A cylinder is a three-dimensional shape formed by extruding a circle vertically. It has three key components: a circular base with radius r, and a height h that extends perpendicular to the base.
Volume represents the amount of space inside a three-dimensional shape. For cylinders, we can think of volume as the base area multiplied by height. Each horizontal slice through the cylinder has the same circular cross-section, so we're essentially stacking identical circular layers.
Let's derive the cylinder volume formula step by step. First, we find the area of the circular base using A equals pi r squared. Then, we multiply this base area by the height h to get the volume. This gives us the final formula: V equals pi r squared h.
Let's work through a concrete example. We have a cylinder with radius 3 units and height 8 units. Using our formula V equals pi r squared h, we substitute the values: V equals pi times 3 squared times 8, which equals pi times 9 times 8, giving us 72 pi cubic units, or approximately 226.19 cubic units.
Let's compare two cylinders to see how radius and height affect volume. Cylinder A has radius 2 and height 3, giving volume 12 pi. Cylinder B has radius 3 and height 2, giving volume 18 pi. Notice that cylinder B has larger volume even though it's shorter, because radius is squared in the formula, making it more impactful than height.