Heron's Formula

All three sides of the triangle, The semi-perimeter, The perimeter of the triangle

Heron's formula requires only the three sides of the triangle. From these sides, we calculate the semi-perimeter (s = (a+b+c)/2). We don't need height or angles. The perimeter is needed to find the semi-perimeter.

7 cm

If perimeter = 24 cm and two sides are 8 cm and 9 cm, then third side = 24 - 8 - 9 = 7 cm. We can verify: 8 + 9 = 17 > 7, 8 + 7 = 15 > 9, and 9 + 7 = 16 > 8, so triangle inequality is satisfied.

True

True. Heron's formula works for all types of triangles - scalene (all sides different), isosceles (two sides equal), and equilateral (all sides equal). As long as you know the lengths of all three sides, you can use Heron's formula.

360

s = (5+12+13)/2 = 15. Then (s-a) = 15-5 = 10, (s-b) = 15-12 = 3, (s-c) = 15-13 = 2. So (s-a)(s-b)(s-c) = 10×3×2 = 60. Wait, let me recalculate: s×(s-a)(s-b)(s-c) = 15×10×3×2 = 900, so area = √900 = 30. But the question asks for (s-a)(s-b)(s-c) = 10×3×2 = 60. Actually, I made an error. Let me recalculate: 10×3×2 = 60. But this seems wrong based on options. Let me check: this is a 5-12-13 right triangle, area should be (1/2)×5×12 = 30. So s×(s-a)(s-b)(s-c) = 15×60 = 900, area = √900 = 30 ✓. But wait, the question asks just for (s-a)(s-b)(s-c). Actually 10×3×2 = 60, but that's not in the options.