30-60-90 and 45-45-90 Triangle Rules

Cut a square in half along its diagonal and you get a right triangle with two equal legs. Triangles like this show up everywhere, in ramps, rooftops, and folded napkins, so it pays to know them by heart. Start with the smallest one and measure its hypotenuse with the Pythagorean Theorem.

Problem

A right triangle has legs

1

and

1

. Use the Pythagorean Theorem to find the square of the hypotenuse. What number is

c^{2}

Show a hint

The theorem says $a^{2} + b^{2} = c^{2}$ , and here both legs are $1$ .
Compute $1^{2} + 1^{2}$ .

Show the full solution

The legs give

c^{2} = 1^{2} + 1^{2} = 2

, so the hypotenuse itself is

\sqrt{2}

. The square of the hypotenuse is

2

Problem

Each leg of a 45-45-90 triangle is

5

. The hypotenuse simplifies to

k \sqrt{2}

. What whole number is

k

Show a hint

The hypotenuse of a 45-45-90 triangle is the leg times $\sqrt{2}$ .
With legs of $5$ , the hypotenuse is $5 \sqrt{2}$ .

Show the full solution

Hypotenuse

= s \sqrt{2} = 5 \sqrt{2}

, so

k = 5

. The theorem confirms it, since

5^{2} + 5^{2} = 50

and

\sqrt{50} = 5 \sqrt{2}

Problem

A wheelchair ramp rises at a

45

degree angle, so its rise and its horizontal run are the two legs of a 45-45-90 triangle. The run is

12

meters. How many meters is the rise?

Show a hint

In a 45-45-90 triangle the two legs are equal.
The rise and the run are the two legs.

Show the full solution

The legs of a 45-45-90 triangle are equal, so the rise matches the run at

12

meters.

Problem

The hypotenuse of a 45-45-90 triangle measures

7 \sqrt{2}

. How long is each leg?

Show a hint

Run the rule backwards. The hypotenuse is leg $\times \sqrt{2}$ , so the leg is hypotenuse $\div \sqrt{2}$ .
If $s \sqrt{2} = 7 \sqrt{2}$ , what is $s$ ?

Show the full solution

The hypotenuse is always

s \sqrt{2}

. Matching

s \sqrt{2} = 7 \sqrt{2}

gives

s = 7

Problem

The short leg of a 30-60-90 triangle is

4

. The hypotenuse is always twice the short leg. How long is the hypotenuse?

Show a hint

Hypotenuse $= 2 \times$ short leg.

Show the full solution

Twice the short leg is

2 \times 4 = 8

Problem

The hypotenuse of a 30-60-90 triangle is

14

. How long is the short leg?

Show a hint

The hypotenuse is twice the short leg, so run it backwards.
Half of $14$ .

Show the full solution

The short leg is half the hypotenuse,

14 \div 2 = 7

Problem

The short leg of a 30-60-90 triangle is

5

. The long leg simplifies to

k \sqrt{3}

. What whole number is

k

Show a hint

The long leg is the short leg times $\sqrt{3}$ .
With a short leg of $5$ , the long leg is $5 \sqrt{3}$ .

Show the full solution

Long leg

=

short leg

\times \sqrt{3} = 5 \sqrt{3}

. So

k = 5

Problem

An equilateral triangle has

10

cm sides. Its height simplifies to

k \sqrt{3}

cm. What whole number is

k

Show a hint

The height cuts the triangle into two 30-60-90 halves, and the short leg of each half is half of $10$ .
Height $=$ short leg $\times \sqrt{3}$ , and the short leg is $5$ .

Show the full solution

The height splits the base, so the short leg is

10 \div 2 = 5

, and the height is the long leg,

5 \sqrt{3}

. So

k = 5

Practice these ideas

Practice

Each leg of a 45-45-90 triangle is

3

. The hypotenuse simplifies to

k \sqrt{2}

. What is

k

Show the solution

Hypotenuse

= 3 \sqrt{2}

, so

k = 3

Practice

The hypotenuse of a 45-45-90 triangle is

11 \sqrt{2}

. How long is each leg?

Show the solution

From

s \sqrt{2} = 11 \sqrt{2}

we get

s = 11

Practice

A square has

7

inch sides. Its diagonal simplifies to

k \sqrt{2}

inches. What is

k

Show the solution

The diagonal is side

\times \sqrt{2} = 7 \sqrt{2}

, so

k = 7

Practice

The short leg of a 30-60-90 triangle is

6

. How long is the hypotenuse?

Show the solution

Hypotenuse

= 2 \times 6 = 12

Practice

The hypotenuse of a 30-60-90 triangle is

20

. How long is the short leg?

Show the solution

Half of

20

10

Practice

The short leg of a 30-60-90 triangle is

8

. The long leg simplifies to

k \sqrt{3}

. What is

k

Show the solution

The long leg is

8 \sqrt{3}

, so

k = 8

Practice

An equilateral triangle has

12

cm sides. Its height simplifies to

k \sqrt{3}

cm. What is

k

Show the solution

The short leg is

12 \div 2 = 6

, so the height is

6 \sqrt{3}

and

k = 6

Practice

Each leg of a 45-45-90 triangle is

9

. Use the Pythagorean Theorem to check the shortcut. What is the square of the hypotenuse?

Show the solution

c^{2} = 9^{2} + 9^{2} = 81 + 81 = 162

. The shortcut agrees, since

(9 \sqrt{2})^{2} = 81 \times 2 = 162

Special Right Triangles

Practice these ideas