What this calculator does
Heron’s formula finds the area of a triangle from its three sides alone, with no need to measure a height. That is enormously practical, because sides are easy to measure with a tape while perpendicular heights rarely are.
Three sides also determine every angle, through the cosine rule. Along with the area this calculator reports the angles, the height on the longest side, and the inradius and circumradius: the circles that fit inside and pass through all three corners.
The formula
Heron’s formula uses the semi-perimeter, half the sum of the sides. The expression under the square root becomes negative whenever the sides cannot form a triangle, which is why the triangle inequality must hold: no side may be longer than the other two combined.
| Term | Meaning |
|---|---|
| s | Semi-perimeter: (a + b + c) ÷ 2. |
| Heron’s formula | Area = √(s(s−a)(s−b)(s−c)). |
| Inradius | Radius of the largest circle fitting inside, equal to Area ÷ s. |
| Circumradius | Radius of the circle through all three corners. |
| Triangle inequality | Each side must be shorter than the sum of the other two. |
The inputs explained
| Field | What to enter |
|---|---|
| Side a | The first side length. |
| Side b | The second side length. |
| Side c | The third side length. If any side exceeds the sum of the other two, no triangle exists and the calculator says so. |
When to use it
Measuring an irregular block of land
Divide the plot into triangles, measure each side with a tape, and add the areas. This is how land was surveyed for centuries before instruments.
Checking a right angle on site
A 3-4-5 triangle is exactly right-angled. Measuring those distances is the standard way to square up a frame or a foundation.
Classifying a triangle
The type line reports acute, right or obtuse by comparing the square of the longest side against the sum of the other two squares.
Finding a height you cannot measure
The height on side a follows from twice the area divided by a. Useful when the apex is out of reach.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Triangles with a growing third side
Two sides fixed at 7 and 8 while the third varies from nearly degenerate to nearly straight.
| Third side | Area | Angles | Type |
|---|---|---|---|
| 2 | 6.437 | 53.58° · 113.13° · 13.29° | Obtuse |
| 5 | 17.321 | 60.00° · 81.79° · 38.21° | Acute |
| 9 | 26.833 | 48.19° · 58.41° · 73.40° | Acute |
| 10.63 | 28.000 | 41.19° · 48.81° · 90.00° | Acute |
| 12 | 26.906 | 34.09° · 39.84° · 106.07° | Obtuse |
| 14.9 | 6.427 | 6.19° · 7.08° · 166.73° | Obtuse |
Classic triangles
A selection of triangles whose properties are worth recognising.
| Third side (with 3 and 4) | Area | Type | Inradius · circumradius | Perimeter |
|---|---|---|---|---|
| 3, 4, 3 | 4.472 | Acute | 0.8944 · 2.012 | 10.000 |
| 3, 4, 4 | 5.562 | Acute | 1.011 · 2.157 | 11.000 |
| 3, 4, 5 | 6.000 | Right-angled | 1.000 · 2.500 | 12.000 |
| 3, 4, 6 | 5.333 | Obtuse | 0.8204 · 3.375 | 13.000 |
| 3, 4, 6.9 | 2.012 | Obtuse | 0.2895 · 10.287 | 13.900 |
Questions
What is the triangle inequality?
Every side must be shorter than the sum of the other two. If it is not, the sides cannot close into a triangle, and Heron’s formula returns a negative value under the square root.
When should I use Heron’s formula?
When you know all three sides but no angles or heights. If you know two sides and the angle between them, ½ab·sin C is simpler.
How do I tell if a triangle is right-angled?
Compare the square of the longest side against the sum of the squares of the other two. Equal means right-angled, smaller means acute, larger means obtuse.
What is the difference between inradius and circumradius?
The inradius is the largest circle that fits inside, touching all three sides. The circumradius is the circle passing through all three corners. The circumradius is always larger.
Can I find the angles from the sides alone?
Yes, using the cosine rule rearranged for the angle. This calculator applies it to all three, and they will always sum to 180°.
For two sides and an angle, see the triangle solver. For right angles specifically, use the right triangle calculator.