surface area of cylinder in terms of pi
Understand the Problem
The question is asking for the formula to calculate the surface area of a cylinder, expressed in terms of pi. The surface area of a cylinder consists of the areas of the two circular bases and the rectangular side (lateral surface area). The formula is typically given as 2πrh + 2πr², where r is the radius and h is the height of the cylinder.
Answer
The formula for the surface area of a cylinder is $A = 2 \pi r (r + h)$.
Answer for screen readers
The formula for the surface area of a cylinder is:
$$ A = 2 \pi r (r + h) $$
Steps to Solve
- Identify the Components of the Surface Area
The surface area of a cylinder consists of two parts: the area of the circular bases and the lateral (side) area.
- Calculate the Area of the Circular Bases
The area of one circular base is given by the formula $A = \pi r^2$. Since there are two bases, the total area for both bases is:
$$ A_{\text{bases}} = 2 \times \pi r^2 $$
- Calculate the Lateral Surface Area
The lateral surface area can be visualized as a rectangle that wraps around the cylinder. The height of the rectangle is the height of the cylinder ($h$), and the width is the circumference of the base ($2\pi r$). The formula for the lateral surface area is:
$$ A_{\text{lateral}} = 2 \pi r h $$
- Combine the Areas to Find Total Surface Area
To find the total surface area ($A_{\text{total}}$), we add the areas of the bases and the lateral surface area together:
$$ A_{\text{total}} = A_{\text{bases}} + A_{\text{lateral}} $$
Substituting the expressions we found:
$$ A_{\text{total}} = 2 \pi r^2 + 2 \pi r h $$
- Final Simplified Formula
We can factor out $2 \pi$ from the total surface area:
$$ A_{\text{total}} = 2 \pi r (r + h) $$
The formula for the surface area of a cylinder is:
$$ A = 2 \pi r (r + h) $$
More Information
The surface area formula for a cylinder is important in various fields including engineering, architecture, and manufacturing. It helps calculate the amount of material needed to cover the cylinder.
Tips
- Forgetting to include both circular bases when calculating the total surface area.
- Misidentifying the dimensions; remember that the radius ($r$) is always the distance from the center to the edge of the base, while the height ($h$) is the vertical distance.
