Special Right Triangles Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Special Right Triangles.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: These two triangle families have exact, memorable side ratios that let you solve problems without a calculator.

Common stuck point: 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.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
The diagonal of a square is 10 cm. Find the side length of the square.

Example 2

hard
An equilateral triangle has side length 12. Find its height and area using the 30-60-90 triangle relationship.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

right triangle trigonometrypythagorean theoremsquare roots