A cylinder is a three-dimensional geometric shape with two identical, parallel circular bases connected by a curved side. The surface area refers to the total area encompassing all its outer surfaces. Calculating this total area helps in understanding the amount of material needed to cover or construct the object.
Cylinder’s Geometric Parts
A cylinder is composed of distinct two-dimensional shapes. It has two flat, circular bases, which are congruent and parallel to each other. Connecting these two bases is a single curved surface. If one were to “unroll” this curved surface, it would form a perfect rectangle.
The dimensions that define a cylinder are its radius and its height. The radius (r) is the distance from the center of either circular base to its outer edge. The height (h) represents the perpendicular distance between the two parallel circular bases.
The Surface Area Formula
To calculate the total surface area (SA) of a cylinder, one must consider the area of its individual components: the two circular bases and the curved rectangular surface. The area of a single circular base is given by the formula πr², where ‘π’ (pi) is a mathematical constant approximately equal to 3.14159, and ‘r’ is the radius. Since a cylinder has two such bases, their combined area is 2πr².
One side of this rectangle corresponds to the height (h) of the cylinder, while the other side corresponds to the circumference of the circular base, which is 2πr. Therefore, the area of this curved surface is calculated as 2πrh. Combining these components, the total surface area of a cylinder is expressed by the formula: SA = 2πr² + 2πrh. This formula can also be factored as SA = 2πr(r + h).
Step-by-Step Calculation Example
Calculating the surface area of a cylinder involves applying the formula with specific measurements for the radius and height. Consider a cylinder with a radius (r) of 3 centimeters and a height (h) of 10 centimeters.
Next, substitute these values into the total surface area formula, SA = 2πr² + 2πrh. For the two circular bases, the area is 2 × π × (3 cm)² = 2 × π × 9 cm² = 18π cm². For the curved surface, the area is 2 × π × 3 cm × 10 cm = 60π cm².
Finally, add the areas of the bases and the curved surface to find the total surface area. SA = 18π cm² + 60π cm² = 78π cm². If an approximate numerical value is needed, using π ≈ 3.14159 yields SA ≈ 78 × 3.14159 cm² ≈ 245.04 cm². Always include the appropriate square units in the final answer.