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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.

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Lesson 9 l.jpg

Lesson 9

Three-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.


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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.


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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.


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Example

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

  • The volume is

  • The surface area is


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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


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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:


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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:


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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:


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