Welcome! Today we'll learn how to calculate the surface area of a cylinder. A cylinder is a three-dimensional shape with two circular bases and one curved lateral surface. To find the total surface area, we need to calculate the area of all three surfaces and add them together. The cylinder has radius r and height h, which are the key measurements we'll use in our formula.
Now let's calculate the area of the circular bases. Each base is a circle with radius r, so its area is π r squared. Since a cylinder has two identical bases - one at the top and one at the bottom - we need to multiply by 2. Therefore, the total area of both bases is 2 π r squared. This is the first part of our surface area formula.
Now let's find the lateral surface area - that's the curved side of the cylinder. The key insight is to imagine unrolling this curved surface into a flat rectangle. When we do this, the width of the rectangle equals the circumference of the circular base, which is 2 π r. The height of the rectangle is the same as the cylinder's height, h. Therefore, the lateral surface area equals width times height, which gives us 2 π r h.
Now we combine everything together to get the complete formula. The total surface area equals the sum of the base areas plus the lateral area. That's 2 π r squared plus 2 π r h. This gives us our final formula: Surface Area equals 2 π r squared plus 2 π r h. This formula works for any cylinder - just substitute the values for radius r and height h, and you can calculate the total surface area.
Let's work through a concrete example to see how this formula works in practice. Suppose we have a cylinder with radius 3 units and height 5 units. Using our formula, Surface Area equals 2 π r squared plus 2 π r h. Substituting our values: 2 π times 3 squared plus 2 π times 3 times 5. This gives us 2 π times 9 plus 2 π times 15, which equals 18 π plus 30 π, for a total of 48 π. Converting to decimal form, this is approximately 150.8 square units. Now you know how to calculate the surface area of any cylinder!