Calculate all sides, area, perimeter, altitudes, and radii of a 45-45-90 isosceles right triangle. Enter any one side and get every property instantly.
The 45-45-90 triangle is one of the two fundamental "special" right triangles in geometry. It is an isosceles right triangle whose two legs are equal and whose angles measure 45°, 45°, and 90°. The fixed side ratio is 1 : 1 : √2, meaning the hypotenuse is always √2 times either leg. This ratio lets you solve the entire triangle from a single known side without a calculator or trig tables.
You can construct a 45-45-90 triangle by cutting a square along its diagonal — each half is an isosceles right triangle. This connection to the square makes the shape ubiquitous in architecture, carpentry, tile work, and drafting. A miter saw set to 45° produces this triangle in every cut, and the geometry is essential for computing diagonal bracing in framing.
In standardized tests (SAT, GRE, ACT, GMAT), knowing the 1 : 1 : √2 ratio lets you solve problems in seconds. In physics, resolving a vector at 45° decomposes into equal x and y components — both V / √2.
This calculator derives every property from either leg or the hypotenuse: area, perimeter, semi-perimeter, all three altitudes, circumradius, inradius, and medians. Visual ratio bars, presets for common values, and a reference table make exploration fast and intuitive.
The 45-45-90 triangle turns up whenever a square, diagonal brace, or 45-degree cut is involved, so it is one of the fastest shapes to recognize and exploit. This calculator helps when you need more than the basic leg-to-hypotenuse conversion by also returning the area, perimeter, altitudes, medians, and circle radii of the isosceles right triangle. It is a practical check for classroom geometry, drafting, woodworking, CNC layouts, and any design task built around square corners and diagonal spans.
Ratios — leg : leg : hypotenuse = 1 : 1 : √2 Leg: a Hypotenuse: c = a√2 Area: A = a² / 2 Perimeter: P = 2a + a√2 = a(2 + √2) Altitude to hypotenuse: h = a / √2 = a√2 / 2 Circumradius: R = c / 2 = a√2 / 2 Inradius: r = a(√2 − 1) / √2 = (2a − c) / 2
Result: Leg ≈ 14.14 cm, Hyp = 20 cm, Area = 100 cm², Perimeter ≈ 48.28 cm
If the known side is the hypotenuse and value = 20 cm, each leg is 20 / √2 ≈ 14.14 cm. The area is 14.14² / 2 = 100 cm², and the perimeter is 14.14 + 14.14 + 20 ≈ 48.28 cm. The altitude to the hypotenuse is also about 10 cm, which is half of the hypotenuse in this special triangle.
A 45-45-90 triangle is what you get when a square is divided along its diagonal. Because both legs come from equal sides of the square, they must match, and the diagonal becomes the hypotenuse. That gives the fixed ratio 1 : 1 : √2. Once you know one side, the rest of the triangle follows immediately, which is why this shape is one of the most heavily memorized patterns in geometry.
The special ratio is only the starting point. The area is half of a square with side length a, so A = a² / 2. The circumradius is half the hypotenuse, and the altitude to the hypotenuse equals a / √2, which is also half the hypotenuse. These shortcuts show up in diagonal bracing, gusset plate layouts, tile patterns, and coordinate-grid problems where a line rises and runs by equal amounts.
Any time you see a 45-degree miter, a square corner split by a brace, or a plan view that uses equal horizontal and vertical offsets, you are effectively working with a 45-45-90 triangle. That makes the calculator useful for checking cut lengths, diagonal spans, and material layouts. It also serves as a fast verification tool in exam settings, where recognizing the pattern is often the difference between a one-line solution and a full trigonometry setup.
The sides are always in the ratio 1 : 1 : √2. Both legs are equal, and the hypotenuse is √2 (≈ 1.414) times either leg.
Multiply the leg by √2. If the leg is 8, the hypotenuse is 8√2 ≈ 11.31. Conversely, divide the hypotenuse by √2 to get the leg.
Area = leg² / 2. For a leg of 6, area = 36 / 2 = 18 square units. Alternatively, area = hypotenuse² / 4.
Cutting a square along its diagonal produces two congruent 45-45-90 triangles. The square side becomes the leg, and the diagonal becomes the hypotenuse.
Yes, they are identical. "Isosceles right triangle" emphasizes the two equal legs and right angle; "45-45-90" emphasizes the three angle measures.
R = hypotenuse / 2 = leg × √2 / 2. For a leg of 10: R = 10 × 1.414 / 2 ≈ 7.07.