A square is inscribed in a circle of diameter [tex]$12 \sqrt{2}$[/tex] millimeters. What is the area of the shaded region?
Recall that in a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle, if the legs each measure [tex]$x$[/tex] units, then the hypotenuse measures [tex]$x \sqrt{2}$[/tex] units.
A. [tex]$(72 \pi-144) mm ^2$[/tex]
B. [tex]$(72 \pi-72) mm ^2$[/tex]
C. [tex]$(288 \pi-288) mm ^2$[/tex]
D. [tex]$(288 \pi-144) mm ^2$[/tex]
Answer (2)
Calculate the radius of the circle: r = f r a c 12 s q r t 2 2 = 6 s q r t 2 .
Calculate the area of the circle: A c i rc l e = p i ( 6 s q r t 2 ) 2 = 72 p i .
Calculate the side length of the square: s = f r a c 12 s q r t 2 s q r t 2 = 12 .
Calculate the area of the shaded region: A s ha d e d = 72 p i − 144 .
( 72 π − 144 ) m m 2
Explanation
Find the radius First, we need to find the radius of the circle. The diameter is given as 12 s q r t 2 millimeters, so the radius is half of that.
Calculate the radius The radius r is calculated as: r = f r a c 12 s q r t 2 2 = 6 s q r t 2
Calculate the area of the circle Next, we calculate the area of the circle using the formula A c i rc l e = p i r 2 .
Area of the circle Substituting the value of r we get: A c i rc l e = p i ( 6 s q r t 2 ) 2 = p i ( 36 c d o t 2 ) = 72 p i
Find the side length of the square Now, we need to find the side length of the square. The diagonal of the square is equal to the diameter of the circle, which is 12 s q r t 2 mm. If s is the side length of the square, then the diagonal is ss q r t 2 .
Calculate the side length So, we have ss q r t 2 = 12 s q r t 2 . Dividing both sides by s q r t 2 , we get s = 12 mm.
Calculate the area of the square The area of the square is A s q u a re = s 2 .
Area of the square Substituting the value of s , we get: A s q u a re = 1 2 2 = 144
Calculate the shaded area Finally, we find the area of the shaded region by subtracting the area of the square from the area of the circle: A s ha d e d = A c i rc l e − A s q u a re = 72 p i − 144
Final Answer Therefore, the area of the shaded region is ( 72 p i − 144 ) m m 2 .
Examples
Imagine you're designing a circular garden with a square seating area in the middle. Knowing how to calculate the shaded area (the garden space around the seating) helps you determine how much soil you need for planting flowers! This calculation ensures you have enough space for your plants while keeping the seating area intact. The area of the shaded region, calculated as the difference between the circle's area and the square's area, is a practical way to optimize space in your garden design.
The area of the shaded region between the inscribed square and the surrounding circle is given by ( 72 π − 144 ) mm 2 . This was calculated by determining the areas of both shapes and subtracting the area of the square from the area of the circle. Thus, the correct choice is option A.
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