# Calculus question?

A square is inscribed in the circle x^2+y^2=32. Find the area of the square.

Relevance

x² + y² = 32

(x - 0)² + (y - 0)² = (√32)²

The typical equation of a circle is: (x - xo)² + (y - yo)² = R² → where:

xo: abscissa of center → 0 in your case

yo: ordinate of center → 0 in your case

R: radius of circle → √32 in your case → then the diameter is: d = 2√32

As the square is inscribed in the circle, the diameter of the circle is the diagonal of the square.

The surface area of the square is:

a = s² ← where s is one of the sides of the square

According the Pythagorean's theorem, you can see that:

s² + s² = d² ← where d is the diagonal of the square, i.e. the diameter of the circle

2s² = d²

s² = d²/2 → recall: a = s²

a = d²/2 → recall: d = 2√32

a = (2√32)²/2

a = (4 * 32)/2

a = 64

• Circle equation: x^2 + y^2 = 32

Circle center: (0, 0)

Radius of circle: square root of 32 = 4 times square root of 2

Diameter of circle: 8 times square root of 2

Sides of square using Pythagoras: 8

Area of inscribed square: 8 times 8 = 64 square units

• The area of a square is the half the square of the diagonal.

The area of a square is the half the square of the diameter of the circumscribed circle.

The area of a square is the double the square of the radius of the circumscribed circle.

The square of the radius of the circle is 32, so the area of a square is 64 square units.

• Since the square is inscribed in the circle,

the diagonal distance between opposite corners is √32.

x^2 + y^2 = √(32)^2, where 'c' is the diagonal (which is √32) across the square,

and forms the hypotenuse of a right triangle.

Since this is a square, we know that x = y.

So we know that x^2 + x^2 = (16)^2.

• Let the apex of the square on the circle be (x, y) in the first quadrant. But x=y for a square.

Area, A = 4yx=4x^2.

x^2+y^2=32.

x^2 + (x)^2 = 32.

4x^2 =64

A = 64.

Or, you may use the diagonal length = diameter of the circle = 2√32 =8√2.

Area = 8√2*8√2/2 = 64.

• The diagonal of the square is the diameter of the circle = 2√32 = 8√2

The square of the diagonal is twice the square of the side of the square:(8√2)² = 2s²

s² = 2(8)²

s² = 64

Area of the square is the square of the side of the square:

A = s² = 64

Ans: 64

• Anonymous
1 month ago