The Universal Rule for All Triangles
Before getting into types, one property applies to every triangle without exception: the three interior angles always add up to exactly 180°. If you know two angles, the third is always 180 − (angle1 + angle2).
If A = 65° and B = 45°, then C = 180 - 65 - 45 = 70°
Classification by Sides
Equilateral Triangle
All three sides are the same length, and all three angles are equal (each is 60°). It's the most symmetric triangle — if you rotate it by 120° or flip it over, it looks identical.
| 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
Two sides are equal, and the two base angles (opposite the equal sides) are also equal. This is one of the most common triangle types you'll see in geometry problems.
| Property | Value |
|---|---|
| Equal sides | a = b (two of three sides) |
| Base | c (the different side) |
| Base angles | Equal to each other |
| Perimeter | 2a + c |
The area formula for an isosceles triangle uses the height (h) to the base: Area = (1/2) × base × height. If you don't know the height directly, use Pythagoras to find it from the equal side and half the base.
Scalene Triangle
All three sides are different lengths, and all three angles are different. No special symmetry, but the same formulas still apply. Most real-world triangles are scalene — perfectly equal sides are actually pretty rare in practice.
Classification by Angles
Acute Triangle
All three angles are less than 90°. An equilateral triangle is a special case of acute (all angles 60°). Acute triangles feel "pointy" — no angle is dominant.
Right Triangle
One angle is exactly 90°. This is by far the most important triangle type in practical math — it's the basis of trigonometry, the Pythagorean theorem, and a huge amount of real-world measurement and construction.
(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)
The area of a right triangle is especially clean because the two legs are already perpendicular — one is the base and the other is the height:
Legs 6 and 8: Area = (1/2) × 6 × 8 = 24
Obtuse Triangle
One angle is greater than 90°. The other two angles must add up to less than 90° to keep the total at 180°. Obtuse triangles look wide and "flattened." Important: a triangle can only have one obtuse angle — if one angle is greater than 90°, there's no room for another.
Combined Classification
A triangle is classified by both its sides and angles at the same time:
| 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° |
Key Triangle 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
Missing Sides and Angles — The Law of Cosines
For any triangle (not just right triangles), the law of cosines finds a missing side or angle when the Pythagorean theorem doesn't apply:
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
Pythagorean Triples
These are sets of whole numbers that satisfy a² + b² = c². Handy to recognize because they often show up in problems:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 6 | 8 | 10 |
| 9 | 12 | 15 |
Multiples of Pythagorean triples also work — 6-8-10 is just 2×(3-4-5), for example.
Using the Triangle Calculator
The SolveCalc triangle calculator works out area, perimeter, angles, and missing sides for any triangle type. Enter what you know — side lengths, angles, or a combination — and it calculates the rest using the appropriate formula.
Conclusion
Triangles are classified by their sides (equilateral, isosceles, scalene) and their angles (acute, right, obtuse). The Pythagorean theorem handles right triangles specifically, while the law of cosines and Heron's formula work for any triangle when you have enough information. The 180° angle rule is the universal anchor — always check your angles sum correctly before trusting any calculation. Right triangles especially are worth being comfortable with; they underpin trigonometry and show up in almost every geometry application you'll ever encounter.