Welcome to our lesson on the surface area of a cylinder. A cylinder is a three-dimensional shape with two circular bases and one curved lateral surface. The surface area is the total area of all these surfaces combined.
Now let's calculate the area of the circular bases. Each base is a circle with radius r. The area of a circle is pi r squared. Since a cylinder has two identical bases, the total area of both bases is two pi r squared.
Now let's find the lateral surface area. If we imagine unrolling the curved side of the cylinder, it forms a rectangle. The height of this rectangle equals the cylinder's height h, and the width equals the circumference of the base, which is two pi r. Therefore, the lateral surface area is two pi r h.
Now we combine everything to get the total surface area formula. We add the area of the two bases, which is two pi r squared, plus the lateral surface area, which is two pi r h. This gives us the total surface area equals two pi r squared plus two pi r h. We can factor this as two pi r times the quantity r plus h.
To summarize what we've learned about cylinder surface area: A cylinder consists of two circular bases and one lateral surface. The total base area is two pi r squared. The lateral surface area is two pi r h. Combining these gives us the total surface area formula: two pi r times the quantity r plus h. This formula is essential for solving real-world problems involving cylindrical objects.