Triangle Calculator
Calculate the area, perimeter, angles, and height of a triangle from three sides using Heron's formula.
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How to use this calculator
Heron's formula computes triangle area from all three side lengths without needing the height.
- 1
Enter the lengths of all three sides of the triangle.
- 2
The calculator verifies the triangle inequality and computes area using Heron's formula.
- 3
All three angles are calculated using the law of cosines.
- 4
Height to side a is also shown (useful for area verification).
Frequently asked questions
What do I need to enter?
All three side lengths, in any one unit. The calculator does not solve triangles from an angle plus sides, or from two angles and a side; if you only know those, you need to find the third side first with the law of cosines or law of sines. Area comes out in square units of whatever you typed, and angles are always in degrees.
Why does it say “Invalid triangle”?
A triangle exists only when each side is shorter than the sum of the other two. Sides 2, 3 and 5 fail because 2 + 3 equals 5, which would flatten the shape into a straight line with zero area. The check is strict, so equal-to counts as invalid, as does any zero or negative length.
Which side is the height measured to?
Only side a, the first field. The output “Height to side a” is 2 × area ÷ a. If you need the height onto b or c, swap the numbers so that side sits in the first field. Area and angles do not change when you reorder the sides, only the labels on them.
How are the angles found and how accurate are they?
Angle A and angle B come from the law of cosines, and angle C is 180° minus those two. They are shown to two decimal places, so the three can differ from 180° by up to a hundredth of a degree if you add the rounded figures yourself.
Does it tell me whether the triangle is right, isosceles or obtuse?
It does not label the type, but you can read it off the output. An angle of 90.00° means right (3, 4, 5 and 7, 24, 25 both give exactly that), two equal angles mean isosceles, and an angle above 90° means obtuse, such as the 104.48° for sides 2, 3 and 4.
What about extremely thin triangles?
They work, but the results are sensitive to rounding in your inputs. Sides 1, 1 and 1.99 give an area of 0.0994 and a 168.54° angle, while a tiny change to the longest side moves both noticeably. Enter measurements with as many digits as you actually have.
Does it give the inradius, circumradius or medians?
No. The outputs are area, perimeter, the three angles and the height to side a. Those other quantities can be derived from them, for example inradius = area ÷ semi-perimeter, but they are not calculated here.
Triangle calculator from three sides
From three sides to everything else
Three side lengths fix a triangle completely, up to reflection. The calculator first checks that the sides can close up, then finds the semi-perimeter s = (a + b + c) ÷ 2 and applies Heron’s formula, area = √(s(s−a)(s−b)(s−c)). The angles follow from the law of cosines, for example cos A = (b² + c² − a²) ÷ 2bc, and the height to side a is 2 × area ÷ a.
| Sides a, b, c | Area | Perimeter | Angles A / B / C |
|---|---|---|---|
| 3, 4, 5 | 6 | 12 | 36.87° / 53.13° / 90.00° |
| 5, 7, 8 | 17.3205 | 20 | 38.21° / 60.00° / 81.79° |
| 13, 14, 15 | 84 | 42 | 53.13° / 59.49° / 67.38° |
| 2, 3, 4 | 2.9047 | 9 | 28.96° / 46.57° / 104.48° |
Worked example: the 13-14-15 triangle
With a = 13, b = 14 and c = 15 the semi-perimeter is 21, and the calculator returns an area of exactly 84 with a perimeter of 42. The height onto side a is 12.9231, which you can confirm by hand: 2 × 84 ÷ 13. This triple is a convenient test case because the area is a whole number even though no angle is a right angle; the angles come out at 53.13°, 59.49° and 67.38°.
Worked example: an uneven plot of land
Suppose a corner lot measures 5 m, 7 m and 8 m along its three boundaries. Entering those gives 17.3205 square metres and a 20 m perimeter, which is the fencing length. Notice angle B, opposite the 7 m side, comes out at exactly 60.00°, a fact that is not obvious from the side lengths alone. Height to the 5 m side is 6.9282 m, useful if you are laying a path perpendicular to that boundary.
Reading the angles
The largest angle always sits opposite the longest side. If it reaches 90.00° the sides satisfy a² + b² = c², as for 3, 4, 5 and for 7, 24, 25. Past 90° the triangle is obtuse, and for sides 1, 1 and 1.99 the apex angle reaches 168.54°. Equal sides give equal opposite angles, so 6, 6, 6 returns 60.00° three times and an area of 15.5885.
Input and rounding limits
Area, perimeter and height are rounded to four decimals and angles to two. Enter all sides in the same unit; the area then carries that unit squared. If you only know two sides and the angle between them, compute the third side first, then come back here for the rest of the figures.
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Results are estimates for informational purposes only and do not constitute professional financial, medical, legal, or technical advice. Read full disclaimer →