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Three-Dimensional Figure A three-dimensional figure is a shape whose points do not all lie in the same plane.

Polyhedron A polyhedron is a closed three-dimensional figure with flat faces that are polygons. Polyhedra is the plural form.

Face A face is any of the polygons making up a polyhedron.

Edge An edge is a line of intersection of two faces of a polyhedron.

Vertex A vertex is any point where three or more faces of a polyhedron intersect (plural form: vertices).

Prism A prism is a polyhedron with two congruent parallel faces.

Bases Bases are the congruent parallel faces of a prism.

Lateral Faces Lateral faces are all the faces of a prism or pyramid except the bases.

Right Prism A right prism is a prism that has lateral edges that are perpendicular to the plane of the base.

Oblique Prism An oblique prism is a prism with lateral edges that are not perpendicular to the plane of the base.

Rectangular Prism A rectangular prism is a prism that has two parallel congruent rectangular bases. bases

Triangular Prism A triangular prism is a prism that has two parallel congruent triangular bases. bases

Cube A cube is a special type of rectangular prism with six square congruent faces. bases

Lateral Surface Area Lateral surface area (L) is the sum of the areas of the lateral faces of a prism or pyramid.

Surface Area Surface area (S) is the sum of the areas of all its surfaces.

Area of bottom: 9 × 4 = 36 in. 2 Area of right side: 4 × 3 = 12 in. 2 Area of front: 9 × 3 = 27 in. 2 9 in. 3 in. 4 in.

S = area of both bases + area of both sides + area of front and back S = 2(36) + 2(12) + 2(27) = 72 + 24 + 54 = 150 in. 2 9 in. 3 in. 4 in.

9 in. 3 in. 4 in. base 3 in. 4 in. 3 in. 9 in. base 4 in.

The area of the long rectangles comprises the four lateral faces of the prism and, therefore, the lateral surface area (L) of the prism. The rectangle’s dimensions are the perimeter of the base (p) and the height of the prism (H). So, L = pH = 26 × 3 = 78 in. 2.

The total surface area (S) can be determined by adding the lateral surface area (L) and the combined area of the two bases (2B). 2B = 2[9(4)] = 72 in. 2 and S = L + 2B = 78 + 72 = 150 in. 2.

Formula: Lateral Surface Area of a Prism L = pHThe lateral surface area of a prism (L) is equal to the product of the perimeter of the base (p) and the height of the prism (H).

Formula: Surface Area of a Prism S = L + 2BThe surface area of a prism (S) is equal to the sum of the lateral surface area (L) and the combined area of the two bases (B).

Example 1 Find the lateral surface area of the prism. L = pH H = 8 cm 5 cm 8 cm 3 cm 4 cm p = 3 + 4 + 5 = 12 L = 12(8) = 96 cm 2

= 108 cm 2 S = L + 2B = 96 + 2(6) = 96 + 12 5 cm 8 cm 3 cm 4 cm

Find the surface area of a box with w = 5 units, l = 8 units, and h = 6 units. 236 units 2 Example

Circular Cylinder A circular cylinder is a three- dimensional figure with two parallel bases that are congruent circles. The lateral surface of a circular cylinder is the curved surface.

bases

Formula: Lateral Surface Area of a Cylinder L = cHThe lateral surface = 2  rH area of a cylinder (L) is equal to the product of the circumference of the base (c) and the height of the cylinder (H). Substitute for c.

Formula: Surface Area of a Cylinder S = L + 2BThe surface area = 2  rH + 2  r 2 of a cylinder (S) is equal to the sum of the lateral surface area (L) and the combined area of the two circular bases (B). Substitute for L and B.

Find the surface area of the cylinder. L = 2  rH = 2  (3)(9) 9 in. = 54  B =  r 2 =  (3 2 ) = 9  3 in. r r Example 2

≈ 226.1 in. 2 S = L + 2B = 54  + 2(9  ) = 72  = 72(3.14) 9 in. 3 in. r r

What is the surface area of a right cylinder with r = 3 units and h = 5 units? 150.72 units 2 Example

What happens to the lateral surface area of a right cylinder if the radius is doubled? Example It doubles.

What happens to the lateral surface area if the height is doubled? Example It doubles.

Draw a conclusion about the total surface area if both the height and radius are doubled. Example It quadruples.

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