Home › Calculators › 45-45-90 Triangle Calculator
45-45-90 Triangle Calculator
Enter a leg or the hypotenuse of a 45-45-90 (isosceles right) triangle to get every side in exact form such as 5√2 and as a decimal, plus the area, perimeter and altitude. You can also start from the area or perimeter.
Last updated:
Enter your values to see the result.
Advertisement
Advertisement
How to use the 45-45-90 Triangle Calculator
- Choose what you know: a leg, the hypotenuse, the altitude, the area or the perimeter.
- Enter the value (decimals and fractions like 7/2 work).
- Pick a unit to label the answers (optional).
- Read the legs and hypotenuse in exact √2 form and as decimals.
- Review the steps and drawing, or download the table as CSV.
45-45-90 triangle formulas
| You know | Leg a | Hypotenuse c | Area |
|---|---|---|---|
| Leg a | a | a√2 | a²/2 |
| Hypotenuse c | c√2/2 | c | c²/4 |
| Altitude h | h√2 | 2h | h² |
| Area A | √(2A) | 2√A | A |
| Perimeter P | P(2 − √2)/2 | P(√2 − 1) | P²(3 − 2√2)/4 |
The perimeter is a(2 + √2) ≈ 3.4142·a, the inradius is a(2 − √2)/2 and the circumradius is c/2, because the hypotenuse is a diameter of the circumscribed circle.
Common 45-45-90 triangles
| Leg | Hypotenuse | Area | Perimeter |
|---|---|---|---|
| 1 | √2 ≈ 1.414 | 1/2 | 2 + √2 ≈ 3.414 |
| 2 | 2√2 ≈ 2.828 | 2 | 4 + 2√2 ≈ 6.828 |
| 3 | 3√2 ≈ 4.243 | 9/2 | 6 + 3√2 ≈ 10.243 |
| 5 | 5√2 ≈ 7.071 | 25/2 | 10 + 5√2 ≈ 17.071 |
| √2 ≈ 1.414 | 2 | 1 | 2 + 2√2 ≈ 4.828 |
| 5√2 ≈ 7.071 | 10 | 25 | 10 + 10√2 ≈ 24.142 |
Where 45-45-90 triangles show up
- Square diagonals. The diagonal of any square is side × √2, so a 12-inch tile measures about 16.97 inches corner to corner.
- Miter cuts and braces. A 45° corner brace with equal legs is a 45-45-90 triangle; its long edge is the leg × 1.4142.
- Unit circle values. With a hypotenuse of 1 the legs are √2/2, which is why sin 45° = cos 45° = √2/2 ≈ 0.7071.
- Quick check. The hypotenuse must be about 41% longer than each leg; if not, recheck which side you entered.
Frequently asked questions
What is the side ratio of a 45-45-90 triangle?
1 : 1 : √2. The two legs are equal, and the hypotenuse is a leg times √2 ≈ 1.4142. A leg of 6 gives a hypotenuse of 6√2 ≈ 8.485.
How do I find the legs from the hypotenuse?
Divide the hypotenuse by √2, which is the same as multiplying by √2/2. A hypotenuse of 10 gives legs of 10/√2 = 5√2 ≈ 7.071.
What is the area of a 45-45-90 triangle?
Area = ½ × leg × leg = a²/2. In terms of the hypotenuse it is c²/4. Legs of 6 give an area of 18 square units.
Why is the hypotenuse a√2?
By the Pythagorean theorem c² = a² + a² = 2a², so c = √(2a²) = a√2. It is also the diagonal of a square with side a, because a diagonal cuts a square into two 45-45-90 triangles.
Is a 45-45-90 triangle isosceles?
Yes. Its two base angles are both 45°, so the sides opposite them, the legs, are equal. It is the only triangle that is both isosceles and right.
How do I rationalize a/√2?
Multiply the top and bottom by √2: a/√2 = a√2/2. For example 8/√2 = 8√2/2 = 4√2. The calculator always shows the rationalized form.
What is the altitude to the hypotenuse?
It is exactly half of the hypotenuse, h = c/2 = a√2/2, and it lands on the midpoint of the hypotenuse, splitting the triangle into two smaller 45-45-90 triangles.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
Related tools
30-60-90 Triangle CalculatorEnter one side of a 30-60-90 triangle to get the other sides, area and perimeter in exact radical form.Right Triangle CalculatorSolve a right triangle from any two values: sides, angles, area, perimeter and altitude.Pythagorean Theorem CalculatorSolve for the hypotenuse or a missing leg of a right triangle, with a diagram and steps.Triangle CalculatorSolve any triangle from SSS, SAS, ASA, AAS or right-triangle values: sides, angles, area, type.Roofing Calculator (Roof Squares)Estimate roof area, squares, shingle bundles and cost from pitch.Square Root CalculatorFind square, cube and nth roots with simplified radical form and steps.Body Surface Area CalculatorCalculate body surface area in m² with the Du Bois, Mosteller, Haycock and other formulas.Area CalculatorFind the area of squares, rectangles, triangles, circles, trapezoids, ellipses and sectors.Volume CalculatorVolume of a cube, box, cylinder, sphere, cone, pyramid or capsule, in liters and gallons too.Surface Area CalculatorSurface area of a cube, box, cylinder, sphere, cone or pyramid, with lateral and base areas.Circle CalculatorEnter a circle's radius, diameter, circumference or area and get all the others.Slope CalculatorSlope, angle, line equation y = mx + b and distance from two points, with a graph.Midpoint CalculatorFind the midpoint of two points in 2D or 3D, or the missing endpoint from a midpoint.Distance Formula CalculatorDistance between two points in 2D or 3D with step-by-step work and an exact radical answer.Vector Calculator2D and 3D vectors: add, subtract, dot and cross product, magnitude, angle, unit vector, projection.Pascal's Triangle GeneratorGenerate Pascal's triangle up to row 30, highlight a row and expand (ax + by)ⁿ.Trigonometry CalculatorSin, cos, tan, csc, sec, cot and inverses in degrees or radians, with exact values.Arc Length CalculatorArc length, sector area and chord from radius and angle, or solve for the radius or angle.Cylinder Volume CalculatorCylinder volume in liters, gallons and cubic units from radius or diameter, plus surface area.Regular Polygon CalculatorArea, perimeter, apothem, radius and angles of a regular polygon from any one measurement.Aquarium Volume CalculatorFish tank gallons, liters and water weight: rectangle, cylinder, bow-front and hexagon tanks.Tank Volume CalculatorTotal and filled volume of cylinder, rectangular and capsule tanks in gallons and liters.Circle Skirt CalculatorWaist radius, fabric yardage and cutting layout for full, half or quarter circle skirts.Ellipse CalculatorArea, perimeter (Ramanujan), eccentricity, foci and equation of an ellipse from its axes.Camera Field of View CalculatorAngle of view and frame size from focal length, sensor size and subject distance.Law of Sines and Cosines CalculatorSolve SSS, SAS, ASA, AAS and SSA triangles with the laws of sines and cosines, steps shown.Miter Angle CalculatorMiter saw angle for any polygon or corner, plus crown molding miter and bevel settings.Simplify Square Root CalculatorSimplify square, cube and nth roots to simplest radical form a√b, with steps.Interactive Unit CircleDrag an angle on the unit circle to see degrees, radians and exact sin, cos and tan. Printable chart.Cone Volume CalculatorVolume, slant height and surface area of a right cone from its radius and height.Perimeter CalculatorPerimeter of a square, rectangle, triangle, circle, trapezoid, polygon or sector, with a diagram.Point Slope Form CalculatorLine equation in point-slope form from a point and slope or two points, plus y = mx + b.Polar to Cartesian ConverterConvert polar coordinates (r, θ) to rectangular (x, y) with steps, a plot and bulk lists.Slope Intercept Form CalculatorGet y = mx + b from two points, a point and slope, or standard form, with steps and a graph.Sphere Volume CalculatorVolume and surface area of a sphere from radius, diameter or circumference, or the reverse.Trapezoid Area CalculatorArea, perimeter, median and missing base or height of a trapezoid, with a diagram.Complementary and Supplementary Angle CalculatorComplement, supplement and explement of any angle in degrees or radians, with diagrams.Coordinate Plane PlotterPlot points, segments and polygons on an interactive coordinate grid, or print blank planes.Coterminal Angle CalculatorList positive and negative coterminal angles and the principal angle, in degrees and radians.Heron's Formula CalculatorFind a triangle's area from its three sides with Heron's formula, semi-perimeter and steps.Implicit Differentiation CalculatorFind dy/dx for any equation in x and y with steps, plus the slope and tangent at a point.Inverse Trig Calculatorarcsin, arccos, arctan, arcsec, arccsc and arccot in degrees and radians, with exact values.Polygon Area from CoordinatesArea, perimeter and centroid of any polygon from its vertex coordinates (shoelace formula).Reference Angle CalculatorFind the reference angle and quadrant of any angle in degrees or radians, with a unit circle sketch.Tangent Line CalculatorFind the tangent and normal line to f(x) at a point, with the derivative, slope and a graph.