MultiConvers

Triangle Calculator

Solve a general triangle from three sides for area, perimeter and all three angles, or skip the full solve and compute area alone from a base and height or from two sides and the included angle.

How to use

  1. Pick SSS when you have all three sides, base-and-height when you have a side and the perpendicular to it, or SAS when you have two sides and the angle between them.
  2. On SSS, check the triangle inequality first: each pair of sides must be longer than the remaining side, or the form rejects the triple (1, 2, 9 never closes).
  3. On SAS, type the included angle in degrees, strictly between 0° and 180°. That angle must sit between the two sides you entered, not opposite one of them.
  4. Read area in unit². Only SSS also lists perimeter and the three opposite angles; the other two modes stop at area.

How it works

A survey stake-out, a roof truss or a triangular garden plot usually arrives as three measured sides. SSS runs Heron’s formula: s is the semi-perimeter and area is √[s(s−a)(s−b)(s−c)]. The angles opposite a, b and c then come from the law of cosines, with angle C finished as 180° minus the other two so they sum to a straight line. A 3-4-5 triple yields area 6, perimeter 12 and a right angle opposite the side of 5.

Base-and-height is the school formula ½bh. It is the right choice when a drawing shows an altitude but not the other two sides. It does not invent those sides or the angles — you only get the enclosed area. The height must be perpendicular to the chosen base; a sloped side used as “height” is the same error as on the area page, but here it is the whole mode.

SAS uses ½ab sin(C). The angle is the one between those two sides. An angle of 0° or 180° collapses the figure to a line and is rejected, as is anything outside that open interval. If you instead know two sides and a non-included angle (SSA), this page will not resolve the ambiguous case; that configuration is not implemented.

This is not the right-triangle specialist. When you already know one corner is 90° and you have two legs, or a leg and the hypotenuse, the right-triangle calculator also returns the missing side, complementary acute angles, area and perimeter without asking for a third side. It is also not the Pythagorean page, which refuses to discuss angles at all and only solves a² + b² = c².

Keep sides in one length unit. Area is that unit squared; angles are degrees. Degenerate or impossible side lists fail the inequality or produce a non-positive Heron radicand and return nothing rather than a negative or imaginary area.

Formula

s = (a + b + c) / 2
Area = √[s(s − a)(s − b)(s − c)]
cos C = (a² + b² − c²) / (2ab)
Area = ½bh  or  ½ab sin(C)

Worked examples

  • A 3-4-5 triangle

    sides 3, 4, 5 → area 6, perimeter 12, right angle opposite 5

  • Base and height

    base 10, height 4 → area 20

Useful notes

  • SSS also reports angles opposite a, b and c.
  • For a right triangle with two legs, the right-triangle page also gives the hypotenuse.
  • Base-and-height and SAS return area only — no perimeter and no angles.
  • SAS rejects included angles of 0° or 180° and anything outside that range.
  • SSA (two sides and a non-included angle) is not a mode on this page.

FAQ

How do I find the area of a triangle?

If you know base and height, area is half their product. If you know all three sides, use Heron. If you know two sides and the included angle, use ½ab sin(C).

Why was my triangle rejected?

The sides must satisfy the triangle inequality: the sum of any two sides is greater than the third. 1, 2 and 9 cannot form a triangle.

What unit is the area in?

The square of the length unit you selected. Sides in cm give area in cm².

Does base-and-height mode also give the angles?

No. That mode only computes ½ × base × height. Switch to three sides when you need perimeter and the angles opposite a, b and c.

What does SAS mean on this calculator?

Two sides and the included angle between them. Area is ½ab sin(C) with C in degrees. The angle must be the one that sits between those sides.

Guides

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