The three sides of a triangle measure 3 cm, 5 cm and 7 cm. Its area is:

Find the angle between the two shorter sides first: cosθ = $(3^2+5^2-7^2)/(2\times 3\times 5)=-15/30=-1/2$, so θ = 120°. The area is then (1/2)·3·5·sin120° = 7.5 × (√3/2) = 15√3/4 ≈ $6.50{\text{cm}}^2$, which Heron's formula confirms: √$(7.5\times 4.5\times 2.5\times 0.5)=6.495$. Forgetting the factor 1/2 gives 15√3/2; applying the formula to sides 5 and 7, which do NOT enclose the 120° angle, gives 35√3/4 ≈ $15.16{\text{cm}}^2$ — larger than the whole 3-5-7 triangle could ever be; and using sin90° = 1 instead of sin120° gives 15/2.