Right Triangle & Pythagorean Calculator

Right Triangle & Pythagorean: worked example

Leg a	5
Leg b	12
Hypotenuse (c)	13
Angle A (degrees)	22.6
Angle B (degrees)	67.4
Area	30

Right Triangle & Pythagorean: formula

c = √(a² + b²); angle A = arctan(a ÷ b); angle B = 90° − A; area = ½ab

Pythagorean Theorem: Wolfram MathWorld

Enter the two legs of the right triangle, a and b. The calculator finds the hypotenuse with the Pythagorean theorem, the two acute angles with the arctangent and the area. Angle A is opposite leg a.

The angles add up to 90 degrees. If you know one leg and the hypotenuse, use the Pythagorean theorem calculator or subtract to find the other leg.

Right Triangle & Pythagorean: changing Leg a

Leg a	Hypotenuse (c)	Angle A (degrees)	Angle B (degrees)	Area
4	12.6	18.4	71.6	24
4.5	12.8	20.6	69.4	27
5 (example)	13	22.6	67.4	30
5.5	13.2	24.6	65.4	33
6	13.4	26.6	63.4	36

Right Triangle & Pythagorean: compare input ranges

Input	Tested from	To	Hypotenuse (c): low input	Hypotenuse (c): high input	Swing
Leg b	11	13	12.1	13.9	1.8 (14%)
Leg a	4.5	5.5	12.8	13.2	0.384 (3.0%)

Right Triangle & Pythagorean: individual input comparisons

Input	Value	Hypotenuse (c)	Angle A (degrees)	Angle B (degrees)	Area
Leg a	4	12.6	18.4	71.6	24
	4.5	12.8	20.6	69.4	27
	5	13	22.6	67.4	30
	5.5	13.2	24.6	65.4	33
	6	13.4	26.6	63.4	36
Leg b	10	11.2	26.6	63.4	25
	11	12.1	24.4	65.6	27.5
	12	13	22.6	67.4	30
	13	13.9	21	69	32.5
	14	14.9	19.7	70.3	35

Right Triangle & Pythagorean: combined input changes

Leg a / Leg b	10	12	14
4	10.8	12.6	14.6
4.5	11	12.8	14.7
5	11.2	13	14.9
5.5	11.4	13.2	15
6	11.7	13.4	15.2

Right Triangle & Pythagorean: which assumptions move the result?

Chart quantity	Displayed settings
Leg b	10 → 14
Hypotenuse (c)	11.2 → 14.9
Leg a	5

Chart quantity	Displayed settings
Leg a	4 → 6
Hypotenuse (c)	12.6 → 13.4
Leg b	12

Right Triangle & Pythagorean: values and definitions

Quantity and definition	Value used
Leg a	5
Leg b	12
Hypotenuse (c)	13
Angle A (degrees)	22.6
Angle B (degrees)	67.4
Area	30

Right Triangle & Pythagorean: half and double Leg b

Leg b	Hypotenuse (c)
6	7.8
12	13
24	24.5

Common error: Mixing up which leg goes with which angle. Angle A sits opposite leg a, so the longer leg faces the larger angle.

The triangle must have a right angle, or the formulas do not apply. Small errors in measuring the legs can shift the angles, especially in thin triangles. Angles are given in degrees.

For other triangles use the law of sines or the triangle side calculator.

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Two legs of a right triangle fix everything else. The hypotenuse follows from the Pythagorean theorem, the angles from the ratio of the legs, and the area from half their product. A 5 by 12 triangle has a 13 unit hypotenuse, angles of about 22.6 and 67.4 degrees, and an area of 30.

That angle is what you set on a saw or a level when you have only tape measure readings. For comparison, a 3 by 4 triangle has a hypotenuse of 5 and an 8 by 15 triangle has 17. These whole-number sets, 3-4-5, 5-12-13 and 8-15-17, are why builders can check a square corner with a tape measure alone.

Confirm the corner really is square. The formulas assume a 90 degree angle, and a slightly off corner gives an angle that only looks right.

See the Pythagorean theorem calculator for the hypotenuse, the trigonometry page for sine and cosine, and the distance and slope calculator for coordinates. The triangle area and equilateral triangle pages cover other triangles, and the stair, roof and ramp calculators in the trades categories apply right triangles.

Geometry for DIY and design: area, volume and angles

Right Triangle & Pythagorean Calculator: application examples

Pythagorean Theorem Calculator

Calculate the hypotenuse of a right triangle from its two legs. Only applies to right triangles: a 90° angle between the two legs.

Pythagorean Theorem Calculator: full formula

Leg A	3
Leg B	4
Hypotenuse	5
Triangle area	6

The hypotenuse is the square root of the sum of the squares of the legs. Legs of 3 and 4 give a hypotenuse of 5 and an area of 6, the classic 3-4-5 right triangle. Builders use the 3-4-5 rule to check that corners are square: measure 3 feet along one wall, 4 along the other and confirm the diagonal is 5.

It also gives the straight-line distance between two points on a grid.

Enter the lengths of the two legs of a right triangle, the sides that meet at the right angle, in the same unit. The hypotenuse is the longest side, opposite the right angle. The theorem says the square of the hypotenuse equals the sum of the squares of the legs.

To find a leg instead, use the right triangle calculator. Make sure the triangle really has a right angle before using the theorem.

The theorem only holds when the angle between the two given sides is ninety degrees. Apply it to a triangle that merely looks roughly square and the hypotenuse comes out wrong, sometimes by a meaningful margin depending on how far the real angle sits from ninety.

Confirm the two known sides meet at a right angle, ideally measured, before running the calculation. A framing square or a measured 3-4-5 check is more reliable than eyeballing it.

Applying the theorem to any triangle with a roughly square corner. Off by even a few degrees, the result stops being the hypotenuse of anything real and becomes a number that only looks like one.

Solve for angles and missing legs on the right triangle calculator, general triangles on the law of sines and triangle side pages, and distance on the distance and slope calculator. The rectangular prism page gives space diagonals, and the roof and stair tools in the carpentry category use right triangles.

What are Pythagorean triples?

Sets of whole numbers that satisfy the theorem, such as 3-4-5, 5-12-13 and 8-15-17, useful for quick checks.

Pythagorean Theorem: changing Leg A

Pythagorean Theorem Calculator: hypotenuse and triangle area and angle a (degrees) and angle b (degrees) across a range of leg a, every other input held at the worked-example value.
Leg A	Hypotenuse	Triangle area	Angle A (degrees)	Angle B (degrees)
2.4	4.7	4.8	31	59
2.7	4.8	5.4	34	56
3 (example)	5	6	36.9	53.1
3.3	5.2	6.6	39.5	50.5
3.6	5.4	7.2	42	48

Pythagorean Theorem: compare input ranges

Pythagorean Theorem Calculator: hypotenuse with each input moved on its own, every other input held at the worked-example value.
Input	Tested from	To	Hypotenuse: low input	Hypotenuse: high input	Swing
Leg B	3.6	4.4	4.7	5.3	0.639 (13%)
Leg A	2.7	3.3	4.8	5.2	0.36 (7.2%)

Pythagorean Theorem: individual input comparisons

Pythagorean Theorem Calculator: each input varied independently; remaining quantities retain their worked-example settings.
Input	Value	Hypotenuse	Triangle area	Angle A (degrees)	Angle B (degrees)
Leg A	2.4	4.7	4.8	31	59
	2.7	4.8	5.4	34	56
	3	5	6	36.9	53.1
	3.3	5.2	6.6	39.5	50.5
	3.6	5.4	7.2	42	48
Leg B	3.2	4.4	4.8	43.2	46.8
	3.6	4.7	5.4	39.8	50.2
	4	5	6	36.9	53.1
	4.4	5.3	6.6	34.3	55.7
	4.8	5.7	7.2	32	58

Pythagorean Theorem: combined input changes

Pythagorean Theorem Calculator: hypotenuse at each combination of leg a (rows) and leg b (columns).
Leg A / Leg B	3.2	4	4.8
2.4	4	4.7	5.4
2.7	4.2	4.8	5.5
3	4.4	5	5.7
3.3	4.6	5.2	5.8
3.6	4.8	5.4	6

Pythagorean Theorem: values and definitions

Pythagorean Theorem Calculator: worked-example quantities, units and calculated results.
Quantity and definition	Value used
Leg A	3
Leg B	4
Hypotenuse	5
Triangle area	6
Angle A (degrees)	36.9
Angle B (degrees)	53.1

Pythagorean Theorem: half and double Leg B

Pythagorean Theorem Calculator: half, original and double leg b, with all other assumptions unchanged.
Leg B	Hypotenuse
2	3.6
4	5
8	8.5

What is the Pythagorean theorem? In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides

How can I check a corner is square? Measure 3 units along one side and 4 along the other. If the distance between the marks is 5 units, the corner is a right angle

The theorem applies only to right triangles, and for other triangles you need the law of cosines. Measurement errors show up in the result, especially for long distances. Units must match.

In three dimensions, the space diagonal of a box extends the formula with a third term.

NIST Digital Library of Mathematical Functions

What is a 5-12-13 triangle?

A right triangle with legs 5 and 12 and hypotenuse 13, one of the Pythagorean triples that fit the theorem with whole numbers.

How steep is a 22 degree slope?

It rises about 0.4 for each unit of run, which is a grade of about 40%.

How do I find the angles of a right triangle?

Use the arctangent of the opposite leg over the adjacent leg for one angle, and subtract it from 90 degrees for the other.