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.

The Formula

45-45-90: 1:1:2\text{45-45-90: } \quad 1 : 1 : \sqrt{2} 30-60-90: 1:3:2\text{30-60-90: } \quad 1 : \sqrt{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 55: hypotenuse=52≈7.07\text{hypotenuse} = 5\sqrt{2} \approx 7.07 A 30-60-90 triangle with short leg 44: long leg=43,hypotenuse=8\text{long leg} = 4\sqrt{3}, \quad \text{hypotenuse} = 8

Notation

Side ratios are written as a:b:ca : b : c where aa is opposite the smallest angle and cc 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 1:3:21 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1:1:21 : 1 : \sqrt{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

45-45-9045\text{-}45\text{-}90: sides a:a:a2a : a : a\sqrt{2}; sin⁡45°=cos⁡45°=22\sin 45° = \cos 45° = \frac{\sqrt{2}}{2}. 30-60-9030\text{-}60\text{-}90: sides a:a3:2aa : a\sqrt{3} : 2a; sin⁡30°=12\sin 30° = \frac{1}{2}, cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2}, sin⁡60°=32\sin 60° = \frac{\sqrt{3}}{2}, cos⁡60°=12\cos 60° = \frac{1}{2}

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= 7\sqrt{2} \approx 9.9.

First step

1
Step 1: Recall the 45-45-90 ratio: if each leg has length aa, the hypotenuse has length a2a\sqrt{2}.

Full solution

  1. 2
    Step 2: The legs are both 7, so a=7a = 7.
  2. 3
    Step 3: Hypotenuse =72≈7×1.414=9.9= 7\sqrt{2} \approx 7 \times 1.414 = 9.9.
In a 45-45-90 triangle, the two legs are equal and the hypotenuse is 2\sqrt{2} times the length of a leg. This ratio (1:1:21:1:\sqrt{2}) comes from applying the Pythagorean theorem: a2+a2=c2a^2 + a^2 = c^2, so c=a2c = a\sqrt{2}.

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°\tan 45° using a 45-45-90 triangle.

Common Mistakes

  • Swapping which leg is opposite 30°30° vs 60°60° — the shorter leg (×1\times1) is opposite the smaller angle.
  • Using the 30-60-90 ratio for a 45-45-90 triangle — an isosceles right triangle is 1:1:21:1:\sqrt2.
  • Leaving the hypotenuse as the '1' — the hypotenuse is the largest ratio number (2 or 2\sqrt2).

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 1:3:21 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1:1:21 : 1 : \sqrt{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:ca : b : c where aa is opposite the smallest angle and cc 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°30° vs 60°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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