Heron’s formula when height is hidden

With three sides known, s = (a+b+c)/2 and area = √[s(s−a)(s−b)(s−c)].

Key formulas

No perpendicular height is marked and no angle is given — only three sides. Heron avoids dropping an awkward altitude.

Sides 5, 5, 6. s=8. Area = √[8(3)(3)(2)] = √144 = 12.

If s−a is negative, the lengths do not form a triangle. Stop there.

Problem. Find the area of a triangle with sides 5, 5 and 6.

Solution. s = (5+5+6)/2 = 8. Area = √[8(8−5)(8−5)(8−6)] = √[8×3×3×2] = √144 = 12.

For 5,5,6 you could also split the base; Heron confirms the same 12.

Sides 7, 8, 9. Semi-perimeter s is:

Yes when three sides are known; show the substitution clearly.