Know two things about a right triangle? Pick them, enter the numbers and get the missing sides, both acute angles, the area, perimeter, altitude to the hypotenuse, inradius and circumradius, plus a labeled diagram drawn to scale and the SOH-CAH-TOA working.
Last updated:
In square units of the side lengths.
Angle unit
Enter your values to see the result.
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How to use the Right Triangle Calculator
Choose what you know from the list, e.g. two legs, or the hypotenuse and an angle.
Enter the values. Angle α is opposite leg a; angle β is opposite leg b.
Pick degrees or radians if you entered an angle.
Press Solve triangle to see every side, angle and measurement.
Check the diagram and steps, then copy the results or download the trig table.
Right triangle formulas
Label the legs a and b, the hypotenuse c, and the acute angles α (opposite a) and β (opposite b). The right angle sits between the legs.
Find
Formula
Example (a = 3, b = 4)
Hypotenuse
c = √(a² + b²)
√25 = 5
Missing leg
a = √(c² − b²)
√(25 − 16) = 3
Angle α
α = arctan(a ÷ b) = arcsin(a ÷ c)
arctan(0.75) ≈ 36.87°
Angle β
β = 90° − α
≈ 53.13°
Leg from angle
a = c · sin α, b = c · cos α, a = b · tan α
5 × sin 36.87° = 3
Area
A = a · b ÷ 2
6
Perimeter
P = a + b + c
12
Altitude to c
h = a · b ÷ c
2.4
Inradius
r = (a + b − c) ÷ 2
1
Circumradius
R = c ÷ 2
2.5
Special right triangles and Pythagorean triples
Triangle
Angles
Side ratio
Where you meet it
45-45-90
45°, 45°, 90°
1 : 1 : √2
Half of a square cut along the diagonal
30-60-90
30°, 60°, 90°
1 : √3 : 2
Half of an equilateral triangle
3-4-5
36.87°, 53.13°, 90°
3 : 4 : 5
Squaring corners in carpentry (also 6-8-10, 9-12-15)
5-12-13
22.62°, 67.38°, 90°
5 : 12 : 13
Whole-number test problems
8-15-17
28.07°, 61.93°, 90°
8 : 15 : 17
Whole-number test problems
7-24-25
16.26°, 73.74°, 90°
7 : 24 : 25
Whole-number test problems
Practical tips
Ladders and ramps: the ladder is the hypotenuse, the wall height is the leg opposite the ground angle. A 12 ft ladder at 75° reaches 12 × sin 75° ≈ 11.59 ft up the wall, with its foot 12 × cos 75° ≈ 3.11 ft out.
Check a square corner: measure 3 units along one side and 4 along the other; if the diagonal is exactly 5, the corner is 90°.
Calculator mode matters: sin 30 is 0.5 in degrees but −0.988 in radians. Choose the angle unit that matches your problem.
Keep precision: round only the final answers; rounding a side early changes the angles you compute from it.
Frequently asked questions
How do you solve a right triangle?
You need two values besides the 90° angle, and at least one must be a side. With two sides, use the Pythagorean theorem a² + b² = c² for the third side and an inverse trig function for the angles. With a side and an angle, use sine, cosine or tangent (SOH-CAH-TOA) for the other sides and β = 90° − α for the other angle.
What does SOH-CAH-TOA mean?
It is a memory aid for the trig ratios of an acute angle in a right triangle: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
Why can't I solve a right triangle from two angles?
Angles alone fix the shape but not the size: a 3-4-5 triangle and a 6-8-10 triangle have the same angles. You need at least one length to get the sides, area and perimeter.
What are the side ratios of 30-60-90 and 45-45-90 triangles?
A 30-60-90 triangle has sides in the ratio 1 : √3 : 2 (short leg, long leg, hypotenuse). A 45-45-90 triangle has two equal legs and a hypotenuse of leg × √2, about 1.4142 × the leg.
How do I find the area of a right triangle?
Multiply the two legs and divide by 2: Area = a × b ÷ 2, because the legs are the base and the height. A 3-4-5 triangle has an area of 3 × 4 ÷ 2 = 6.
What is the altitude to the hypotenuse?
It is the perpendicular height from the right angle to the hypotenuse: h = a × b ÷ c. It splits the hypotenuse into two segments, a² ÷ c and b² ÷ c, and creates two smaller triangles similar to the original.
Which side is the hypotenuse?
The hypotenuse is the side opposite the right angle and is always the longest side. If a value you enter as a leg is longer than the hypotenuse, the triangle is impossible and the calculator will tell you.
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