show how to calculate area between two concentric circles

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To calculate the area between two concentric circles, we first need to understand what concentric circles are. Concentric circles share the same center point but have different radii. In this example, we have two circles with center O. The larger circle has radius R, and the smaller circle has radius r. To find the area between these two circles, we subtract the area of the smaller circle from the area of the larger circle. Now let's derive the formula for the area between two concentric circles. The area of the larger circle with radius R is pi times R squared. The area of the smaller circle with radius r is pi times r squared. To find the area between the two circles, we subtract the area of the smaller circle from the area of the larger circle. This gives us pi times R squared minus pi times r squared, which can be factored as pi times the difference of the squares: R squared minus r squared. So the formula for the area between two concentric circles is pi times R squared minus r squared. Let's work through an example. Find the area between two concentric circles with radii 5 centimeters and 3 centimeters. We'll use our formula: area equals pi times the difference of the squares of the radii. The larger radius R is 5 centimeters, and the smaller radius r is 3 centimeters. Substituting these values into our formula, we get pi times 5 squared minus 3 squared, which is pi times 25 minus 9, which equals pi times 16. This gives us 16 pi square centimeters, which is approximately 50.27 square centimeters. The green shaded region represents the area between the two concentric circles. There's an alternative method to calculate the area between concentric circles using the difference of squares formula. We know that R squared minus r squared can be factored as R plus r times R minus r. So the area between the circles can be written as pi times R plus r times R minus r. For our example with R equals 5 centimeters and r equals 3 centimeters, we can calculate the area as pi times 5 plus 3 times 5 minus 3, which is pi times 8 times 2, giving us 16 pi square centimeters. This approach is often more efficient for calculations. Geometrically, we can visualize this as transforming the annular region into a rectangle with width R minus r and height R plus r, while preserving the area. To summarize what we've learned about calculating the area between two concentric circles: First, concentric circles share the same center but have different radii. The area between these circles is found by subtracting the area of the smaller circle from the area of the larger circle. This gives us the formula: Area equals pi times R squared minus r squared, where R is the radius of the larger circle and r is the radius of the smaller circle. We can also use the difference of squares formula to express this as pi times R plus r times R minus r. This concept has many practical applications, including calculating washer areas in calculus, designing ring-shaped structures in engineering, and determining the area of circular pathways in architecture and landscaping.