functions word problem

Question

A square with side length  𝑎 is inscribed inside a circle. Denote by  𝑆  the area of the region inside the circle, and outside the square (the shaded region in the diagram).

How would you express  𝑆  as a function of  𝑎 ?

answer choice 1: 𝑆=𝜋𝑎²

answer choice 2: 𝑆=(𝜋−2)𝑎²

answer choice 3:𝑆=𝜋𝑎²-𝑎²

answer choice 4:𝑆=(𝜋−1)𝑎²/2

answer choice 5:𝑆=(𝜋−1)𝑎²/4

Answer 1

If you draw the picture like this does it help?:

The segment labeled "r" is both the radius and the legs of the right triangle whose hypotenuse is "a".

r² + r² = a²

2r² = a²

r² = a²/2

So the area of the circle = πr² = πa²/2

And the area of the square is a²

So the area of the shaded region is πa²/2 - a² and then simplify

Answer 2

If you divide the square into two right triangles, the length of the hypotenuse is also the diameter of the circle.

The area of the circle is A = πr², where r is the diameter divided by 2.

You can obtain the area of the circle outside the square by subtracting the area of the square from the area of the circle.