A hollow cylinder is a three-dimensional shape. It consists of two parallel circular bases and a curved lateral surface. The area of the hollow cylinder is the total surface area of the cylinder, including both the inside and outside surfaces.
Area of a Hollow Cylinder
The area of a hollow cylinder refers to the total surface area of the cylinder with its both the inside and outside surfaces. To calculate the area of a hollow cylinder, you need to know its height, inner radius, and outer radius.
The formula to calculate
the area of a hollow cylinder is: A = 2πh(r1 + r2)
where A is the area of the hollow cylinder, h is the height of the cylinder, r1 is the inner radius, and r2 is the outer radius.
volume of a hollow cylinder is: V = πh( r2² – r1² )
Area Percentage = (A/V) x 100%
Examples of Area of a Hollow Cylinder
Example 1: Find the area percentage of a hollow cylinder with a height of 6 cm, an inner radius of 3 cm, and an outer radius of 5 cm.
Solution: Using the formula for the volume of a hollow cylinder, we can find the volume:
V = π x 6 x (5^2 – 3^2) = 150π cm^3
Using the formula for the area of a hollow cylinder, we can find the area:
A = 2π x 6 x (3 + 5) = 96π cm^2
Therefore, the area percentage of the hollow cylinder would be:
Area Percentage = (96π/150π) x 100% = 64%
So, the area of the hollow cylinder represents 64% of its total volume.
Example 2: A hollow cylinder has a volume of 1000 cm^3 and a height of 10 cm. The inner radius is 3 cm. What is the outer radius, and what is the area percentage of the cylinder?
Solution: Using the formula for the volume of a hollow cylinder, we can find the outer radius:
1000 = π x 10 x (r2^2 – 3^2) r2^2 – 9 = 100/π r2^2 = 109/π r2 ≈ 5.25 cm
Using the formula for the area of a hollow cylinder, we can find the area:
A = 2π x 10 x (3 + 5.25) ≈ 347.97 cm^2
Therefore, the area percentage of the hollow cylinder would be:
Area Percentage = (347.97/1000) x 100% ≈ 34.8%
So, the area of the hollow cylinder represents about 34.8% of its total volume.
Example 3: A hollow cylinder has an area percentage of 50% and a height of 8 cm. The inner radius is 2 cm. What is the outer radius, and what is the volume of the cylinder?
Solution: Using the formula for the volume of a hollow cylinder and the given area percentage, we can find the outer radius:
50/100 = (2π x 8 x (2 + r2))/(π x 8 x (r2^2 – 2^2)) r2^2 – 4 = 4(2 + r2)/(2 + r2) r2^2 – 4 = 4 – 16/(2 + r2) r2^2 + 16/(2 + r2) – 8 = 0 (r2 + 4)(r2 – 2) = 0 r2 = 2 or -4 (discard)
Therefore, the outer radius is 2 cm.
Using the formula for the volume of a hollow cylinder, we can find the volume:
V = π x 8 x (2^2 – 2^2) = 0 cm^3
Since the volume is zero, this means that the area percentage cannot be 50% for a cylinder with these dimensions. Therefore, there must be an error in the given information.