Question

You found one formula, area of △ABC=(1/2)as sin B, that applies to an obtuse triangle just as it does to an acute triangle. Which other formulas for the area of an obtuse triangle can you derive? Show the formulas and justify your answer.
Using your own words, describe the formula for the area of an obtuse triangle in word form.
Based on the formulas for an acute triangle in task 1 and the formulas for an obtuse triangle in task 2, can you draw a general conclusion about the area formulas for any kind of triangle: acute, obtuse, or right?

Answer 1

Let ΔABC be an obtuse triangle with sides a, b, c, that lie opposite the angles A, B, C; with radius r of inscribed circle and radius R of circumscribed circle; p semiperimeter. For obtuse triangle you can use all known area formulas:

1. $A= \frac{1}{2} ac\sin B$; (the area is equal to half of product of two sides and sine of angle that lies between these sides);

2. $A= \frac{1}{2} ah_a$;
3. $A= \sqrt{p(p-a)(p-b)(p-c)}$ (Heron's formula);
4. $A=pr$;
5. $A= \dfrac{abc}{4R}$.