Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is pi h (2r+h)
A(r) = πr^2
A(r+h) = π(r+h)^2
now subtract.
Prove that the area of circular path of uniform width h surrounding a circular region at radius r is pi h 2r+h
2pieh (2r+h)
To prove the formula for the area of a circular path of uniform width h surrounding a circular region of radius r, we can follow these steps:
Step 1: Start with the formula for the area of a larger circle with radius (r + h). Let's call this Circle A.
- The area of Circle A is given by A_A = π * (r + h)^2.
Step 2: Next, subtract the area of the smaller circle with radius r from the area of Circle A. Let's call this Circle B.
- The area of Circle B is given by A_B = π * r^2.
Step 3: The difference in area between Circle A and Circle B will give us the area of the circular path of uniform width h surrounding the circular region.
- The area of the circular path is given by A_path = A_A - A_B.
Now, let's simplify and calculate the area of the circular path.
Step 4: Substitute the values of A_A and A_B into the equation for A_path.
- A_path = π * (r + h)^2 - π * r^2.
- Expanding (r + h)^2, we get A_path = π * (r^2 + 2rh + h^2) - π * r^2.
Step 5: Simplify further by distributing π to each term within the parentheses.
- A_path = π * r^2 + π * 2rh + π * h^2 - π * r^2.
- The r^2 and -r^2 terms cancel out, leaving us with A_path = π * 2rh + π * h^2.
Step 6: Factor out π from the expression.
- A_path = π * (2rh + h^2).
This gives us the final formula for the area of a circular path of uniform width h surrounding a circular region of radius r as A_path = π * (2rh + h^2).
Therefore, we have proven that the area of the circular path is πh(2r + h).