Surface Area of a Cylinder Formula
The surface area of a cylinder is the total area of its curved surface and the two circular bases. Calculating the surface area requires knowing the dimensions of the cylinder, such as the radius or diameter of the base and the height. Here are all the possible formulas for calculating the surface area of a cylinder, with detailed explanations and examples for each method.
Formulas for the Surface Area of a Cylinder
Using Radius and Height:
If the radius \( r \) of the base and the height \( h \) of the cylinder are known,
The surface area \( S \) can be calculated with the formula:
\( S = 2\pi r (r + h) \)
Using Diameter and Height:
If the diameter \( d \) of the base and the height \( h \) are known,
The surface area \( S \) can be calculated as:
\( S = \pi d (d/2 + h) \)
Using Circumference and Height:
If the circumference \( C \) of the base and the height \( h \) are known,
The surface area \( S \) can be calculated as:
\( S = C \left( \dfrac{C}{2\pi} + h \right) \)
In these formulas:
- \( r \) is the radius of the base of the cylinder
- \( d \) is the diameter of the base of the cylinder, where \( d = 2r \)
- \( C \) is the circumference of the base, where \( C = 2\pi r \)
- \( h \) is the height of the cylinder
- \( \pi \) (Pi) is approximately equal to 3.14159
Detailed Explanation of Each Formula
1. Formula for Surface Area Using Radius and Height
The formula \( S = 2\pi r (r + h) \) calculates the surface area by adding the area of the two circular bases \( 2\pi r^2 \) to the area of the curved surface \( 2\pi r h \). This formula is useful when both the radius and height of the cylinder are known.
Example 1: Calculating Surface Area with Radius and Height
Problem: Find the surface area of a cylinder with a radius of \( r = 5 \, \text{cm} \) and a height of \( h = 10 \, \text{cm} \).
Solution:
- Write down the formula: \( S = 2\pi r (r + h) \).
- Substitute \( r = 5 \) and \( h = 10 \): \( S = 2\pi \times 5 \times (5 + 10) \).
- Calculate inside the parentheses: \( S = 2\pi \times 5 \times 15 \).
- Multiply: \( S = 150\pi \approx 471.24 \, \text{cm}^2 \) (using \( \pi \approx 3.14159 \)).
The surface area of the cylinder is approximately \( 471.24 \, \text{cm}^2 \).
2. Formula for Surface Area Using Diameter and Height
The formula \( S = \pi d (d/2 + h) \) calculates the surface area by using the diameter of the base. Since \( d = 2r \), this formula is equivalent to the radius-based formula but is useful if only the diameter is given.
Example 2: Calculating Surface Area with Diameter and Height
Problem: A cylinder has a diameter of \( d = 10 \, \text{cm} \) and a height of \( h = 15 \, \text{cm} \). Find its surface area.
Solution:
- Write down the formula: \( S = \pi d (d/2 + h) \).
- Substitute \( d = 10 \) and \( h = 15 \): \( S = \pi \times 10 \times (10/2 + 15) \).
- Calculate \( d/2 \): \( S = \pi \times 10 \times (5 + 15) \).
- Multiply: \( S = \pi \times 10 \times 20 = 200\pi \approx 628.32 \, \text{cm}^2 \).
The surface area of the cylinder is approximately \( 628.32 \, \text{cm}^2 \).
3. Formula for Surface Area Using Circumference and Height
The formula \( S = C \left( \dfrac{C}{2\pi} + h \right) \) calculates the surface area using the circumference of the circular base. This approach is convenient if the circumference and height are known, as the formula uses the relationship \( C = 2\pi r \).
Example 3: Calculating Surface Area with Circumference and Height
Problem: A cylinder has a circumference of \( C = 20 \, \text{cm} \) and a height of \( h = 12 \, \text{cm} \). Find its surface area.
Solution:
- Write down the formula: \( S = C \left( \dfrac{C}{2\pi} + h \right) \).
- Substitute \( C = 20 \) and \( h = 12 \): \( S = 20 \left( \dfrac{20}{2\pi} + 12 \right) \).
- Approximate \( \pi \approx 3.14159 \): \( S = 20 \left( \dfrac{20}{6.28318} + 12 \right) \).
- Divide: \( S = 20 \left(3.183 + 12\right) \).
- Multiply: \( S = 20 \times 15.183 = 303.66 \, \text{cm}^2 \).
The surface area of the cylinder is approximately \( 303.66 \, \text{cm}^2 \).
These examples show how to calculate the surface area of a cylinder using different known measurements, whether it’s the radius and height, diameter and height, or circumference and height.