1. 1 Solids Three-Dimensional Geometry

2. 2 Prisms A prism is a three-dimensional solid with two congruent and parallel polygons called the bases. The lateral faces are rectangular and are perpendicular to the bases. The bases can be any polygon and the name of the prism is based on the name of the bases. For example, the prism shown at right is a right triangular prism. The lateral edges are parallel, congruent and perpendicular to the bases. The surface area of a prism is found by adding the areas of all of its polygonal faces including its bases.

3. 3 Boxes A box (also called a rectangular right prism) is formed by two parallel and congruent polygons called the bases. It has rectangular lateral faces that are perpendicular to the faces. A box has ___ rectangular faces, ___ edges, and ___ 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

4. 4 Surface Area of a Box The surface area of a box is found by adding the areas of its six rectangular faces. We already know how to find the area of a rectangle. (L x W) L W H

5. 5 Surface Area of a Right Prism The surface area of a three-dimensional object is, as the name suggests, the area of its surface. The total area of a right prism = 2 X the area of the base + the area of the lateral sides.

6. Example The surface area is: Area of the base= (8 X 5) = 40 Area of lateral sides = (4 X 5) + (8 X 4) = 20 + 32 = 52 Total area = 2(40 + 52) = 184m² 6 8m 5m 4m

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

8. 8 Pyramids A pyramid is a three-dimensional solid with one polygonal base and with lateral faces that are triangular. There are different kinds of pyramids depending on what shape the base is. To the right is a rectangular pyramid. To find the surface area of a pyramid, add the areas of all of its faces.

9. 9 Cylinders A cylinder is a prism in which the bases are circles. The lateral side is a rectangle and its length is the circumference of the base. The surface area of a cylinder is: h r

10. 10 Cones A cone is like a pyramid but with a circular base instead of a polygonal base. The lateral area of a right circular cone with radius r and slant height a is equal to πra. The surface area of a cone is: A = πr² + πra h r

11. 11 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 surface area of a sphere is: r

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