Special Right Triangles Formula

Special right triangles are two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1: √3: 2, and the 45-45-90 triangle with sides in ratio 1: 1: √2.

The Formula

45-45-90: 1:1:2 30-60-90: 1:3:2

When to use: Cut an equilateral triangle in half and you get a 30-60-90 triangle. Cut a square along its diagonal and you get a 45-45-90 triangle. These two cuts give you exact side ratios you can memorize forever—no calculator needed.

Quick Example

A 45-45-90 triangle with legs of length 5: hypotenuse=52≈7.07 A 30-60-90 triangle with short leg 4: long leg=43,hypotenuse=8

Notation

Side ratios are written as a:b:c where a is opposite the smallest angle and c is the hypotenuse

What This Formula Means

Two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1:3:2, and the 45-45-90 triangle with sides in ratio 1:1:2.

Cut an equilateral triangle in half and you get a 30-60-90 triangle. Cut a square along its diagonal and you get a 45-45-90 triangle. These two cuts give you exact side ratios you can memorize forever—no calculator needed.

Formal View

45-45-90: sides a:a:a2; sin⁡45°=cos⁡45°=22. 30-60-90: sides a:a3:2a; sin⁡30°=12, cos⁡30°=32, sin⁡60°=32, cos⁡60°=12

Worked Examples

Example 1

easy
A 45-45-90 triangle has legs of length 7. Find the length of the hypotenuse.

Answer

Hypotenuse =72≈9.9.

First step

1
Step 1: Recall the 45-45-90 ratio: if each leg has length a, the hypotenuse has length a2.

Full solution

  1. 2
    Step 2: The legs are both 7, so a=7.
  2. 3
    Step 3: Hypotenuse =72≈7×1.414=9.9.
In a 45-45-90 triangle, the two legs are equal and the hypotenuse is 2 times the length of a leg. This ratio (1:1:2) comes from applying the Pythagorean theorem: a2+a2=c2, so c=a2.

Example 2

medium
In a 30-60-90 triangle, the hypotenuse is 16. Find the lengths of both legs.

Example 3

easy
Find tan⁡45° using a 45-45-90 triangle.

Common Mistakes

  • Swapping which leg is opposite 30° vs 60° — the shorter leg (×1) is opposite the smaller angle.
  • Using the 30-60-90 ratio for a 45-45-90 triangle — an isosceles right triangle is 1:1:2.
  • Leaving the hypotenuse as the '1' — the hypotenuse is the largest ratio number (2 or 2).

Why This Formula Matters

These two triangles come from cutting an equilateral triangle in half and a square along its diagonal, so they appear constantly and give exact radical answers instead of rounded decimals. Recognizing them saves the full sine-cosine computation. Recognizing it by "Are the right triangle's angles exactly 30-60-90 or 45-45-90 so I can use a memorized ratio?" — rather than by familiar numbers — is what lets a student tell it apart from right triangle trigonometry and pythagorean theorem and pythagorean triples in a mixed problem set.

Frequently Asked Questions

What is the Special Right Triangles formula?

Two families of right triangles whose side ratios can be determined exactly: the 30-60-90 triangle with sides in ratio 1:3:2, and the 45-45-90 triangle with sides in ratio 1:1:2.

How do you use the Special Right Triangles formula?

Cut an equilateral triangle in half and you get a 30-60-90 triangle. Cut a square along its diagonal and you get a 45-45-90 triangle. These two cuts give you exact side ratios you can memorize forever—no calculator needed.

What do the symbols mean in the Special Right Triangles formula?

Side ratios are written as a:b:c where a is opposite the smallest angle and c is the hypotenuse

Why is the Special Right Triangles formula important in Math?

These two triangles come from cutting an equilateral triangle in half and a square along its diagonal, so they appear constantly and give exact radical answers instead of rounded decimals. Recognizing them saves the full sine-cosine computation. Recognizing it by "Are the right triangle's angles exactly 30-60-90 or 45-45-90 so I can use a memorized ratio?" — rather than by familiar numbers — is what lets a student tell it apart from right triangle trigonometry and pythagorean theorem and pythagorean triples in a mixed problem set.

What do students get wrong about Special Right Triangles?

The procedure for special right triangles is the easy part; the trap is swapping which leg is opposite 30° vs 60°. Asking "Are the right triangle's angles exactly 30-60-90 or 45-45-90 so I can use a memorized ratio?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Special Right Triangles formula?

Before studying the Special Right Triangles formula, you should understand: right triangle trigonometry, pythagorean theorem, square roots.

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