The 45-45-90 Triangle Calculator computes all sides, area, perimeter, and diagonal of an isosceles right triangle from one known leg or hypotenuse. Based on the exact 1:1:√2 ratio, it delivers instant results for geometry, construction, and engineering problems.
5
units
7.0711
units
12.5
sq units
17.0711
units
3.5355
units
1.4645
units
5
units
7.0711
units
12.5
sq units
17.0711
units
3.5355
units
1.4645
units
The calculator for the 45-45-90 triangle computes every property of this isosceles right triangle — both legs, hypotenuse, area, and perimeter — from a single measurement. Because the two legs are always equal and the hypotenuse is always leg × √2, one known value determines the entire triangle instantly.
The 45-45-90 triangle is one of the two special right triangles in Euclidean geometry. It arises directly from cutting a square along its diagonal — each half is a 45-45-90 triangle with legs equal to the square's side and hypotenuse equal to the square's diagonal. By the Pythagorean theorem:
hypotenuse = leg × √2 leg = hypotenuse / √2 = hypotenuse × (√2/2)
This relationship explains why √2 appears constantly in geometry involving squares, diagonals, and 45° angles. The 30-60-90 triangle calculator covers the complementary special right triangle with unequal legs.
The 45-45-90 triangle is the source of the exact trigonometric values for 45°, which appear throughout mathematics, physics, and engineering:
sin 45° = cos 45° = √2/2 ≈ 0.7071 tan 45° = 1
The fact that sin 45° = cos 45° reflects the triangle's symmetry — both acute angles are identical, so the opposite and adjacent sides are interchangeable. These exact values are used in Fourier analysis, signal processing, vector decomposition, and anywhere 45° angles appear. The triangle calculators category covers all standard triangle types.
With leg length a as input, the area and perimeter follow directly from the fixed ratios:
Area = a2 / 2 Perimeter = a × (2 + √2)
The area formula reflects the fact that the 45-45-90 triangle is exactly half a square — its area is half the square with side a. This geometric identity makes it useful for quickly computing diagonal areas in architecture and tile work. The online calculator handles both leg-input and hypotenuse-input cases automatically.
The 45-45-90 triangle is ubiquitous in applied design. Miter cuts at 45° in carpentry and framing create perfect corner joints because two 45° cuts combine to form a 90° corner. Roof trusses using equal-pitch designs follow 45-45-90 geometry. In structural engineering, diagonal bracing at 45° maximizes shear resistance while the √2 relationship determines brace length from bay width. The right triangle calculator handles general right triangles when angles are not fixed at 45°.
On the unit circle, the point at 45° has coordinates (√2/2, √2/2) — both coordinates equal because of the triangle's symmetry. This means any vector at 45° has equal horizontal and vertical components, a property exploited in projectile motion (maximum range at 45° launch angle), antenna design, and coordinate transformations. The Pythagorean theorem verifies: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1. ✓
A 45-45-90 triangle has angles 45°, 45°, and 90°, with sides in the exact ratio $$1 : 1 : \sqrt{2}$$. Let $$x$$ be the length of each leg.
Side Lengths:
$$\text{Each leg} = x \quad (\text{opposite each } 45° \text{ angle})$$
$$\text{Hypotenuse} = x\sqrt{2} \quad (\text{opposite } 90°)$$
This follows directly from the Pythagorean theorem: $$c = \sqrt{x^2 + x^2} = x\sqrt{2}$$.
Area: Since both legs are perpendicular:
$$A = \frac{x \cdot x}{2} = \frac{x^2}{2}$$
Perimeter:
$$P = 2x + x\sqrt{2} = x(2 + \sqrt{2})$$
Altitude to Hypotenuse: Using the area relationship:
$$h = \frac{x \cdot x}{x\sqrt{2}} = \frac{x}{\sqrt{2}} = \frac{x\sqrt{2}}{2}$$
Inradius:
$$r = \frac{A}{s} = \frac{x^2/2}{x(2 + \sqrt{2})/2} = \frac{x}{2 + \sqrt{2}} = x\left(\frac{2 - \sqrt{2}}{2}\right)$$
Each Leg is one of the two equal sides forming the right angle. The legs are perpendicular to each other and equal in length, which is why this triangle is also called an isosceles right triangle.
The Hypotenuse is the side opposite the 90° angle, equal to the leg times $$\sqrt{2} \approx 1.4142$$. It is always the longest side.
The Area is half the square of the leg length. Since the two legs serve as base and height, the area formula simplifies to $$x^2/2$$, which is exactly half the area of the square with side $$x$$.
The Perimeter is the sum of two legs plus the hypotenuse: $$x(2 + \sqrt{2})$$.
The Altitude to Hypotenuse is the perpendicular from the right-angle vertex to the hypotenuse. It equals $$x/\sqrt{2}$$, which is also half the hypotenuse.
The Inradius is the radius of the inscribed circle. For the 45-45-90 triangle, it simplifies to $$x(2 - \sqrt{2})/2$$.
Inputs
Results
Hypotenuse = 5√2 = 7.0711. Area = 25/2 = 12.5. Perimeter = 10 + 7.0711 = 17.0711. Altitude = 25/7.0711 = 3.5355 (also = 5/√2). Inradius = 12.5 / 8.5355 = 1.4645. Verification: sin(45°) = 5/7.0711 = 0.7071 = √2/2. ✓
Inputs
Results
This is the unit 45-45-90 triangle: sides 1, 1, √2. Area = 1/2 = 0.5. The hypotenuse √2 ≈ 1.4142 — the diagonal of a unit square. sin(45°) = cos(45°) = 1/√2 = √2/2 ≈ 0.7071. Inradius = (2-√2)/2 ≈ 0.2929.
Divide the hypotenuse by √2 (approximately 1.4142) to get the leg length. For example, if the hypotenuse is 10, the leg is 10/1.4142 ≈ 7.071. Enter 7.071 as the leg length.
Because it has two equal sides (the legs) making it isosceles, and one 90° angle making it a right triangle. It is the only triangle that is both isosceles and right-angled (besides degenerate cases).
Cutting a square along its diagonal produces two congruent 45-45-90 triangles. The diagonal of a square with side s is s√2, which becomes the hypotenuse. This is why the area of each triangle is exactly half the area of the square.
The speed square (rafter square) is a 45-45-90 triangle used to mark 45° and 90° angles, guide circular saw cuts, and measure roof pitches. Miter joints at 90° corners require 45° cuts, directly using 45-45-90 geometry.
sin(45°) = cos(45°) = √2/2 ≈ 0.7071, and tan(45°) = 1. These are exact values memorized in trigonometry. The equality sin(45°) = cos(45°) reflects the triangle's symmetry (both legs are equal).
At 45°, the horizontal and vertical velocity components are equal (forming a 45-45-90 triangle with the velocity vector). The range formula R = v²sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at θ = 45°. The equal split between horizontal reach and vertical hang time gives the optimal combination.
The 45-45-90 triangle is exactly half of a square, created by cutting along the diagonal. If a square has side length a, the diagonal has length a√2 — which becomes the hypotenuse of each 45-45-90 triangle. This means the area of the triangle is exactly a²/2, half the area of the square. This relationship is why 45° angles appear so frequently in tiling patterns, diagonal bracing, and any design based on square grids.
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