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 1:3:21 : \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.

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: Special right triangles let you write exact side ratios for 30-60-90 and 45-45-90 with no calculator.

Common stuck point: 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.

Sense of Study hint: Ask: Are the right triangle's angles exactly 30-60-90 or 45-45-90 so I can use a memorized ratio?

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.

Example 4

medium
A baseball diamond is a square with 9090 ft sides. Find the distance from home plate to second base exactly.

Example 5

hard
From a point 5050 ft away, the angle of elevation to a flagpole's top is 30°30°. Find the pole's height exactly.

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.

Example 3

easy
What is the side ratio of a 45-45-90 triangle?

Example 4

easy
What is the side ratio of a 30-60-90 triangle?

Example 5

easy
A 45-45-90 triangle has legs of length 5. Find the hypotenuse.

Example 6

easy
A 30-60-90 triangle has its shortest side (opposite 30°) equal to 4. Find the hypotenuse.

Example 7

easy
A 30-60-90 triangle comes from cutting which shape in half?

Example 8

easy
A 45-45-90 triangle comes from cutting which shape along its diagonal?

Example 9

easy
In a 30-60-90 triangle with short side 1, what is the length of the side opposite the 60° angle?

Example 10

easy
Why are 45-45-90 and 30-60-90 called 'special' right triangles?

Example 11

medium
A 45-45-90 triangle has a hypotenuse of 1010. Find the length of each leg.

Example 12

medium
A 30-60-90 triangle has its hypotenuse equal to 12. Find the short side and the medium side.

Example 13

medium
Find the exact value of sin⁡45°\sin 45° using a 45-45-90 triangle.

Example 14

medium
Find the exact value of cos⁡30°\cos 30° using a 30-60-90 triangle.

Example 15

medium
A square has side 8. Find the length of its diagonal exactly.

Example 16

medium
An equilateral triangle has side 6. Find its height exactly.

Example 17

medium
Why is the short side of a 30-60-90 triangle exactly half the hypotenuse?

Example 18

medium
Find the exact value of tan⁡60°\tan 60° using a 30-60-90 triangle.

Example 19

challenge
A 45-45-90 triangle is inscribed so its hypotenuse is the diameter of a circle of radius 5. Find the legs.

Example 20

challenge
Find the area of an equilateral triangle with side 10, using the 30-60-90 height.

Example 21

challenge
Verify the 30-60-90 ratio 1:3:21 : \sqrt 3 : 2 satisfies the Pythagorean theorem.

Example 22

challenge
A regular hexagon has side 6. Using special triangles, find its area.

Example 23

easy
A 45-45-90 triangle has legs of length 33. Find the hypotenuse.

Example 24

easy
A 30-60-90 triangle has its short leg =6= 6. Find the long leg.

Example 25

easy
A 45-45-90 triangle has hypotenuse 626\sqrt 2. Find the leg length.

Example 26

easy
A 30-60-90 triangle has long leg =53= 5\sqrt 3. Find the hypotenuse.

Example 27

easy
A square has side 44. Find the diagonal length exactly.

Example 28

medium
A right triangle with a 30°30° angle has hypotenuse 2020. Find both legs.

Example 29

medium
An equilateral triangle has side 88. Find its altitude exactly.

Example 30

medium
Find sin⁡60°\sin 60° exactly.

Example 31

medium
Find tan⁡30°\tan 30° exactly.

Example 32

medium
A ramp rises at 30°30° to a height of 44 ft. Find the ramp length.

Example 33

medium
A square has diagonal 1414 in. Find its perimeter exactly.

Example 34

medium
A 30-60-90 triangle has perimeter 6+636 + 6\sqrt 3. Find the short leg.

Example 35

hard
Find the area of an equilateral triangle with perimeter 1818.

Example 36

hard
The diagonal of a square equals the long leg of a 30-60-90 with short leg 55. Find the side of the square.

Example 37

hard
An isosceles right triangle has area 5050. Find its hypotenuse.

Example 38

hard
A regular hexagon has area 54354\sqrt 3. Find its side length.

Example 39

hard
A square is inscribed in a circle of radius 55. Find its area.

Example 40

hard
An equilateral triangle with side ss is inscribed in a circle of radius r=4r = 4. Find ss exactly.

Example 41

challenge
In a right triangle with ∠A=30°\angle A = 30° and BC=10BC = 10 (the hypotenuse), find the length of the altitude from CC to ABAB.

Example 42

challenge
A regular octagon is inscribed in a circle of radius 11. Find the length of one side exactly.
← Back to Special Right Triangles Formula Explained Practice Mode

Background Knowledge

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

right triangle trigonometrypythagorean theoremsquare roots