A circle measuring 50 millimeters across lies inside a larger circle 150 millimeters across. The area between them is shaded.

50 mm
150 mm

the shaded area is the difference between the two areas:

pi(150^2-50^2) = 20000pi mm^2

oops. Divide that by 4, since the sizes were diameters, not radii.

5000pi mm^2

To find the area between the two circles, we need to subtract the area of the smaller circle from the area of the larger circle.

First, let's find the area of the smaller circle. The formula to calculate the area of a circle is:

Area = π * (radius^2)

Since we are given the diameter of the smaller circle, we can find its radius by dividing the diameter by 2:

Radius of smaller circle = 50 mm / 2 = 25 mm

Now, we can calculate the area of the smaller circle:

Area of smaller circle = π * (25 mm)^2

Next, let's find the area of the larger circle. Using the same process, we can find its radius by dividing the diameter by 2:

Radius of larger circle = 150 mm / 2 = 75 mm

Now, we can calculate the area of the larger circle:

Area of larger circle = π * (75 mm)^2

Finally, to find the shaded area between the two circles, we subtract the area of the smaller circle from the area of the larger circle:

Shaded area = Area of larger circle - Area of smaller circle

You can substitute the values we calculated into these formulas to find the answer in square millimeters.