Three-Dimensional Geometry Lesson 9 Three-Dimensional Geometry

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.

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. L W H

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.

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. L W H

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

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

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.

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: h r

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.

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: h 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: r