Finding the region inside the circle but outside the rectangle?

Question

A rectangle with length 16 cm and 12 cm is inscribed in a circle. Find the area of the region inside the circle but outside the rectangle.

Answer 1

I encourage you draw a picture of the situation and label everything. Draw a diagonal from the upper left corner of the rectangle to the lower right.

Our strategy will be to find the area of the circle and then subtract the area of the rectangle.

The area of the rectangle is easy:

area = base * height = 16 * 12 = 192

The area of the circle is a bit harder. area of circle = pi * radius^2, so we need to find the radius of the circle.

The diagonal line I encouraged you to draw earlier is a diameter of the circle, and half of that is the radius. To find the length of the diameter, we use the Pythagorean theorem.

a^2 + b^2 = c^2
12^2 + 16^2 = c^2
144 + 256 = c^2
400 = c^2
20 = c = our diameter

radius = diameter / 2 = 20 / 2 = 10
area of circle = pi * radius^2 = pi * 10^2 = 100pi

So the area is

area of circle - area of rectangle = 100pi - 192

Answer 2

The diagonal of the rectangle is sqrt(16^2 + 12^2) = 20. This is the diameter of the circle. The radius then is 10. The area inside the circle is then 100pi (pi * r^2).

The area inside the rectangle is 16 * 12 = 192.

The area inside the circle but not inside the rectangle is 100pi - 192, or about 122.16

Answer 3

The diameter of the rectangle is that for the circle :
= sqrt(12^2+16^2) =20 cm
The area of the region inside the circle but outside the rectangle.
= circle area - recyangle area = pi (r^2) - 12*16
= 3.1415926536 * 100 - 192 = 314.15926536 - 192 = 122.15926536 cm^2

Answer 4

you need to find the area of the circle and rectangle individually..

area of rectangle..
l * w

16 * 12
= 192cm^2

to get the area of the circle, we need to find the radius..
the rectangle is inscribed in the circle..
therefore the diagonal of the rectangle is the diameter of the circle..
diagonal = diameter
diagonal = squareroot of (l^2 + w^2)

diagonal = squareroot of (16^2 + 12^2)
= squareroot of (256 + 144)
= squareroot of 400
= 20

diameter = 20cm
radius = 10cm

area = pir^2, pi = 3.14
= 3.14(10^2)
= 3.14(100)
= 314cm^2

to get the area of the region inside the circle but outside the rectangle, we need to subtract the area of the rectangle from the value of the circle..

314cm^2 - 192cm^2
= 122cm^2

Answer 5

Use the pythagorean formula to find the diameter of your circle (the diagonal of the rectangle). From there, subtract the area of the rectangle from the area of the circle.

Answer 6

Draw the picture.