Triangle Perimeter and Area

Triangle perimeter: basic formula and example

The perimeter (O) is the sum of all three sides: O = a + b + c. Always include units (e.g., cm). Example: a triangle with sides 3 cm, 4 cm, and 5 cm has O = 3 + 4 + 5 = 12 cm. If you know only two sides and an angle or a height, you can’t find the perimeter directly—first determine the missing side (with the Law of Cosines or with Pythagoras for a right triangle).

Area: from the basic formula to Heron’s

The basic area formula (P) is P = (a · h_a)/2, where h_a is the height to base a. If you know two sides and the included angle γ, use P = (1/2)ab sin γ. When only the three sides are known, apply Heron’s formula: compute the semiperimeter s = (a + b + c)/2, then P = √(s(s − a)(s − b)(s − c)). This is especially handy when heights are unknown but side lengths are given.

Special cases: right, isosceles, equilateral

For a right triangle with legs p and q, P = (p · q)/2 and O = p + q + √(p² + q²). In an isosceles triangle, you can get the height to the base by halving the base and applying Pythagoras. For an equilateral triangle with side a, O = 3a and P = (a²√3)/4. These “special” forms save time and offer quick plausibility checks.

Common mistakes and quick checks

(1) Check the triangle inequality: for any pair of sides, a + b > c (cyclically). (2) Don’t forget units and sensible rounding. (3) With Heron’s formula, the expression under the root must not be negative—if it is, the data are inconsistent. (4) Quick test: for sides 7, 8, and 5, O = 20 cm, s = 10 cm, and P = √(10 · 3 · 2 · 5) = √300 ≈ 17.32 cm². The result is reasonably smaller than a rough base–height estimate, so it’s plausible. For exams: sketch first, mark the given data, choose the right formula, then compute.