Special Right Triangles Formula

The Formula

\text{45-45-90: } \quad 1 : 1 : \sqrt{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 5: \text{hypotenuse} = 5\sqrt{2} \approx 7.07 A 30-60-90 triangle with short leg 4: \text{long leg} = 4\sqrt{3}, \quad \text{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 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1 : 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\text{-}45\text{-}90: sides a : a : a\sqrt{2}; \sin 45° = \cos 45° = \frac{\sqrt{2}}{2}. 30\text{-}60\text{-}90: sides a : a\sqrt{3} : 2a; \sin 30° = \frac{1}{2}, \cos 30° = \frac{\sqrt{3}}{2}, \sin 60° = \frac{\sqrt{3}}{2}, \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.

Solution

  1. 1
    Step 1: Recall the 45-45-90 ratio: if each leg has length a, the hypotenuse has length a\sqrt{2}.
  2. 2
    Step 2: The legs are both 7, so a = 7.
  3. 3
    Step 3: Hypotenuse = 7\sqrt{2} \approx 7 \times 1.414 = 9.9.

Answer

Hypotenuse = 7\sqrt{2} \approx 9.9.
In a 45-45-90 triangle, the two legs are equal and the hypotenuse is \sqrt{2} times the length of a leg. This ratio (1:1:\sqrt{2}) comes from applying the Pythagorean theorem: a^2 + a^2 = c^2, so c = a\sqrt{2}.

Example 2

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

Common Mistakes

  • Swapping the \sqrt{3} and 2 in the 30-60-90 ratio
  • Applying the 45-45-90 ratio to a triangle that isn't isosceles
  • Forgetting to multiply all sides by the same scale factor

Why This Formula Matters

Appear constantly in standardized tests, architecture, and physics. They make exact computation possible where other triangles require approximation.

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 : \sqrt{3} : 2, and the 45-45-90 triangle with sides in ratio 1 : 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 : c where a is opposite the smallest angle and c is the hypotenuse

Why is the Special Right Triangles formula important in Math?

Appear constantly in standardized tests, architecture, and physics. They make exact computation possible where other triangles require approximation.

What do students get wrong about Special Right Triangles?

In a 30-60-90, the side opposite 30° is the shortest (1), the side opposite 60° is the middle (\sqrt{3}), and the hypotenuse is the longest (2). Students often mix up which ratio goes with which angle.

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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