Special Right Triangles Calculator - Sides, Area, and Angles

A special right triangles calculator is a geometry tool that solves the two right triangles whose interior angles give clean, predictable side ratios: the 45-45-90 isosceles right triangle and the 30-60-90 half-equilateral right triangle. Enter one side and the calculator returns the other two sides, perimeter, area, and the three fixed interior angles using the 1:1:√2 and 1:√3:2 ratio families.

• • Geometry and trigonometry students: Confirm exact radical answers and see how the 1:1:√2 and 1:√3:2 families scale with a chosen side.
• • Carpenters and roof framers: Lay out square-cut miter joints, equal-pitch rafters, and diagonal braces.
• • Drafters and 3D modelers: Resolve hypotenuse and leg lengths for triangular profiles and trusses.
• • Surveyors and field technicians: Convert a measured leg or hypotenuse into the missing sides for slope, distance, or elevation checks.

Special right triangles appear in classrooms, construction sites, and any problem that expects a clean answer. The 45-45-90 family has two equal legs with hypotenuse = leg × √2; the 30-60-90 family is half an equilateral triangle.

If you need to solve a right triangle that is not 45-45-90 or 30-60-90, the Right Triangle Calculator accepts any two known sides or one side and one angle and returns the remaining values.

The calculator does not solve a system of equations; it maps your single known side to the correct leg or hypotenuse using the fixed ratio family for the chosen triangle type, then sums the sides for perimeter and halves the product of the legs for area.

• triangleType: Either '45-45-90' or '30-60-90'.
• knownSide: The numeric length of the single side you supply, in the chosen unit.
• knownSideRole: Which side the known length represents: hypotenuse, leg, short leg, or long leg.
• unit: Linear unit label (cm, m, in, or ft).

Internally, the calculator stores two reference ratios: for 45-45-90 the leg-to-hypotenuse factor is √2 ≈ 1.4142, and for 30-60-90 the long-leg-to-short-leg factor is √3 ≈ 1.7321 with a hypotenuse-to-short-leg factor of 2. When you enter a hypotenuse, the calculator divides by these factors to recover the legs; when you enter a leg, it multiplies to recover the other two sides. Perimeter is the simple sum of the three sides, and area is half the product of the two legs because the legs are perpendicular.

According to Khan Academy, a 45-45-90 right triangle has two equal legs and a hypotenuse equal to one leg times the square root of 2, which follows from the Pythagorean theorem a² + a² = c².

When the answer needs to come out as integer sides, the Pythagorean Triples Calculator generates and verifies triples like 3-4-5 that satisfy the same underlying Pythagorean relationship the special right triangles use.

The 45-45-90 and 30-60-90 families are the most commonly taught special right triangles, and they have the cleanest closed-form side ratios: 1:1:√2 and 1:√3:2.

Spot the family at a glance: two equal sides in a right triangle means 45-45-90 with hypotenuse = leg × √2. A 30-60-90 triangle is half an equilateral, and a 45-45-90 triangle is half a square.

A 45-45-90 triangle is an isosceles right triangle, so the Isosceles Triangle Area Calculator is the broader tool to reach for when the apex angle is not fixed at 90° but the two base sides are still equal.

Working with the special right triangles calculator only takes four steps because the math has already been encoded in the ratio table.

1. 1 Pick the triangle type: Choose 45-45-90 for an isosceles right triangle or 30-60-90 for a half-equilateral right triangle.
2. 2 Enter your single known side: Type the value of any one side. Use a positive number greater than zero with three or more significant figures for a high-precision answer.
3. 3 Tell the calculator which side you supplied: Set Known Side Role to Hypotenuse, Leg (45-45-90 only), Short Leg (across from 30°), or Long Leg (across from 60°).
4. 4 Choose the unit and read the results: Pick cm, m, in, or ft. The results panel fills in with the other two sides, perimeter, area, and the three fixed angles.

A student is given a 30-60-90 triangle with a hypotenuse of 12 cm and needs the long leg, perimeter, and area. Selecting 30-60-90, entering 12, choosing Hypotenuse, and reading cm produces short leg = 6 cm, long leg = 10.3923 cm, hypotenuse = 12 cm, perimeter = 28.3923 cm, and area = 31.1769 cm² — all the values a problem set would ask for.

Because a 30-60-90 triangle is half of an equilateral triangle, the Equilateral Triangle Area Calculator is the right companion when you already know the full equilateral side and want to split it into the 1:√3:2 family.

The special right triangles calculator is built for speed and accuracy on a small but very common subset of geometry problems, so the benefits focus on what the ratio families actually save you.

• • Solve any special right triangle from one side: Enter a single value and get all three sides, perimeter, area, and the three angles, with no need to set up the Pythagorean theorem or remember which factor to apply.
• • Avoid decimal drift in long radicals: The calculator rounds to 4 decimal places, so √2 and √3 stay accurate to the 4th digit even when the legs themselves are not whole numbers, which prevents the rounding error that creeps in during multi-step textbook problems.
• • Switch between 45-45-90 and 30-60-90 in one step: Change the triangle type without retyping values; the form remaps the known side to the correct ratio family and refreshes every output field as you type.
• • Use it for class, site, and studio work: Switch units from cm to in or ft without redoing the math, so the same tool covers metric homework, imperial shop drawings, and mixed-unit design reviews.
• • Built-in sanity checks for the angle sum: The three angles always add to 180° (45 + 45 + 90 or 30 + 60 + 90), so the displayed angles let you verify the chosen triangle type before committing a layout to paper or a model to CAD.

Because the ratio families are short, the calculator also doubles as a teaching aid. Practitioners keep a special right triangle reference on the workbench for cases where the Pythagorean theorem would be overkill.

For triangles that do not fit the special-ratio shortcut, the Triangle Calculator solves any triangle from three known sides, two sides and an included angle, or other common inputs.

Special right triangles have a handful of inputs, but the choices you make in the form change the answer by the ratio family itself, and a few approximations are baked into any decimal output.

• • This calculator only handles the two special right triangles (45-45-90 and 30-60-90). It does not solve a general right triangle with arbitrary side or angle input; for that, use a full right-triangle solver that accepts any two known values.
• • Results are 2D side, perimeter, and area only. Surface area, volume, or any 3D construction built from a special right triangle needs additional formulas layered on top of the side lengths.
• • The page does not solve for a triangle when the only given information is the area or the perimeter; you need at least one side in linear units to back-solve the rest through the ratio family.

In a classroom, treat the 4-decimal output as a sanity check and the radical form as the source of truth. The fixed angles come from the triangle type, which avoids drift from arctangent of an approximated ratio.

According to Math Is Fun, a 45-45-90 right triangle has two equal legs and a hypotenuse equal to one leg times the square root of 2, while a 30-60-90 right triangle has the side across from 30° as the shortest leg, a hypotenuse twice that length, and the side across from 60° equal to the short leg times the square root of 3.

When you already know two perpendicular legs but want the area expressed for a wider range of triangle types, the Triangle Area Calculator returns the same ½ × leg₁ × leg₂ result without committing to a special ratio.