45-45-90 Triangle (Rules, Formula & Theorem)

Written by

Malcolm McKinsey

January 15, 2023

Fact-checked by

Paul Mazzola

45-45-90 triangle

45-45-90 triangles are special right triangles with one 90 degree angle and two 45 degree angles. All 45-45-90 triangles are considered special isosceles triangles. The 45-45-90 triangle has three unique properties that make it very special and unlike all the other triangles.

45-45-90 triangle ratio

There are two ratios for 45-45-90 triangles:

• The ratio of the sides equals $1:1:\sqrt{2}$

• The ratio of the angles equals $1:1:2$

Properties of 45-45-90 triangles

To identify 45-45-90 special right triangle, check for these three identifying properties:

1. The polygon is an isosceles right triangle

2. The two side lengths are congruent, and their opposite angles are congruent

3. The hypotenuse (longest side) is the length of either leg times square root (sqrt) of two, $\sqrt{2}$

All 45-45-90 triangles are similar because they all have the same interior angles.

45-45-90 triangle theorem

To solve for the hypotenuse length of a 45-45-90 triangle, you can use the 45-45-90 theorem, which says the length of the hypotenuse of a 45-45-90 triangle is the $\sqrt{2}$ times the length of a leg.

45-45-90 triangle formula

You can also use the general form of the Pythagorean Theorem to find the length of the hypotenuse of a 45-45-90 triangle.

Here is a 45-45-90 triangle. Let's use both methods to find the unknown measure of a triangle where we only know the measure of one leg is 59 yards:

We can plug the known length of the leg into our 45-45-90 theorem formula:

Using the Pythagorean Theorem:

Both methods produce the same result!

45-45-90 triangle rules

The main rule of 45-45-90 triangles is that it has one right angle and while the other two angles each measure 45°. The lengths of the sides adjacent to the right triangle, the shorter sides have an equal length.

Another rule is that the two sides of the triangle or legs of the triangle that form the right angle are congruent in length.

Knowing these basic rules makes it easy to construct a 45-45-90 triangle.

Constructing a 45-45-90 triangle

The easiest way to construct a 45-45-90 triangle is as follows:

1. Construct a square four equal sides to the desired length of the triangle's legs

2. Construct either diagonal of the square

Striking the diagonal of the square creates two congruent 45-45-90 triangles. Half of a square that has been cut by a diagonal is a 45-45-90 triangle.

The diagonal becomes the hypotenuse of a right triangle.

You can also construct the triangle using a straightedge and drawing compass:

1. Construct a line segment more than twice as long as the desired length of your triangle's leg

2. Open the compass to span more than half the distance of the line segment

3. Use the compass to construct a perpendicular bisector of the line segment by scribing arcs from both endpoints above and below the line segment; this will produce two intersecting arcs above and two intersecting arcs below the line segment

4. Use the straightedge to draw the perpendicular bisector by connecting the intersecting arcs

5. Reset the compass with the point on the intersection of the two line segments and the span of the compass set to your desired length of the triangle's leg

6. Strike two arcs, one on the line segment and one on the perpendicular bisector

7. Connect the intersections of the arcs and segments

This method takes more time than the square method but is elegant and does not require measuring.

How to solve a 45-45-90 triangle

The length of the hypotenuse, which is the leg times $\sqrt{2}$, is key to calculating the missing sides:

• If you know the measure of the hypotenuse, divide the hypotenuse by $\sqrt{2}$ to find the length of either leg

• If you know the length of one leg, you know the length of the other leg (legs are congruent)

• If you know either leg's length, multiply the leg length times $\sqrt{2}$ to find the hypotenuse

45-45-90 triangle example problems

Here we have a 45-45-90 triangle with a hypotenuse of $3\sqrt{2}$ meters, and each leg is 3 meters.

Take is a 45-45-90 triangle with sides measuring 9.5 feet. What is the length of the hypotenuse?

You can answer either with 13.435 feet, or with $9.5\sqrt{2}$ feet.