Volume

Transcription

1 Volume Volume is a measure of how much space a three-dimensional object takes up. Area only looked at surface; we counted how many little flat squares could fit on a surface. Volume measures how many 1 cm cubes will fit into a space. Imagine a cube: This cube is 3 units on an edge. Let's assume that they are centimeters. So, each edge is 3 cm long. The area of one of the faces of this cube is A = L 2 A = (3 cm) 2 = 9 cm 2 But that's just the area of one face of the cube. How much space does it fill? Well, we can imagine smaller cubes of 1 cm 3, joined together to make the larger cube. Let's remove one and see what it looks like. We've taken one single cubic centimeter out of the block. Can you tell how many cubic centimeters are in this block? There are many ways to figure this out. You can clearly see that the top layer of this block has nine cubic centimeters in it. There are three layers, so we can easily see that there are 27 cubic centimeters. This seems like a large number, but it is correct. We don't have to take objects apart, though. We can calculate the volume very easily. Cylinders 1 of 6 11/15/07 4:24 PM

2 You probably think of a cylinder as shaped like a soup can. Certainly, that is one kind of cylinder. However, a cylinder doesn't have to be round. There are many other cylinders of many other shapes. We start with a face. The face is the "end" of the cylinder. A soup can has a round face. But we can have any shape; let's build a cylinder with a hexagonal face. Here's a hexagon, lying flat on a plane: It looks squished because it's lying down; you're looking at it as if it were printed on a piece of paper lying on your desk. Let's "grow" some edges upward from the vertices of this hexagon. And, finally, let's finish it off with another face on the top. This is a hexagonal cylinder. What are the parts of a cylinder? A face. Sides which extend upwards, perpendicular to the face. The sides are all exactly the same length. A second face, identical to the first, at the top of the sides. Is a cube also a cylinder? Calculating the Volume of a Cylinder: 2 of 6 11/15/07 4:24 PM

3 The volume of a cylinder is given by the formula: V = (Area of face) x (height of cylinder) For a soup can, with a round face, the area of the face is given by π r 2 So, for a soup can, the formula for volume is V = π r 2 h where h is the height of the can. Questions: 1. What is the formula for the volume of a cylinder with a triangular face? What is the formula for the area of a triangular face? What is the formula for the volume of a triangular cylinder? 2. What is the formula for the volume of a cylinder with a rectangular face? What is the formula for the area of a rectangular face? 3 of 6 11/15/07 4:24 PM

4 What is the formula for the volume of a rectangular cylinder? Volumes of Cones and Spheres: Cones do not have to be shaped like ice cream cones. We are most familiar with cones that have round faces, but a cone can have any shape for its face. Look at these examples of cones. This cone has a round face. This cone has a square face. 4 of 6 11/15/07 4:24 PM

5 This cone has a triangular face. The formula for the volume of any cone is: V = (1/3) (Area of the face) (Height of the cone) So, how would you write the formula for the volume of a cone with a round face? The volume of a cone with a square face? With a triangular face? With a pentagonal face? Spheres A sphere is a ball. It is the collection of all points exactly the same distance from one center point. The volume of a sphere is given by this formula: V = (4/3) π r 3 5 of 6 11/15/07 4:24 PM

6 "r" is the radius of the sphere, the distance from the center of the sphere to any point on the surface. 6 of 6 11/15/07 4:24 PM