Triangle Calculator

Area Formulas

There are several ways to find a triangle's area, and which you use depends on what you know. The most familiar: given the base and the perpendicular height, area = ½ × base × height — the height must be the perpendicular distance from the base to the opposite vertex, not a slanted side, which is a common slip. Given all three side lengths, Heron's formula applies: area = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter (half the perimeter). This is handy when you can measure the sides but not the height. Given two sides and the angle between them (the 'included' angle), area = ½ × a × b × sin(C), which is widely used in trigonometry and surveying. A worked example with Heron's formula: a triangle with sides 5, 6, and 7 has s = 9, so area = √(9 × 4 × 3 × 2) = √216 ≈ 14.7 square units. Choosing the right formula for the information you have saves you from trying to find a height you can't easily measure. This calculator accepts different combinations of inputs and applies the appropriate method, returning the area along with other properties of the triangle.

Pythagoras' Theorem

For right-angled triangles, Pythagoras' theorem is one of the most useful relationships in all of mathematics: a² + b² = c², where c is the hypotenuse — the longest side, always opposite the right angle — and a and b are the two shorter sides. Rearranged, the hypotenuse is c = √(a² + b²), and either shorter side can be found as a = √(c² − b²). A worked example: a right triangle with shorter sides of 3 and 4 has a hypotenuse of √(9 + 16) = √25 = 5 — the famous 3-4-5 triangle. The theorem only applies to right-angled triangles, so always confirm there's a 90° angle before using it. Its applications are vast: builders use it to check that corners are square (the 3-4-5 method — measure 3 units along one wall, 4 along the other, and the diagonal should be exactly 5 if the corner is a true right angle), navigators and surveyors use it for distances, and it underpins the distance formula in coordinate geometry (the straight-line distance between two points). It also generalises: for non-right triangles, the cosine rule (c² = a² + b² − 2ab·cos C) extends Pythagoras to any angle. The theorem is genuinely ancient yet remains essential in construction, engineering, design, and computing.

Special Triangles

Certain triangles have special properties worth recognising, as they simplify calculations and appear constantly in practice. An equilateral triangle has all three sides equal and consequently all three angles equal to 60° — it's perfectly symmetrical. An isosceles triangle has two equal sides and, correspondingly, two equal base angles, which is useful because knowing one tells you the other. A right triangle has one 90° angle and is where Pythagoras and trigonometry apply most directly. A scalene triangle has all sides and angles different. Two right triangles are especially common in maths and construction: the 3-4-5 triangle (and its multiples like 6-8-10 or 9-12-15) is a right triangle with whole-number sides, making it ideal for checking square corners on building sites; and the 5-12-13 is another whole-number right triangle. In trigonometry, the 45-45-90 triangle (sides in ratio 1:1:√2) and the 30-60-90 triangle (sides in ratio 1:√3:2) give exact values for common angles and are worth memorising. Recognising a special triangle often lets you find missing sides or angles instantly without a full calculation. The angle sum is always 180° in any triangle, so knowing two angles gives the third — a simple but constantly useful fact this calculator uses alongside the side and area relationships.

Exam Tips and Common Errors

A few habits prevent the most common triangle mistakes. With the area = ½ × base × height formula, ensure the height is the perpendicular height to the chosen base, not the length of a sloping side — using a slant length instead of the true height is the classic error. Check that Pythagoras is only applied to right-angled triangles, and that you've correctly identified the hypotenuse as the longest side opposite the right angle (a frequent slip is solving for the wrong side). When using trigonometric formulas, make sure your calculator is in the right angle mode — degrees or radians — as a mismatch gives completely wrong answers; most everyday problems use degrees. Remember the angle sum is 180°, a quick check: if your three angles don't add to 180°, there's an error. Watch the order of operations (BIDMAS/BODMAS) in Heron's formula — compute the bracket and the semi-perimeter first. Keep units consistent (don't mix cm and m), and remember area comes out in squared units. When a problem gives more information than you need, choose the formula that uses your most reliable measurements. Finally, sanity-check the answer: a side can't be longer than the sum of the other two (the triangle inequality), and the largest angle is always opposite the longest side — quick checks that catch many errors.