Does any obtuse triangle have a greater area than any acute triangle?

Answer 1

No.

Finding an obtuse triangle that does not have a greater area than any acute triangle will show the statement is false:

Consider the obtuse triangle with sides 25, 25, 40 cm; and

the acute triangles with sides 2, 2, 2 cm and 40, 40, 40 cm

Area of (A) obtuse triangle = 40 x 12 ÷ 2 = 240 cm2

Area of (B) acute 2, 2, 2 cm triangle = 2 x √3 ÷ 2 = √3 cm2 ~= 1.7 cm2

Area of (C) acute 40, 40, 40 cm triangle = 40 x (20 x √3) ÷ 2 = 400 x √3 cm2 ~= 692.8 cm2

So you can clearly see that area of acute triangle (C) is greater than that of obtuse triangle (B) which is greater than acute triangle (A).

Thus the area of obtuse triangle (B) is not greater than that of any acute triangle.

(Obtuse triangle A has angles approx 37o, 37o, 106o, whereas triangles B and C are equilateral acute triangles with angles 60o, 60o, 60o.)