30 60 90 Triangle Calculator

Find the missing sides of a 30°–60°–90° triangle from one known side. Calculate its area and perimeter, compare radical expressions with decimal results and convert units.

Calculate Your Triangle

Choose the side you know and enter its length. The angles are fixed at 30°, 60° and 90°.

The short leg is opposite the 30° angle, not the hypotenuse.

60° 30° 90° a b c
a = short leg · b = long leg · c = hypotenuse. Side ratio: 1 : √3 : 2.

Changing the input unit converts the entered length. Changing the known side clears the input for a new measurement.

This tool assumes a 30°–60°–90° triangle; it does not determine the angles of an arbitrary triangle. For a three-dimensional solid with triangular ends, use the triangular prism calculator.

QUICK START

How to Use the 30 60 90 Triangle Calculator

One side length determines the whole triangle. Identify the side you know, enter its measurement and review the missing sides, area and perimeter.

  1. Identify your known side

    Choose the short leg opposite 30°, the long leg opposite 60°, or the hypotenuse opposite 90°. Identify sides by their opposite angles rather than their position on the page.

  2. Enter its length and unit

    Enter a positive number, including decimals, and select its measurement unit. Changing the input unit converts the entered length. Choosing a different known side clears the field.

  3. Compare radical and decimal results

    Choose your result units and decimal places. Decimal results use the selected output unit. Radical expressions retain √3 and use your entered value in its input unit.

  4. Follow the side ratio and copy your answer

    Review how the calculator uses the 1 : √3 : 2 ratio to find both missing sides. Check area and perimeter, then select Copy Result to copy the answers and radical expressions.

Confirm the angles first: Having a 90° angle alone is not enough to use this special ratio. The other two angles must be 30° and 60°, even if the triangle is rotated or reflected.

WORKED EXAMPLES

Find Missing Sides Step by Step

Use the known side to establish the short leg, then apply the 1 : √3 : 2 ratio. Keep radical expressions until the final step when a decimal answer is needed.

EXAMPLE 01 · SHORT LEG

Find both missing sides from the short leg

The side opposite 30° measures 6 centimeters. Multiply this length by √3 for the long leg and by two for the hypotenuse.

GIVEN MEASUREMENT
Short leg (a)
6 cm
Opposite angle
30°
  1. Find the long leg

    b = a√3 = 6√3 cm

    b ≈ 10.3923 cm

  2. Find the hypotenuse

    c = 2a = 2 × 6 = 12 cm

  3. Calculate the area

    Area = ab ÷ 2

    Area = 6 × 6√3 ÷ 2 = 18√3 cm²

RESULT Long leg = 6√3 cm · Hypotenuse = 12 cm

Area ≈ 31.1769 cm². Perimeter = 18 + 6√3 ≈ 28.3923 cm.

EXAMPLE 02 · HYPOTENUSE

Work backward from the longest side

The hypotenuse measures 20 inches. Halve this measurement to find the short leg, then multiply the short leg by √3.

GIVEN MEASUREMENT
Hypotenuse (c)
20 in
Opposite angle
90°
  1. Find the short leg

    a = c ÷ 2 = 20 ÷ 2 = 10 in

  2. Find the long leg

    b = a√3 = 10√3 in

    b ≈ 17.3205 in

  3. Add the three sides for perimeter

    P = 10 + 10√3 + 20

    P = 30 + 10√3 in

RESULT Short leg = 10 in · Long leg = 10√3 in

Perimeter ≈ 47.3205 in. Area = 50√3 ≈ 86.6025 in².

What if the long leg is given?

Divide it by √3 to find the short leg. For a long leg of 9 cm, a = 9 ÷ √3 = 3√3 cm, and c = 2a = 6√3 cm. Select Long leg in the triangle calculator and enter 9 cm to see the decimal results.

UNDERSTAND THE SIDE RATIO

30 60 90 Triangle Formulas

The short leg, long leg and hypotenuse always follow the ratio 1 : √3 : 2. This relationship comes from dividing an equilateral triangle into two equal right triangles.

60° 30° 90° a a√3 2a
Match each side to its opposite angle The side opposite 30° is shortest. The side opposite 60° is √3 times longer, and the hypotenuse is twice the short leg.
30° 30° 60° 60° 2a a a a√3
Half an equilateral triangle The altitude halves the base and the top angle. Each resulting triangle has angles of 30°, 60° and 90°.
a = short leg opposite 30°, b = long leg opposite 60°, and c = hypotenuse opposite 90°. Use consistent length units in each formula.
Known measurement Short leg (a) Other sides
Short leg a a b = a√3
c = 2a
Long leg b b ÷ √3 b is given
c = 2b ÷ √3
Hypotenuse c c ÷ 2 b = c√3 ÷ 2
c is given

Why does the long leg contain √3?

Apply the Pythagorean theorem using short leg a and hypotenuse 2a: a² + b² = (2a)². Subtracting a² gives b² = 3a². Taking the positive square root gives b = a√3.

Area and perimeter from the short leg

The legs are perpendicular, so area = ab ÷ 2 = a²√3 ÷ 2. Adding all three sides gives perimeter = a + a√3 + 2a = a(3 + √3). Area uses square units; perimeter uses length units.

Equivalent expressions can look different: b ÷ √3 equals b√3 ÷ 3. Multiplying the numerator and denominator by √3 removes the radical from the denominator without changing the value.

COMMON QUESTIONS

Frequently Asked Questions

Answers about identifying sides, using √3, entering measurements and checking your triangle results.

Which side is the short leg?

The short leg is opposite the 30° angle. The long leg is opposite 60°, and the hypotenuse is opposite 90°. These relationships stay the same when the triangle is rotated or reflected. A side’s position on the page does not determine its name.

Why is the side ratio 1 : √3 : 2?

Bisect an equilateral triangle with an altitude. Each half has angles of 30°, 60° and 90°, and its short leg is half the original side. If that leg is 1, the hypotenuse is 2. The Pythagorean theorem gives the remaining leg as √(2² − 1²) = √3. The 30-60-90 triangle reference on Wolfram MathWorld also gives the leg lengths in terms of the hypotenuse.

How do I find the sides when the long leg is given?

Divide the long leg by √3 to find the short leg, then double the short leg for the hypotenuse. For example, if the long leg is 12 cm, the short leg is 12 ÷ √3 = 4√3 cm and the hypotenuse is 8√3 cm.

Can I enter a radical such as 5√3?

The input field accepts numbers and decimals, not radical notation. If your long leg is exactly 5√3, select Short leg and enter 5 in the same unit. This preserves the intended relationship without rounding √3 first. Entering a decimal approximation of 5√3 produces results based on that approximate input.

Why do the radical expressions use the input unit?

The expressions show the algebra directly from your entered measurement. Decimal results use your selected output unit instead. For example, a 5 cm short leg gives a long-leg expression of 5√3 cm, even when the decimal result is displayed in meters.

Does this work for every right triangle?

No. This calculator assumes angles of 30°, 60° and 90°. A 45°–45°–90° triangle has the ratio 1 : 1 : √2. Other right triangles need their own known side or angle information; one side alone does not determine them.

How can I check the calculated side lengths?

The hypotenuse should be twice the short leg, and the long leg should be √3 times the short leg. The sides also satisfy a² + b² = c². Use radical expressions or unrounded values when checking; rounded decimal lengths can introduce a small difference.

How are area and perimeter calculated?

Area is half the product of the two perpendicular legs: A = ab ÷ 2. Perimeter is the sum of all three sides: P = a + b + c. With short leg a, these become A = a²√3 ÷ 2 and P = a(3 + √3). Area uses square units and perimeter uses length units.

ONE SIDE IS ALL YOU NEED

Find Your Triangle’s Missing Sides

Enter the short leg, long leg or hypotenuse. Get the remaining lengths, area and perimeter, with radical expressions and calculation steps.

Use 30 60 90 Calculator