The Rule Every Triangle Obeys
However strange a triangle's proportions look, its three interior angles will always add up to 180 degrees, with no exceptions. That single fact is useful on its own: give someone two of the three angles, and they can find the missing one just by subtracting.
If A = 65° and B = 45°, then C = 180 - 65 - 45 = 70°
Classifying Triangles by Side Length
Equilateral Triangle
When all three sides match in length, all three angles are forced to match too, landing at exactly 60° each. An equilateral triangle is the most symmetric shape in this whole category - rotate it a third of a turn, or flip it over, and it looks completely unchanged.
| Property | Formula |
|---|---|
| All sides equal | a = b = c |
| All angles | 60° each |
| Area | (√3 / 4) × a² |
| Perimeter | 3a |
Area = (√3 / 4) × 36 ≈ 0.433 × 36 ≈ 15.59
Perimeter = 3 × 6 = 18
Isosceles Triangle
An isosceles triangle has exactly two equal sides, and the two angles sitting opposite those sides end up equal as well. These turn up constantly in geometry coursework, probably because they strike a middle ground - predictable enough to work with, but without the total symmetry of an equilateral triangle.
| Property | Value |
|---|---|
| Equal sides | a = b (two of three sides) |
| Base | c (the different side) |
| Base angles | Equal to each other |
| Perimeter | 2a + c |
Finding the area still comes down to Area = (1/2) × base × height, but you need the height dropped from the peak to the base first. When that height isn't given directly, cut the triangle down the middle - you'll get two right triangles, and the Pythagorean theorem will get you the height from the equal side and half the base.
Scalene Triangle
A scalene triangle has three different side lengths and three different angle measures - no shortcuts from symmetry here, though every general formula below still applies without modification. If you measured a random triangle out in the real world, say a torn scrap of paper or a bent street sign, it would almost certainly turn out scalene. Sides matching up exactly is really more of a textbook thing.
Classifying Triangles by Angle Size
Acute Triangle
All three angles measure less than 90°. Since 60° sits comfortably under that limit, every equilateral triangle technically qualifies as acute too. Picture an acute triangle as evenly pointed - no single corner dominates the shape.
Right Triangle
One angle sits at exactly 90°, no more and no less. Of the six types covered here, this is the one you'll use the most: trigonometry is built around right triangles, and the Pythagorean theorem alone turns up in construction, navigation, and no shortage of physics problems.
(where c is the hypotenuse - the side opposite the right angle)
Legs of 3 and 4: c = √(3² + 4²) = √(9 + 16) = √25 = 5
(The 3-4-5 right triangle is the most famous Pythagorean triple)
Area is easier to compute for a right triangle than for any other kind, since the two legs already meet at a right angle - one leg simply doubles as the base, the other as the height, with no separate height calculation needed:
Legs 6 and 8: Area = (1/2) × 6 × 8 = 24
Obtuse Triangle
One angle exceeds 90°. Because the full set still has to total 180°, the other two angles get squeezed into a combined total under 90°, which is exactly why obtuse triangles look stretched out and flat. Worth remembering: a triangle can never have two obtuse angles at the same time. Once one angle passes 90°, there simply isn't enough of the 180° budget left over to fit a second one.
Combining Both Classifications
Most of the time, a full triangle description uses side type and angle type together:
| Type | Sides | Angles |
|---|---|---|
| Equilateral | All equal | All 60° (always acute) |
| Isosceles Acute | Two equal | All < 90° |
| Isosceles Right | Two equal | One = 90°, two = 45° |
| Isosceles Obtuse | Two equal | One > 90° |
| Scalene Acute | All different | All < 90° |
| Scalene Right | All different | One = 90° |
| Scalene Obtuse | All different | One > 90° |
Area and Perimeter Formulas
Area
If you know two sides (a, b) and the angle between them (C):
Area = (1/2) × a × b × sin(C)
If you know all three sides (Heron's formula):
s = (a + b + c) / 2 (semi-perimeter)
Area = √(s(s-a)(s-b)(s-c))
Perimeter
When the Triangle Isn't Right-Angled: The Law of Cosines
The Pythagorean theorem only applies when a right angle is involved. For any other triangle, the law of cosines takes over and can find a missing side or angle no matter the triangle's shape:
Find side c when a=5, b=7, C=60°:
c² = 25 + 49 − 2(5)(7)(0.5)
c² = 74 − 35 = 39
c = √39 ≈ 6.24
Common Pythagorean Triples
These are integer triples that satisfy a² + b² = c² with no rounding needed. Knowing a few of them by sight means you can skip the square-root arithmetic entirely when one shows up in a problem:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
| 9 | 12 | 15 |
Any multiple of a known triple works too - 6-8-10 is simply the 3-4-5 triple scaled up by a factor of 2.
Solving Triangles With a Calculator
When you'd rather not run the numbers by hand, SolverCalc's triangle calculator, a calc solver built specifically for this job, handles area, perimeter, angles, and missing sides regardless of which triangle type you're working with. Enter whatever information you already have, whether that's sides, angles, or a mix of both, and it applies whichever formula fits.
The Short Version
Every triangle falls into one of three side-based categories (equilateral, isosceles, scalene) and one of three angle-based categories (acute, right, obtuse). Right triangles get the Pythagorean theorem to themselves; anything else needs the law of cosines or Heron's formula once you know enough measurements. And whatever you're working with, the 180° angle sum is always worth checking - if your three angles don't add up to that, something earlier in the problem went wrong. Right triangles especially are worth knowing cold, since so much of trigonometry rests on them and they come up constantly once you start noticing them.