30-60-90 Triangle Calculator

Enter any one side of a 30-60-90 triangle to find all sides, area, and perimeter instantly.

30° What is a 30-60-90 Triangle?

A 30-60-90 triangle is a right triangle with angles of exactly 30°, 60°, and 90°. It is one of the two special right triangles in geometry (the other being the 45-45-90 triangle). What makes it special is that the side lengths are always in the exact ratio 1 : √3 : 2 — no matter how large or small the triangle is. This means knowing just one side lets you find all three sides precisely, without any trigonometric calculation beyond the ratio.

The three sides correspond to: the short leg (opposite the 30° angle) = a; the long leg (opposite the 60° angle) = a√3; and the hypotenuse (opposite the 90° angle) = 2a. These ratios come directly from trigonometry: sin(30°) = 1/2, so the short leg is half the hypotenuse; sin(60°) = √3/2, so the long leg is (√3/2) × hypotenuse.

The 30-60-90 triangle has a beautiful geometric origin: it is exactly half of an equilateral triangle. Draw an equilateral triangle with side s and drop an altitude from one vertex to the midpoint of the opposite side. The altitude bisects the equilateral triangle into two congruent 30-60-90 triangles, each with short leg = s/2, long leg = s√3/2 (the altitude), and hypotenuse = s. This connection explains why the ratio involves √3.

In practical applications, 30-60-90 triangles appear in architecture (roof pitches at standard angles), engineering (force decomposition at 30° or 60°), electrical engineering (three-phase AC power uses 120° phase differences, which involve 60° geometry), drafting (30-60-90 set squares are standard tools), and in all problems involving equilateral hexagonal structures like honeycomb patterns.

📐 Formulas

Sides: a : b : c = 1 : √3 : 2

a = short leg (opposite 30°) · b = a√3 = long leg (opposite 60°) · c = 2a = hypotenuse

From long leg b: a = b/√3 = b√3/3, c = 2b/√3

From hypotenuse c: a = c/2, b = c√3/2

Area = ½ × a × a√3 = a²√3/2 · Perimeter = a + a√3 + 2a = a(3 + √3)

📖 How to Use This Calculator

Steps

Identify the known side — is it the short leg (opposite 30°), the long leg (opposite 60°), or the hypotenuse?

Select the matching tab and enter the length. Any unit works (metres, cm, inches) as long as you use it consistently.

Click Calculate to see all three sides, area, and perimeter, with the complete formula in the note.

💡 Example Calculations

Example 1 — Short Leg = 5 cm

Given: short leg = 5 cm

Long leg = 5 × √3 = 5 × 1.7321 ≈ 8.660 cm

Hypotenuse = 2 × 5 = 10 cm

Area = ½ × 5 × 8.660 ≈ 21.65 cm²

Short leg 5, long leg ≈8.660, hypotenuse 10. Area ≈ 21.65 cm². Perimeter ≈ 23.66 cm.

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Example 2 — Hypotenuse = 1 (Unit Triangle)

Hypotenuse = 1 (standard unit triangle)

Short leg = 1/2 = 0.5

Long leg = √3/2 ≈ 0.8660

This is the unit triangle showing sin(30°)=0.5, cos(30°)=0.866, tan(30°)=0.577 directly from the sides.

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Example 3 — Long Leg = 12 m

Given: long leg = 12 m

Short leg = 12 / √3 = 12√3/3 = 4√3 ≈ 6.928 m

Hypotenuse = 2 × 4√3 = 8√3 ≈ 13.856 m

Area = ½ × 6.928 × 12 ≈ 41.57 m². Perimeter ≈ 32.78 m.

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Example 4 — Equilateral Triangle Side = 8 (half-triangle)

Equilateral triangle side = 8; altitude creates a 30-60-90 with hypotenuse = 8

Short leg = 8/2 = 4 (half the base)

Long leg = 4√3 ≈ 6.928 (this is the altitude of the equilateral triangle)

Altitude of equilateral triangle side 8 = 6.928. Area of equilateral triangle = 2 × area of 30-60-90 = 2 × (½ × 4 × 6.928) ≈ 27.71.

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Frequently Asked Questions

What are the side ratios of a 30-60-90 triangle?

The sides of a 30-60-90 triangle are always in the ratio 1 : √3 : 2, corresponding to the short leg (opposite 30°) : long leg (opposite 60°) : hypotenuse (opposite 90°). If the short leg is a, then the long leg is a√3 and the hypotenuse is 2a. For example, if the short leg is 5, the long leg is 5√3 ≈ 8.66 and the hypotenuse is 10.

How do you find the sides of a 30-60-90 triangle?

Start from whichever side you know: (1) From short leg a: long leg = a√3, hypotenuse = 2a. (2) From long leg b: short leg = b/√3 = b√3/3, hypotenuse = 2b/√3. (3) From hypotenuse c: short leg = c/2, long leg = c√3/2. All derived from the fundamental 1:√3:2 ratio.

What is the area of a 30-60-90 triangle?

Area = ½ × base × height = ½ × (short leg) × (long leg) = ½ × a × a√3 = a²√3/2. In terms of the hypotenuse c: Area = c²√3/8. Example: short leg = 6 → Area = 6² × √3/2 = 36√3/2 = 18√3 ≈ 31.18 square units.

How is a 30-60-90 triangle related to an equilateral triangle?

A 30-60-90 triangle is exactly half of an equilateral triangle. When you draw the altitude of an equilateral triangle with side s, it bisects the triangle into two congruent 30-60-90 triangles. Each has: short leg = s/2 (half the base), long leg = s√3/2 (the altitude), hypotenuse = s (the original side).

What are the trigonometric values for 30° and 60°?

From the 30-60-90 ratio (1:√3:2): sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3. For 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3. These are exact values, not approximations, and are required for many geometry and trigonometry problems.

What is the perimeter of a 30-60-90 triangle with hypotenuse 10?

Short leg = 10/2 = 5. Long leg = 10√3/2 = 5√3 ≈ 8.660. Perimeter = 5 + 5√3 + 10 = 15 + 5√3 ≈ 15 + 8.660 = 23.66 units. Area = ½ × 5 × 5√3 = 25√3/2 ≈ 21.65 square units.

How do you construct a 30-60-90 triangle?

Method 1 (from equilateral triangle): Draw an equilateral triangle with any side length. Draw the altitude from one vertex to the midpoint of the opposite side. This creates two 30-60-90 triangles. Method 2 (compass and straight edge): Draw a line segment, construct a 60° angle at one end using equilateral construction, then drop a perpendicular from the other end.

Can all sides of a 30-60-90 triangle be integers?

Yes. The simplest integer example is the (1, √3, 2) ratio — but √3 is irrational. Pythagorean triples don't include this exact ratio. However, for practical purposes, (5, 8.66, 10) is very close. For exact integers, there is no perfect 30-60-90 right triangle with integer sides, because the 1:√3 ratio is irrational. Contrast with 45-45-90 triangles, which also have no integer-side versions.

Where do 30-60-90 triangles appear in real life?

Common applications: (1) Architecture — pitched roofs at 30° or 60° angles. (2) Engineering — force component decomposition at 30° or 60°. (3) Equilateral honeycomb structures in materials science. (4) Crystal lattice geometries. (5) Musical instrument design (triangular resonators). (6) Navigation — compass bearings at 30°, 60° increments. (7) Drafting — 30-60-90 set squares are standard drafting tools.

What is the height of a 30-60-90 triangle?

There are three heights (altitudes), one from each vertex. (1) Altitude to hypotenuse: h = a × √3/2 × (a/(2a)) = a√3/2 × a/(c) ... For a right triangle with legs a and b and hypotenuse c, altitude to hypotenuse = ab/c = a × a√3/(2a) = a√3/2. Wait, that's the long leg b itself. The altitude from the right angle to the hypotenuse = (a × b)/c = a²√3/(2a) = a√3/2. (2) The short leg is the altitude to the long leg. (3) The long leg is the altitude to the short leg.

🔗 Related Calculators

📌 Quick Tips

💡The 30-60-90 triangle has sides in ratio 1 : √3 : 2. Memorize this: if the short leg is 1, the long leg is √3 ≈ 1.732, and the hypotenuse is 2.

💡The short leg is opposite the 30° angle. The long leg is opposite the 60° angle. The hypotenuse is opposite the 90° angle.

💡A 30-60-90 triangle is half of an equilateral triangle. If you draw an altitude in an equilateral triangle with side s, you get two 30-60-90 triangles each with short leg = s/2.

💡The area of a 30-60-90 triangle with short leg a is: Area = (√3/2) × a². This simplifies to ½ × a × a√3.

💡In unit circle trigonometry: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 = √3/3 — all derived from the 30-60-90 ratio.