45-45-90 Triangle Calculator
Calculate the leg, hypotenuse, area, and perimeter of any 45-45-90 right triangle.
📊 What is a 45-45-90 Triangle?
A 45-45-90 triangle is a special right triangle with interior angles of exactly 45 degrees, 45 degrees, and 90 degrees. Because both acute angles are equal, the two legs opposite them are also equal, making it an isosceles right triangle. This combination of right angle and isosceles property gives the 45-45-90 triangle its unique and predictable behaviour: the sides are always in the fixed ratio 1 : 1 : √2, no matter what size the triangle is.
This means that if you know any one side, you can calculate every other dimension without needing any trigonometric tables. The hypotenuse is always exactly √2 (approximately 1.41421) times each leg, and the two legs are always equal to each other. This predictability makes the 45-45-90 triangle one of the most commonly encountered shapes in geometry, architecture, engineering, and everyday design work.
The 45-45-90 triangle appears naturally whenever a square is divided diagonally. If a square has side length s, its diagonal measures s√2, and the two triangles formed are both 45-45-90 right triangles with legs equal to s and hypotenuse equal to s√2. This is also the origin of the √2 factor in the diagonal formula for a square.
In trigonometry, the 45-degree angle is fundamental because sin(45°) = cos(45°) = √2/2 ≈ 0.7071, and tan(45°) = 1. These are among the most frequently used exact trigonometric values, derived directly from the 1 : 1 : √2 ratio of this triangle. Students who memorise the 45-45-90 ratios can solve many geometry and trigonometry problems mentally, without a calculator.
Practically, 45-degree angles appear everywhere: roof pitch calculations, mitre cuts in woodworking, diagonal bracing in structural engineering, and perspective drawing in art all rely on 45-degree geometry. The drafting set square is a physical 45-45-90 triangle used by architects and engineers daily.
📐 Formula and Derivation
The 45-45-90 triangle formulas follow directly from the Pythagorean theorem applied to an isosceles right triangle.
Given leg a:
Given hypotenuse h:
Derivation: For a right triangle with two equal legs a, the Pythagorean theorem gives h² = a² + a² = 2a², so h = a√2. Solving for a: a = h/√2 = h√2/2.
📖 How to Use This Calculator
Steps to Calculate
💡 Example Calculations
Example 1 - Leg of 5 units
Find the hypotenuse, area, and perimeter of a 45-45-90 triangle with legs of 5 cm each.
Example 2 - Hypotenuse of 10 units
A hypotenuse of 10 m is given. Find the legs, area, and perimeter.
Example 3 - Leg of 8 units (tile diagonal problem)
A square ceramic tile has a side of 8 inches. A cut along the diagonal creates two 45-45-90 triangles. Find all dimensions of each triangle.
Example 4 - Hypotenuse of 14.142 units
A ramp forms a 45-degree angle and has a slant length (hypotenuse) of 14.142 m. Find the horizontal run and vertical rise.
Frequently Asked Questions
🔗 Related Calculators
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle with angles of 45 degrees, 45 degrees, and 90 degrees. Because the two acute angles are equal, the two legs opposite them are also equal in length, making it an isosceles right triangle. The sides are always in the ratio 1 : 1 : root(2), meaning if each leg is a, then the hypotenuse is a times root(2) approximately 1.4142 times a.
What is the formula for the hypotenuse of a 45-45-90 triangle?
If each leg has length a, then the hypotenuse h = a times root(2). For example, if the leg is 5 cm, the hypotenuse = 5 times 1.41421 = 7.0711 cm. This follows from the Pythagorean theorem: h squared = a squared + a squared = 2a squared, so h = a times root(2).
How do I find the leg length from the hypotenuse in a 45-45-90 triangle?
Divide the hypotenuse by root(2). Equivalently, multiply by root(2)/2 = 1/root(2). Formula: a = h / root(2) = h times root(2) / 2. Example: if hypotenuse = 10, each leg = 10 / 1.41421 = 7.0711 units.
What is the area of a 45-45-90 triangle?
Area = leg squared divided by 2, i.e. A = a squared / 2. This is because the two legs serve as base and height: Area = (1/2) times base times height = (1/2) times a times a = a squared / 2. If the hypotenuse is h, then Area = h squared / 4.
What is the perimeter of a 45-45-90 triangle?
Perimeter = 2a + a times root(2) = a(2 + root(2)), where a is the leg length. If you know the hypotenuse h, perimeter = h times (root(2) + 1) = h times 2.41421. Example: leg = 5, perimeter = 5 times (2 + 1.41421) = 5 times 3.41421 = 17.0711 units.
Why is the 45-45-90 triangle called a special right triangle?
Special right triangles are right triangles whose side ratios are constant regardless of size. The 45-45-90 triangle always has sides in ratio 1 : 1 : root(2), and the 30-60-90 triangle always has sides in ratio 1 : root(3) : 2. These fixed ratios allow you to calculate all sides from just one measurement, which makes them extremely useful in geometry, trigonometry, and engineering.
What are the trigonometric values for 45 degrees?
sin(45) = cos(45) = root(2)/2 approximately 0.7071. tan(45) = 1. These values come directly from the 45-45-90 triangle: in a right triangle with two equal legs a, sin(45) = opposite / hypotenuse = a / (a times root(2)) = 1/root(2) = root(2)/2. The fact that sin(45) equals cos(45) reflects the symmetry of the isosceles right triangle.
How is a 45-45-90 triangle related to a square?
Cutting a square diagonally produces exactly two 45-45-90 triangles. The diagonal of the square becomes the hypotenuse of each triangle, and the two sides of the square become the two equal legs. If the square has side s, the diagonal = s times root(2). This is also why the diagonal formula for a square uses root(2).
Can a 45-45-90 triangle be used in real life?
Yes, frequently. Architects use 45-degree angles for roof pitches, staircases, and bracing. Carpenters use 45-degree mitre cuts to join wood at corners. Engineers use the 1:1:root(2) ratio to calculate diagonal distances. In art and design, 45-degree diagonals create visual balance. The 45-degree set square is a standard drafting tool based on this triangle.
What is the relationship between a 45-45-90 triangle and a unit circle?
On the unit circle (radius = 1), the point at 45 degrees has coordinates (root(2)/2, root(2)/2), approximately (0.7071, 0.7071). These coordinates are the cosine and sine of 45 degrees respectively, and they come directly from the 45-45-90 triangle inscribed in the unit circle.
How do I calculate the 45-45-90 triangle with a leg of 8?
Leg a = 8. Hypotenuse = 8 times root(2) = 8 times 1.41421 = 11.3137 units. Area = 8 squared / 2 = 64 / 2 = 32 sq units. Perimeter = 2 times 8 + 11.3137 = 16 + 11.3137 = 27.3137 units. All values are exact multiples of root(2).