Lesson 9

1. Lesson 9 Three-Dimensional Geometry

2. Planes • A plane is a flat surface (think tabletop) that extends forever in all directions. • It is a two-dimensional figure. • Three non-collinear points determine a plane. • So far, all of the geometry we’ve done in these lessons took place in a plane. • But objects in the real world are three-dimensional, so we will have to leave the plane and talk about objects like spheres, boxes, cones, and cylinders.

3. H W L Boxes • A box (also called a right parallelepiped) is just what the name box suggests. One is shown to the right. • A box has six rectangular faces, twelve edges, and eight vertices. • A box has a length, width, and height (or base, height, and depth). • These three dimensions are marked in the figure.

4. Volume and Surface Area • The volume of a three-dimensional object measures the amount of “space” the object takes up. • Volume can be thought of as a capacity and units for volume include cubic centimeters cubic yards, and gallons. • The surface area of a three-dimensional object is, as the name suggests, the area of its surface.

5. H W L Volume and Surface Area of a Box • The volume of a box is found by multiplying its three dimensions together: • The surface area of a box is found by adding the areas of its six rectangular faces. Since we already know how to find the area of a rectangle, no formula is necessary.

6. 4 5 8 Example • Find the volume and surface area of the box shown. • The volume is • The surface area is

7. Cubes • A cube is a box with three equal dimensions (length = width = height). • Since a cube is a box, the same formulas for volume and surface area hold. • If s denotes the length of an edge of a cube, then its volume is and its surface area is

8. Prisms • A prism is a three-dimensional solid with two congruent bases that lie in parallel planes, one directly above the other, and with edges connecting the corresponding vertices of the bases. • The bases can be any shape and the name of the prism is based on the name of the bases. • For example, the prism shown at right is a triangular prism. • The volume of a prism is found by multiplying the area of its base by its height. • The surface area of a prism is found by adding the areas of all of its polygonal faces including its bases.

9. h r Cylinders • A cylinder is a prism in which the bases are circles. • The volume of a cylinder is the area of its base times its height: • The surface area of a cylinder is:

10. Pyramids • A pyramid is a three-dimensional solid with one polygonal base and with line segments connecting the vertices of the base to a single point somewhere above the base. • There are different kinds of pyramids depending on what shape the base is. To the right is a rectangular pyramid. • To find the volume of a pyramid, multiply one-third the area of its base by its height. • To find the surface area of a pyramid, add the areas of all of its faces.

11. h r Cones • A cone is like a pyramid but with a circular base instead of a polygonal base. • The volume of a cone is one-third the area of its base times its height: • The surface area of a cone is:

12. r Spheres • Sphere is the mathematical word for “ball.” It is the set of all points in space a fixed distance from a given point called the center of the sphere. • A sphere has a radius and diameter, just like a circle does. • The volume of a sphere is: • The surface area of a sphere is: