An isosceles triangle has an area A, an apex angle 2 theta, and a height of h. Determine an expression for theta in terms of A and h.
If the base of the triangle is 2b, then
b/h = tanθ, so
b = h tanθ
The area
A = 1/2 (2b)(h)
= bh
= h^2 tanθ
So, θ = arctan(A/h^2)
To determine an expression for theta in terms of A and h, we can start by recalling the formula for the area of a triangle. The area of a triangle is given by the formula:
A = (base * height) / 2
For an isosceles triangle, the base is given by the formula:
base = 2 * (height / tan(theta))
where theta is the apex angle and tan(theta) represents the tangent of theta.
Substituting the base into the formula for the area, we have:
A = (2 * (height / tan(theta)) * height) / 2
Simplifying:
A = height^2 / tan(theta)
Multiplying both sides of the equation by tan(theta):
A * tan(theta) = height^2
Dividing both sides of the equation by A:
tan(theta) = height^2 / A
Finally, to determine an expression for theta, we can take the inverse tangent of both sides:
theta = arctan(height^2 / A)
So, the expression for theta in terms of A and h is:
theta = arctan(height^2 / A)