Triangle 30 60 90 Calculator - Sides, Area, and Perimeter

A triangle 30 60 90 calculator is a right-triangle tool that solves the 30-60-90 right triangle from any one side, using the fixed 1 : sqrt(3) : 2 side ratio. Pick which side you know, type the value, and get the other two sides, perimeter, area, and the three fixed interior angles.

• • Geometry and trigonometry students: Confirm the 1 : sqrt(3) : 2 side ratio and see how one known side recovers the rest of the triangle.
• • Carpenters and roof framers: Lay out a 30 degree miter cut or a triangular brace where one side is the known measurement.
• • Drafters and 3D modelers: Resolve the missing sides for a 30-60-90 facet in a truss or surface model.

A 30-60-90 right triangle is one of two special right triangles. The ratio between the three sides is always 1 : sqrt(3) : 2: the hypotenuse is twice the short leg, and the long leg is the short leg times sqrt(3). Once you know any one side the whole triangle is fixed.

For a tool that also covers the 45-45-90 family from the same one-side input, the special right triangles calculator handles both 1:1:sqrt(2) and 1:sqrt(3):2 right triangles in one form.

The 30-60-90 calculator does not solve a system of equations. It maps your known side to the short leg a, then multiplies a by sqrt(3) and 2 to recover the long leg and the hypotenuse. Perimeter is the sum of the three sides, and area is half the product of the two legs.

• a (short leg): Side across from the 30 degree angle. Equal to half the hypotenuse.
• sqrt(3): Factor that scales the short leg to the long leg (about 1.7320508).
• hypotenuse (2a): Side across from the 90 degree angle. Always twice the short leg.

The only number the calculator needs is one side in linear units. Once the role is known, the short leg a is recovered by dividing the supplied side by the appropriate ratio factor (2 for the hypotenuse, sqrt(3) for the long leg, or 1 when the side is the short leg).

According to Wikipedia: Special right triangle, a 30-60-90 right triangle has sides in the ratio 1 : sqrt(3) : 2, with the side across from 30 degrees equal to a, the side across from 60 degrees equal to a * sqrt(3), and the hypotenuse equal to 2a.

When the triangle is not a clean 30-60-90 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.

These four ideas cover why the 30-60-90 family is a clean, closed-form shape, and why a single side is enough to recover the whole triangle.

Together those four ideas explain why a 30-60-90 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 rational (1 and 2) or a single radical.

Because a 30-60-90 triangle is half of an equilateral triangle cut along its altitude, 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:sqrt(3):2 family.

Pick which side of the 30-60-90 triangle you know, type its value, choose the unit, and read the result panel.

1. 1 Choose which side you know: Pick 'short leg' (across from 30 degrees), 'long leg' (across from 60 degrees), or 'hypotenuse' (across from 90 degrees).
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, and the three fixed angles (30, 60, 90).
5. 5 Verify the angle set: The three angles always add to 180 degrees. Use this as a quick sanity check.

A carpenter lays out a 30 degree miter cut with a measured hypotenuse of 12 inches. Selecting 'Hypotenuse', entering 12, and choosing 'in' gives short leg = 6 in, long leg = 10.3923 in, area = 31.1769 in^2, and the angles 30 - 60 - 90.

When the side you know came from a 30 or 60 degree angle rather than a measured length, the sine triangle calculator takes the angle and returns the sine ratio, which is the missing link between the side inputs and the underlying 1:sqrt(3):2 family.

The 30-60-90 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 30-60-90 triangle from one side: Enter a single value and get the other two sides, perimeter, area, and the three fixed angles, with no need to set up the Pythagorean theorem.
• • Avoids radical-handling mistakes: sqrt(3) is kept at 4 decimal places, so long legs like 5 * sqrt(3) come out as 8.6603 instead of a hand calculation that drifts on the 4th digit.
• • Accepts short leg, long leg, or hypotenuse input: The side role selector covers all three sides, so the same tool works whether the known length came from a slope measurement, a staked short leg, or a triangle on paper.
• • 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 and perimeter alongside the sides: The panel reports the area and perimeter, so the calculator doubles as a quick area and perimeter tool for the same triangle.

Because the ratio family is short, the calculator is also useful as a teaching aid. The 30, 60, 90 angle set and the 1 : sqrt(3) : 2 side family show up in the answer panel automatically, which is the cleanest way to introduce a class to the half-equilateral relationship.

The 1:sqrt(3):2 ratio comes from the underlying trig values, so the cosine triangle calculator is a useful follow-up when a problem gives you the angle (30 or 60 degrees) rather than a side and asks for the cosine or sine ratio.

Three choices change what the calculator returns, and a few limitations of the 1 : sqrt(3) : 2 family are worth knowing before you commit the answer to a layout.

• • The calculator only solves the 30-60-90 right triangle. It cannot solve a triangle where the 30, 60, 90 angles do not all appear, nor a general right triangle with arbitrary side or angle input.
• • Results are 2D side, perimeter, and area only. Surface area, volume, or any 3D construction built from a 30-60-90 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 : sqrt(3) : 2 ratio also shows up in equilateral-triangle altitude problems and regular-hexagon layouts.

According to Wikipedia: Trigonometric functions, sin(30 degrees) is exactly 0.5 and cos(60 degrees) is exactly 0.5, while sin(60 degrees) and cos(30 degrees) are both sqrt(3)/2, which is what produces the 1 : sqrt(3) : 2 side ratio in a 30-60-90 right triangle.

According to Wikipedia: Equilateral triangle, an equilateral triangle has all three sides equal and all three interior angles equal to 60 degrees; cutting it along an altitude produces two congruent 30-60-90 right triangles, with the altitude equal to the side times sqrt(3)/2.

When you already know two perpendicular legs but want the area expressed for any triangle type rather than only the 30-60-90 family, the triangle area calculator returns the same half-leg-product result without committing to a specific angle set.