Calculate the perimeter of a triangle from three sides with automatic triangle inequality validation. Also computes area, angles, semi-perimeter, classification, radii, and altitudes.
The perimeter of a triangle is the total length of its three sides: P = a + b + c. While the formula itself is simple, this calculator goes far beyond a basic sum. It validates that your three side lengths actually form a valid triangle (the triangle inequality: the sum of any two sides must exceed the third), classifies the triangle by its sides (equilateral, isosceles, or scalene) and by its angles (acute, right, or obtuse), and derives a full set of properties from the three sides alone.
Using Heron's formula with the semi-perimeter s = P/2, the area is A = √[s(s−a)(s−b)(s−c)]. The law of cosines recovers all three angles: cos A = (b² + c² − a²) / (2bc). From these, the calculator computes all three altitudes (h = 2A/side), the circumradius R = abc / (4A), the inradius r = A / s, and the medians.
Knowing the perimeter is essential in practical contexts: fencing a triangular plot of land, trimming fabric, welding triangular frames, or estimating material length for construction projects. Architects use semi-perimeter extensively in structural calculations.
This calculator includes a unit selector, presets for common triangles (Pythagorean triples and named triangles), a classification badge, visual side-comparison bars, and a reference table of well-known triangles with their perimeters and areas.
Perimeter is often the first quantity you need, but it is rarely the only one that matters. If you are cutting trim, estimating fencing, or checking whether measured sides make sense in a design drawing, you usually also need validation, area, and triangle classification at the same time. This calculator bundles those checks so the side lengths become immediately useful instead of just producing a total.
It also helps when you are comparing candidate triangles. Two sets of sides can have similar perimeters but very different angle structure, area, and radius values. Seeing perimeter alongside those derived measures gives a much better picture of the triangle than a simple sum alone.
Perimeter: P = a + b + c Semi-perimeter: s = P / 2 Area (Heron): A = √[s(s−a)(s−b)(s−c)] Angles (law of cosines): cos A = (b² + c² − a²) / (2bc) Altitude: h_a = 2A / a Circumradius: R = abc / (4A) Inradius: r = A / s
Result: Perimeter = 22, Area ≈ 16.25, Angles ≈ 40.5°, 111.8°, 27.7° (obtuse scalene)
Sides 7, 10, 5 cm: P = 22, s = 11. Area = √(11 × 4 × 1 × 6) = √264 ≈ 16.25 cm². cos B = (49 + 25 − 100) / 70 = −0.371 → B ≈ 111.8° (obtuse). Classification: scalene obtuse triangle.
Adding three sides gives the boundary length, but perimeter alone does not tell you whether the triangle is narrow, wide, acute, or obtuse. That is why side validation is important. Before using any perimeter in a real problem, make sure the measurements satisfy the triangle inequality and actually correspond to a possible figure.
Triangle perimeter appears in material estimates more often than students expect. It is used for edge banding, frame construction, border trim, fencing, and cable length around triangular layouts. In those settings, a bad side measurement affects both cost and fit, so it helps to see the perimeter together with a geometric sanity check.
Once the perimeter is known, the semi-perimeter becomes the gateway to Heron's formula and many classical triangle identities. If your computed area seems unusually small relative to the perimeter, that usually indicates a long, thin triangle. If the largest angle turns obtuse, the triangle is spreading out rather than staying compact. Those comparisons help you move beyond the arithmetic and understand the geometry behind the numbers.
The perimeter is the total length of all three sides: P = a + b + c. It represents the distance around the triangle.
The sum of any two sides must be greater than the third side: a + b > c, a + c > b, and b + c > a. If any condition fails, the three lengths cannot form a triangle.
Use Heron's formula: compute s = (a+b+c)/2, then Area = √[s(s−a)(s−b)(s−c)]. No angles or heights needed.
Equilateral: all sides equal. Isosceles: exactly two sides equal. Scalene: all sides different.
Acute: all angles < 90°. Right: one angle = 90°. Obtuse: one angle > 90°. Use the law of cosines to find angles from sides.
The semi-perimeter s = P/2 is half the perimeter. It's the central variable in Heron's formula and simplifies many triangle computations.