To determine the angle θ that maximizes the area of a triangle when the lengths of two sides are known (let’s call them ‘a’ and ‘b’), we can use the formula for the area of the triangle:
Area = (1/2) * a * b * sin(θ)
The sine function, sin(θ), reaches its maximum value of 1 when θ is equal to 90 degrees (or π/2 radians). Therefore, the area of the triangle is maximized when the angle between the two sides measures:
θ = 90 degrees
In this scenario, with θ equal to 90 degrees, the triangle formed is a right triangle, which provides the largest possible area for given side lengths ‘a’ and ‘b’.
To summarize, if you want to maximize the area of a triangle with two sides of lengths ‘a’ and ‘b’, you should have the angle between them be:
θ = 90 degrees