A square is inscribed in a circle of diameter 12 millimeters. What is the area of the shaded region? A square is inscribed in a circle with a diameter of 12 StartRoot 2 EndRoot millimeters. Everything outside of the square is shaded. Recall that in a 45 – 45 – 90 triangle, if the legs each measure x units, then the hypotenuse measures x units. (72π – 144) mm2 (72π – 72) mm2 (288π – 288) mm2 (288π – 144) mm2

[tex]A_{shaded}=A_{circle}-A_{square}\\\\\\A_{circle}=\pi \cdot r^2\\.\qquad \ =\pi \bigg(\dfrac{12\sqrt2}{2}\bigg)^2\\\\.\qquad \ =\pi (6\sqrt2)^2\\.\qquad \ =72\pi\\\\\\A_{square}=side^2\\.\qquad \quad =\dfrac{12\sqrt2}{\sqrt2}^2\\\\.\qquad \quad =12^2\\\\.\qquad \quad =144\\\\\\\large\boxed{A_{shaded}=72\pi-144}[/tex]

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ibiscollingwood ibiscollingwood · Answer 2 · 2022-04-08T15:34:13+02:00

The area of shaded region is (72π – 144) square millimeters.

To understand more, check below explanation.

Area of shaded region:

It is given that,

The diameter of circle is [tex]12\sqrt{2}[/tex] millimeters.

Since, radius = diameter/2

So that, radius of circle[tex]=12\sqrt{2}/2=6\sqrt{2}[/tex]

now, we have to find area of circle,

[tex]Area=\pi *r^{2} \\\\Area=\pi *(6\sqrt{2} )^{2} \\\\[/tex]

Area = 72π square millimeters

The side of inscribed square[tex]=12\sqrt{2} /\sqrt{2}[/tex] = 12mm

Since, area of square= (side)^2

Area of square= 12 * 12 = 144 square millimeters

To find the area of shaded region, subtract area of square from area of circle.

Area of shaded region = area of circle - area of square

Area of shaded region = (72π – 144) square millimeters.

Learn more about the area of circle here:

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