How does it find the angles from three sides?

By the law of cosines, once for each corner. The cosine of the angle opposite side a is the sum of the squares of the other two sides less the square of a, all over twice their product, and the inverse cosine turns that back into an angle. Doing it three times gives all three angles, and they add to 180 degrees. It is the general form of Pythagoras: when the angle is 90 its cosine is 0, the last term vanishes and a squared equals b squared plus c squared.

Why does it say the sides cannot make a triangle?

Because of the triangle inequality: any two sides must add to more than the third, or the two short ones never reach across the long one. Try 1, 2 and 10 and the tool says so, naming the three numbers rather than showing an error code. If two sides add to exactly the third, say 1, 2 and 3, the corners fall on one straight line: the shape is flat, it has no height and no area, and that is reported as its own case.

Is this triangle right, acute or obtuse?

The largest angle decides, and the tool prints the answer. If it is 90 degrees the triangle is right, and the corner mark on the figure squares off; below 90 every angle is acute; above 90 the triangle is obtuse. Any Pythagorean triple comes back right, so 3, 4, 5 and 8, 15, 17 and 5, 12, 13 all square up exactly. An equilateral triangle comes back acute with three 60 degree angles.

How is the area worked out without a height?

Heron's formula, which needs the three sides and nothing else. The tool uses the numerically stable ordering of it, with the sides sorted before they are subtracted, so a long thin triangle still gets an area instead of a rounding error. The integer triples come out as integers: 3, 4, 5 gives exactly 6 and 8, 15, 17 gives exactly 60. The area is in the square of whatever unit you typed the sides in, and the tool does not know or assume that unit.

What if I know two sides and an angle instead?

Use the right triangle mode when a right angle is known: two sides or a side and one acute angle solve it. For a general triangle, Triangle Calculator accepts SAS, ASA, AAS and SSA combinations and returns both valid solutions when SSA is ambiguous. This page specifically takes all three side lengths.

Side Side Side Triangle Calculator

Give all three sides of a triangle and read its three angles, its area and its perimeter, with no right angle assumed. The law of cosines gives each angle from the side opposite it, Heron's formula gives the area from the sides alone, and the largest angle decides whether the triangle is right, acute or obtuse: 8, 15 and 17 come back right with an area of exactly 60, while 2, 2 and 3 come back obtuse. Three lengths that cannot meet, or that lie flat in a straight line, are named rather than answered, and the figure is redrawn to scale as you type. The Calculation steps region shows the cosine-law angles, semiperimeter, Heron area and perimeter for the current sides. Keep all lengths in the same unit; area uses that unit squared. Use Triangle Calculator when the known values mix sides and angles.

Solvedright angled triangle

A: 28.0725°; opposite side a
B: 61.9275°; opposite side b
C: 90°; opposite side c
shape: right angled; from the largest angle
area: 60; Heron's formula
perimeter: 40; a + b + c

Known

a

b

c

Angles in

Give all three sides of any triangle, right or not. The law of cosines gives the three angles, Heron's formula gives the area, and the tool says whether the triangle is right, acute or obtuse.

Working for your triangle.

Given sides: a = 8, b = 15, c = 17.
A = arccos((b² + c² − a²) / (2bc)) = 28.0724869359 degrees.
B = arccos((a² + c² − b²) / (2ac)) = 61.9275130641 degrees.
C = arccos((a² + b² − c²) / (2ab)) = 90 degrees.
s = (a + b + c) / 2 = 20.
Heron's area = √(s(s − a)(s − b)(s − c)) = 60.
Perimeter = a + b + c = 40.

Three sides8, 15, 17
right angled · 28.0725° and 61.9275° · area 60
Leg and anglea = 10, A = 30°
b = 17.3205, c = 20 · area 86.6025
Angle sumone acute angle is 34°
90° + 34° + 56° = 180°

How the three modes work. A right triangle has five values: the legs a and b, the hypotenuse c, and the acute angles A and B, with C fixed at a right angle. Any two of them fix the triangle, so two boxes are what the tool asks for: sides come from Pythagoras, angles from the inverse sine, cosine and tangent, and the altitude is the perpendicular dropped onto the hypotenuse, a times b over c. Two angles are refused, because they set the shape and not the size. Three sides go to the law of cosines instead, which works on any triangle and says whether it is right, acute or obtuse, with the area from Heron’s formula. Two angles of any triangle give the third from the 180 degree sum. Exact trigonometry to the precision of the numbers you enter, with every side and angle shown in degrees or radians. It cannot know your units or your measurement error, and it is geometry only, not a structural calculation.

Open Right Triangle Calculator, Three Side Calculator, or Triangle Angle Sum Calculator. For a general triangle with a mixture of sides and angles, use Triangle Calculator. Convert a pitch ratio such as 6:12 to an angle with Roof Pitch Calculator, then enter that angle and an actual length here for geometric rise or sloping length.

Common questions

How does it find the angles from three sides?: By the law of cosines, once for each corner. The cosine of the angle opposite side a is the sum of the squares of the other two sides less the square of a, all over twice their product, and the inverse cosine turns that back into an angle. Doing it three times gives all three angles, and they add to 180 degrees. It is the general form of Pythagoras: when the angle is 90 its cosine is 0, the last term vanishes and a squared equals b squared plus c squared.
Why does it say the sides cannot make a triangle?: Because of the triangle inequality: any two sides must add to more than the third, or the two short ones never reach across the long one. Try 1, 2 and 10 and the tool says so, naming the three numbers rather than showing an error code. If two sides add to exactly the third, say 1, 2 and 3, the corners fall on one straight line: the shape is flat, it has no height and no area, and that is reported as its own case.
Is this triangle right, acute or obtuse?: The largest angle decides, and the tool prints the answer. If it is 90 degrees the triangle is right, and the corner mark on the figure squares off; below 90 every angle is acute; above 90 the triangle is obtuse. Any Pythagorean triple comes back right, so 3, 4, 5 and 8, 15, 17 and 5, 12, 13 all square up exactly. An equilateral triangle comes back acute with three 60 degree angles.
How is the area worked out without a height?: Heron's formula, which needs the three sides and nothing else. The tool uses the numerically stable ordering of it, with the sides sorted before they are subtracted, so a long thin triangle still gets an area instead of a rounding error. The integer triples come out as integers: 3, 4, 5 gives exactly 6 and 8, 15, 17 gives exactly 60. The area is in the square of whatever unit you typed the sides in, and the tool does not know or assume that unit.
What if I know two sides and an angle instead?: Use the right triangle mode when a right angle is known: two sides or a side and one acute angle solve it. For a general triangle, Triangle Calculator accepts SAS, ASA, AAS and SSA combinations and returns both valid solutions when SSA is ambiguous. This page specifically takes all three side lengths.

Exact trigonometry to the precision of the numbers you enter, with every side and angle shown in degrees or radians. It cannot know your units or your measurement error, and it is geometry only, not a structural calculation.