Let's consider a common object, a cylindrical bottle as an example. To calculate its surface area and volume, we'll follow these steps:

Measurements

Let's assume the following dimensions:

• Height ($h$) of the bottle = 20 cm
• Radius ($r$) of the bottle = 5 cm

1. Surface Area of the Cylinder

The surface area of a cylinder consists of:

• The lateral surface area, which is the side part of the cylinder.
• The top and bottom circular areas.

The formula for the lateral surface area is: $A_{\text{lateral}} = 2 \pi r h$ The formula for the area of the top and bottom (two circles) is: $A_{\text{circles}} = 2 \pi r^2$ Thus, the total surface area is: $A_{\text{total}} = A_{\text{lateral}} + A_{\text{circles}} = 2 \pi r h + 2 \pi r^2 = 2 \pi r (h + r)$ Substitute the values: $A_{\text{total}} = 2 \pi \times 5 \times (20 + 5) = 2 \pi \times 5 \times 25 = 250 \pi \, \text{cm}^2$ Approximating $\pi \approx 3.1416$: $A_{\text{total}} = 250 \times 3.1416 \approx 785.4 \, \text{cm}^2$

2. Volume of the Cylinder

The formula for the volume of a cylinder is: $V = \pi r^2 h$ Substitute the values: $V = \pi \times 5^2 \times 20 = \pi \times 25 \times 20 = 500 \pi \, \text{cm}^3$ Approximating $\pi \approx 3.1416$: $V = 500 \times 3.1416 \approx 1570.8 \, \text{cm}^3$

Conclusion:

• Surface area ≈ 785.4 cm²
• Volume ≈ 1570.8 cm³

Comparison with Actual Measurements

You can measure your cylindrical object using a ruler or measuring tape and compare the actual dimensions with the results from these formulas. Slight variations can occur due to measurement rounding or the accuracy of your tools.

Do you have an object in mind, or should I help with another example?

Questions to explore:

1. How would the surface area change if the radius were doubled?
2. What happens to the volume if the height is halved but the radius remains the same?
3. Can you find the surface area of an open cylinder (no top)?
4. How does the surface area formula differ for a rectangular box?
5. What is the relationship between volume and surface area for different shapes?

Tip:

Always double-check measurements using consistent units, such as cm or inches, for accuracy in real-life applications!