Triangle Calculator

Frequently Asked Questions

What is the Pythagorean theorem?

For a right triangle: a² + b² = c² where c is the hypotenuse. So legs of 3 and 4 give a hypotenuse of 5.

How do I find the area of a triangle?

Area = ½ × base × height. If you know all three sides, use Heron's formula: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.

When do I use the Law of Sines vs Law of Cosines?

Use Law of Sines (a/sin A = b/sin B) when you know an angle and its opposite side. Use Law of Cosines (c² = a² + b² − 2ab·cos C) for SAS or SSS.

What is the triangle inequality?

The sum of any two sides must be greater than the third side. Sides 2, 3, 6 cannot form a triangle because 2 + 3 < 6.

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How This Calculator Works

Three side lengths fully determine a triangle - this is the SSS (side-side-side) case, and from those three numbers alone you can recover everything else about the shape. Enter sides a, b and c, and this tool first checks the triangle inequality to confirm the sides can actually close into a triangle, then computes the area with Heron's formula (which needs no height), each interior angle with the law of cosines, the perimeter, and the triangle's type (acute, right or obtuse). Because it derives the angles directly from the sides, it works for any triangle, not just right triangles, and never requires you to measure a perpendicular height.

Formula / Method

First the triangle inequality: each side must be less than the sum of the other two, or no triangle exists. The semi-perimeter is s = (a + b + c) ÷ 2, and Heron's formula gives area = √(s(s−a)(s−b)(s−c)). Each angle comes from the law of cosines rearranged for the angle - for the angle A opposite side a: A = arccos((b² + c² − a²) ÷ (2bc)), converted from radians to degrees. Angle C is taken as 180° − A − B since the interior angles of any planar triangle sum to 180°. The type is read from the largest angle: exactly 90° is right, above 90° is obtuse, otherwise acute.

Worked Example

Sides a = 3, b = 4, c = 5 - the classic 3-4-5 triangle. The inequality holds (3 + 4 > 5, etc.). Semi-perimeter s = 12 ÷ 2 = 6, so area = √(6 × 3 × 2 × 1) = √36 = 6. Angle C opposite the longest side: arccos((3² + 4² − 5²) ÷ (2 × 3 × 4)) = arccos(0 ÷ 24) = arccos(0) = 90°. So this is a right triangle - confirming a² + b² = 9 + 16 = 25 = c² - with the other two angles about 36.87° and 53.13°, and a perimeter of 12.

Common Mistakes

Entering sides that violate the triangle inequality - sides like 1, 2 and 10 cannot form a triangle, and the tool will flag them.

Assuming a² + b² = c² for every triangle - the Pythagorean theorem holds only for right triangles.

Mismatching an angle with the wrong side - each angle sits opposite the side of the same letter (A opposite a).

Working in radians when you expect degrees - this tool reports angles in degrees, summing to 180°.

Trying to use base × height without a known perpendicular height - Heron's formula sidesteps this by using only the sides.

Tips & FAQ

What are the triangle types? By sides: equilateral (all three equal, all angles 60°), isosceles (two equal sides and two equal base angles), and scalene (all different). By angles: acute (all under 90°), right (one exactly 90°), and obtuse (one over 90°). Which area formula does this use? Heron's, area = √(s(s−a)(s−b)(s−c)) - ideal for SSS because it needs no height; the alternatives are ½ × base × height (needs a height) and ½ab sin(C) (needs two sides and the included angle). What is the law of cosines? c² = a² + b² − 2ab cos(C), the generalisation of Pythagoras to any triangle - this tool inverts it to solve for the angles. And the law of sines? a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C), useful when you instead know an angle and its opposite side. Why check the inequality first? If the two shortest sides do not exceed the longest, the "triangle" collapses to a line or cannot close at all, and area and angles are undefined.

Triangle Calculator

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