About

A 45-45-90 triangle is the only isosceles right triangle. Its interior angles are fixed at 45°, 45°, and 90°, producing a rigid side ratio of 1 : 1 : √2. This ratio is a direct consequence of the Pythagorean theorem applied to two equal legs: c = a√2. The triangle appears constantly in structural engineering (diagonal bracing), woodworking (miter joints cut at 45°), and navigation (bearing offsets). Errors in computing the hypotenuse propagate into material cuts, producing waste or structural weakness.

This calculator derives every metric of a 45-45-90 triangle from a single known side. It computes both legs, hypotenuse, area, perimeter, altitude to the hypotenuse, inradius, and circumradius. The tool assumes Euclidean geometry on a flat plane. Results are approximate for physical materials subject to kerf, thermal expansion, or curvature. Pro tip: when cutting lumber for a 45° brace, add your saw's kerf width to the computed hypotenuse before marking.

Formulas

All measurements of a 45-45-90 triangle derive from a single known side through the constant ratio 1 : 1 : √2. The Pythagorean theorem applied to two equal legs a gives the fundamental relationship:

c = a ⋅ √2

Conversely, given the hypotenuse:

a = c√2 = c ⋅ √22

Area of the triangle (half the product of the two equal legs):

A = a²2

Perimeter:

P = 2a + a√2 = a(2 + √2)

Altitude from the right-angle vertex to the hypotenuse:

h = a√2 = a√22

Inradius (radius of the inscribed circle):

r = a + b − c2 = a(2 − √2)2

Circumradius (radius of the circumscribed circle, always half the hypotenuse for a right triangle):

R = c2 = a√22

Where a = b = leg length, c = hypotenuse, h = altitude to hypotenuse, r = inradius, R = circumradius, A = area, P = perimeter.

Reference Data

Property	Formula	Leg = 1	Leg = 5	Leg = 10	Leg = 25	Leg = 50	Leg = 100
Leg a	a	1	5	10	25	50	100
Leg b	a	1	5	10	25	50	100
Hypotenuse c	a√2	1.4142	7.0711	14.1421	35.3553	70.7107	141.4214
Area	a²2	0.5	12.5	50	312.5	1250	5000
Perimeter	2a + a√2	3.4142	17.0711	34.1421	85.3553	170.7107	341.4214
Altitude to hypotenuse h	a√2	0.7071	3.5355	7.0711	17.6777	35.3553	70.7107
Inradius r	a(2 − √2)2	0.2929	1.4645	2.9289	7.3223	14.6447	29.2893
Circumradius R	a√22	0.7071	3.5355	7.0711	17.6777	35.3553	70.7107
Angle A	Fixed	45° (0.7854 rad)
Angle B	Fixed	45° (0.7854 rad)
Angle C	Fixed	90° (1.5708 rad)
Diagonal ratio	c ÷ a	√2 ≈ 1.41421356
Leg from hypotenuse	c√22	Multiply hypotenuse by 0.70711

Frequently Asked Questions

The Pythagorean theorem states c² = a² + b². In a 45-45-90 triangle, a = b, so c² = 2a², giving c = a√2. This is an algebraic identity, not an approximation. The ratio holds for any positive leg length in Euclidean space.

The irrational constant √2 ≈ 1.41421356 cannot be represented exactly in decimal. Truncating to 1.414 introduces an error of about 0.01%. For a leg of 3000 mm, that is 0.4 mm of deviation on the hypotenuse. This is within tolerance for rough carpentry but may exceed limits for precision metalwork (ISO 2768-m tolerances start at ±0.1 mm). This calculator provides results to 4 decimal places, sufficient for most practical work.

No. The 1 : 1 : √2 ratio is strictly valid in Euclidean (flat) geometry. On a sphere, the sum of interior angles exceeds 180°, and the Pythagorean theorem does not hold. For geodetic calculations on Earth's surface, you need spherical trigonometry or Vincenty's formulae. This tool assumes a flat plane.

The altitude h from the right-angle vertex to the hypotenuse equals a ÷ √2, which is the same as the circumradius R. This altitude bisects the hypotenuse into two equal segments of length a√2 ÷ 2 each. In a general right triangle, the altitude to the hypotenuse is the geometric mean of the two hypotenuse segments. In the 45-45-90 case, because the segments are equal, h equals the segment length.

For any triangle, the inradius is r = A ÷ s, where A is area and s is the semi-perimeter. For a right triangle specifically, a shortcut exists: r = (a + b − c) ÷ 2. Substituting b = a and c = a√2 yields r = a(2 − √2) ÷ 2 ≈ 0.2929a. The inscribed circle is notably small relative to the triangle.

Thales' theorem states that any triangle inscribed in a semicircle with the diameter as one side is a right triangle. The converse also holds: the circumscribed circle of any right triangle has the hypotenuse as its diameter. Therefore R = c ÷ 2. This applies to all right triangles, not just the 45-45-90 case. It means the midpoint of the hypotenuse is equidistant from all three vertices.