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Lesson 9Three-Dimensional Geometry

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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.

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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.

LW

H

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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.

3( ),cm

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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.

LW

H

V L W H

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Example

• Find the volume and surface area of the box shown.

• The volume is

• The surface area is

85

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8 5 4 40 4 160

8 5 8 5 5 4 5 4 8 4 8 4

40 40 20 20 32 32

184

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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

3s26 .s

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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.

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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

2V r h

22 2A r rh

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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.

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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

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3V r h

2 2 2A r r r h

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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

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3V r

24A r