特殊直角三角形计算器

Two triangles appear so often in geometry, trigonometry, and engineering that they have memorable exact side ratios and earn the name "special right triangles": the 30-60-90 and the 45-45-90. Knowing these ratios by heart lets you solve a large class of problems without a calculator — useful for tests, mental estimation, and recognizing patterns in proofs. This tutorial derives both ratios from first principles, walks through how to use them in either direction (from any given side to the other two), and shows where they appear in trigonometry.

The two ratios at a glance

"Short leg" is the side opposite the smallest angle (30° in the first, either 45° in the second). "Long leg" is opposite the next angle. "Hypotenuse" is opposite the right angle and is always longest.

Why 30-60-90 sides are 1 : √3 : 2

Take an equilateral triangle with side length 2. All three angles are 60°. Drop a perpendicular from one vertex to the opposite side. This splits the equilateral triangle into two congruent halves — each is a 30-60-90 triangle.

The hypotenuse of each half is the original side of the equilateral triangle, length 2. The short leg is half of the opposite side, length 1. The long leg is the perpendicular height, which we get from the Pythagorean theorem:

long leg² = 2² − 1² = 3, so long leg = √3.

Hence the 30-60-90 ratio is 1 : √3 : 2. Scaling up: a 30-60-90 with short leg s has long leg s√3 and hypotenuse 2s.

Why 45-45-90 sides are 1 : 1 : √2

Take a square with side length 1. Draw one of its diagonals. The diagonal splits the square into two congruent right triangles, each isosceles with both legs equal to 1.

By the Pythagorean theorem, the diagonal (hypotenuse of each half-square) is √(1² + 1²) = √2.

Hence the 45-45-90 ratio is 1 : 1 : √2. Scaling up: a 45-45-90 with leg L has hypotenuse L√2.

Solving from any given side — 30-60-90

Pick which side you know, then use the ratio:

• Given short leg s: long leg = s√3, hypotenuse = 2s.
• Given long leg L: short leg = L/√3 = L√3/3, hypotenuse = 2L/√3 = 2L√3/3.
• Given hypotenuse h: short leg = h/2, long leg = h√3/2.

Example: hypotenuse h = 10. Short leg = 10/2 = 5. Long leg = 10·√3/2 = 5√3 ≈ 8.660.

Solving from any given side — 45-45-90

• Given a leg L: the other leg is also L, hypotenuse = L√2.
• Given the hypotenuse h: each leg = h/√2 = h√2/2.

Example: leg L = 5. Hypotenuse = 5√2 ≈ 7.071.

How these triangles power trigonometry

The exact trigonometric values for 30°, 45°, and 60° all come directly from the special right triangles. Read each ratio as sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent:

These exact values are why 30, 45, and 60 degrees appear in the answers to so many "evaluate without a calculator" trig problems. The 30-60-90 and 45-45-90 triangles are literally the source of the table.

Worked example — combining special triangles

A common test problem: a 60° angle is drawn from a horizontal baseline. From its apex, a 30° angle is then drawn off the original 60° hypotenuse. Find ratios of the resulting segments.

Setup: the first 30-60-90 has its 60° at the baseline. The second 30-60-90 nests inside it, sharing the hypotenuse of the first. Walking through this with exact ratios (no decimals) lets you express all segments in terms of one chosen length plus √3 — far easier than calculator arithmetic and far more elegant when written up.

Real-world applications

• Drafting tools. The two standard set squares used in technical drawing are exactly the 30-60-90 and 45-45-90 triangles.
• Carpentry. A "miter cut" at 45° produces two 45-45-90 corners that mate flush — used for picture frames, door casings, and crown molding.
• Roofing. Many residential roof pitches use 30° or 45° for aesthetic and structural reasons; their rafters' length ratios come directly from these triangles.
• Navigation. Bearings of N30°E, N45°E, etc. lead to course calculations that simplify when the triangle is a special one.

Common mistakes

• Confusing the short and long legs of 30-60-90. The short leg is opposite the 30° angle (smallest), the long leg is opposite the 60° angle (medium). Easy to swap if you do not draw the triangle.
• Treating 30-60-90 ratio as 1 : 2 : 3. The ratio is 1 : √3 : 2, NOT 1 : 2 : 3. √3 ≈ 1.732, which is between 1 and 2.
• Rationalizing too aggressively. Expressing 1/√3 as √3/3 is mathematically equivalent and often preferred. Either form is correct, but a textbook may insist on one. Check your style guide.
• Forgetting that "special" is exact only for these two triangles. A right triangle with angles 31-59-90 is NOT a 30-60-90 and does not have the 1 : √3 : 2 ratio. Stick to the named angles.