)', matching the way classroom problems are usually stated.

When you need a single form that handles both the 1:1:sqrt(2) and the 1:sqrt(3):2 families in one place, the special right triangles calculator returns the missing sides for either special right triangle from one input.

Pick which side you know, type its value, choose the unit, and read the result panel.

1. 1 Choose which side you know: Pick 'Leg' (one of the two equal legs across from a 45 degree angle) or 'Hypotenuse' (the long side across from the 90 degree angle).
2. 2 Type the side length: Enter the numeric value in the value box. Use three or more significant figures for a high-precision answer.
3. 3 Pick the unit: Select cm, m, in, or ft. The chosen unit is shown next to every output.
4. 4 Read the result panel: The panel shows the other two sides, perimeter, area, inradius, circumradius, and the three fixed angles (45, 45, 90).
5. 5 Verify the angle sum: The three angles always add to 180 degrees. Use this as a quick sanity check whenever you change inputs.

A carpenter cuts a 45 degree miter on a baseboard with a measured leg of 8 inches. Selecting 'Leg', entering 8, and choosing 'in' gives second leg = 8 in, hypotenuse = 11.3137 in, perimeter = 27.3137 in, area = 32 in^2, and the angles 45 - 45 - 90.

When the triangle is not a clean 45-45-90 family and the input is two sides or one side and one angle, the right triangle calculator solves the full general right triangle and returns the missing sides, angles, and area.

The 45-45-90 calculator turns a one-side input into a complete triangle, with no need to set up the Pythagorean theorem or remember which radical to apply.

• • Solves any 45-45-90 triangle from one side: Enter a single value and get the other two sides, perimeter, area, inradius, circumradius, and the three fixed angles.
• • Avoids radical-handling mistakes: sqrt(2) is kept at 4 decimal places, so hypotenuses like 5 * sqrt(2) come out as 7.0711 instead of a hand calculation that drifts on the 4th digit.
• • Accepts leg or hypotenuse input: The side role selector covers both legs and the hypotenuse, so the same tool works whether the known length came from a measured side or from the diagonal of a square.
• • Switches between metric and imperial units: The unit selector covers cm, m, in, and ft, so the calculator covers metric homework, imperial shop drawings, and mixed-unit reviews.
• • Returns area, perimeter, inradius, and circumradius: The panel reports the area, perimeter, inradius, and circumradius alongside the sides, so the calculator doubles as a quick special-triangle solver for the same triangle.

Because the ratio family is short, the calculator is also useful as a teaching aid: the 45, 45, 90 angle set and the 1 : 1 : sqrt(2) side family show up in the answer panel automatically.

When the triangle is a general scalene or isosceles shape rather than a 45-45-90 right triangle, the triangle perimeter calculator recovers the perimeter from any three known sides.

Three choices change what the calculator returns, and a few limitations of the 1 : 1 : sqrt(2) family are worth knowing before you commit the answer to a layout.

• • The calculator only solves the 45-45-90 right triangle. It cannot solve a triangle where the 45, 45, 90 angles do not all appear, nor a general right triangle with arbitrary side or angle input.
• • Results are 2D side, perimeter, area, inradius, and circumradius only. Surface area, volume, or any 3D construction built from a 45-45-90 triangle needs additional formulas.
• • The calculator cannot recover the triangle from the area or perimeter alone. You need at least one side in linear units to back-solve the rest.

In a classroom, treat the 4-decimal output as a sanity check and the radical form as the source of truth. The 1 : 1 : sqrt(2) ratio shows up in any square-diagonal problem.

According to Wikipedia: Isosceles right triangle, the area of a 45-45-90 triangle is a^2 / 2, which is half the product of its two equal legs.

According to Wikipedia: Square root of 2, the square root of 2 equals about 1.4142135623730951, which is the exact factor that scales a 45-45-90 leg into its hypotenuse.

When you already know two perpendicular legs and want the area expressed for any triangle type rather than only the 45-45-90 family, the triangle area calculator returns the same half-leg-product result without committing to a specific angle set.