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45-45-90 Triangle Calculator

Calculate all sides of a 45-45-90 special right triangle from one known leg. Enter the leg length to find the hypotenuse using the fixed ratio 1:1:sqrt(2), fundamental for trigonometry, diagonal calculations, and standardized testing preparation.

Reviewed by Chase FloiedUpdated

This free online 45-45-90 triangle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Length of either leg (both legs are equal)

Results

Hypotenuse

7.0711

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the 45-45-90 Triangle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The 45-45-90 Triangle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

45-45-90 Triangle Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the 45-45-90 Triangle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The 45-45-90 Triangle Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate all sides of a 45-45-90 special right triangle from one known leg. Enter the leg length to find the hypotenuse using the fixed ratio 1:1:sqrt(2), fundamental for trigonometry, diagonal calculations, and standardized testing preparation. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About 45-45-90 Triangle Calculator

The 45-45-90 Triangle Calculator computes the hypotenuse and area of the special 45-45-90 isosceles right triangle from one leg length. This triangle has two 45-degree angles and sides in the ratio 1 : 1 : sqrt(2). It is one of the two special right triangles essential for trigonometry and standardized tests. The 45-45-90 triangle appears whenever you bisect a square along its diagonal: the diagonal of a unit square is sqrt(2), and each resulting triangle is a 45-45-90. Applications include computing screen diagonals, roof rafter calculations, architectural symmetry, compass heading decomposition, and any situation involving 45-degree angles.

The Math Behind It

The 45-45-90 triangle is an isosceles right triangle: two legs are equal and the angles are 45, 45, and 90 degrees. The side ratios 1 : 1 : sqrt(2) follow directly from the Pythagorean theorem: if each leg has length s, then the hypotenuse is sqrt(s^2 + s^2) = s*sqrt(2). The exact trigonometric values follow: sin(45) = cos(45) = 1/sqrt(2) = sqrt(2)/2, and tan(45) = 1. The fact that tan(45) = 1 means a 45-degree line has slope 1, which is the diagonal direction. The 45-45-90 triangle arises naturally from the diagonal of a square. If a square has side length s, its diagonal has length s*sqrt(2), and the diagonal divides the square into two congruent 45-45-90 triangles. This is why TV screens are measured diagonally: a 50-inch diagonal screen with 16:9 aspect ratio has specific width and height determined by the relationship between diagonal and sides. In coordinate geometry, the 45-45-90 triangle appears in rotation by 45 degrees. The rotation matrix for 45 degrees uses cos(45) and sin(45), both equal to sqrt(2)/2. In physics, when forces or velocities act at 45 degrees, their horizontal and vertical components are equal, each being the magnitude times sqrt(2)/2. The irrationality of sqrt(2) was one of the first mathematical discoveries of the ancient Greeks and is attributed to the Pythagorean school. The proof that sqrt(2) is irrational (by contradiction, assuming it equals p/q in lowest terms) was a foundational moment in the history of mathematics, revealing that not all lengths are expressible as ratios of whole numbers.

Formula Reference

45-45-90 Side Ratios

leg : leg : hypotenuse = 1 : 1 : sqrt(2)

Variables: Both legs are equal; hypotenuse = leg * sqrt(2)

Worked Examples

Example 1: Leg = 5

Find the hypotenuse and area of a 45-45-90 triangle with leg 5.

Step 1:Hypotenuse = 5 * sqrt(2) = 5 * 1.4142 = 7.0711
Step 2:Area = 5^2 / 2 = 25 / 2 = 12.5

Hypotenuse = 7.07, Area = 12.5 square units.

Example 2: Square Diagonal

A square has side length 10. Find its diagonal.

Step 1:The diagonal creates two 45-45-90 triangles with leg = 10
Step 2:Diagonal = 10 * sqrt(2) = 14.142

The square's diagonal is 14.14 units.

Common Mistakes & Tips

  • !Using sqrt(3) instead of sqrt(2). The 45-45-90 triangle uses sqrt(2); the 30-60-90 triangle uses sqrt(3).
  • !Applying the formula to non-isosceles right triangles. Both legs must be equal for the 1:1:sqrt(2) ratio to apply.
  • !Dividing the hypotenuse by 2 instead of sqrt(2) to find a leg. If you know the hypotenuse, the leg = hypotenuse / sqrt(2) = hypotenuse * sqrt(2) / 2.

Related Concepts

30-60-90 Triangle

The other special right triangle with ratio 1:sqrt(3):2, used for 30-degree and 60-degree angles.

Right Triangle Trig

The general right triangle solver for triangles that are not special right triangles.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

30-60-90 Triangle Calculator

Right Triangle Trigonometry Calculator

Frequently Asked Questions

Why is the diagonal of a square irrational?

Because the diagonal of a unit square is sqrt(2), which was proven irrational by the ancient Greeks. This means no matter how you try to express it as a fraction, it cannot be done. This discovery shocked the Pythagoreans, who believed all numbers were rational.

How do I find the leg from the hypotenuse?

Divide the hypotenuse by sqrt(2). Equivalently, multiply the hypotenuse by sqrt(2)/2. For example, if the hypotenuse is 10, each leg is 10/sqrt(2) = 10*sqrt(2)/2 = 7.07.

Where does the 45-45-90 triangle appear in everyday life?

Computing screen diagonals, measuring the distance from corner to corner of a square room, cutting plywood along a 45-degree miter joint, designing 45-degree roof pitches, and determining how far an object moves diagonally when it moves equal distances horizontally and vertically.