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

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

PolyhedronPolyhedron

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

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

FaceFace

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

EdgeEdge

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

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

VertexVertex

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

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

PrismPrism

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

BasesBases

Bases are the congruent parallel faces of a prism.Bases are the congruent parallel faces of a prism.

Lateral FacesLateral Faces

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

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

Right PrismRight Prism

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

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

Oblique PrismOblique Prism

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

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

Rectangular PrismRectangular Prism

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

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

basesbases

Triangular PrismTriangular Prism

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

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

basesbases

CubeCube

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

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

basesbases

Lateral Surface AreaLateral Surface Area

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

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

Surface AreaSurface Area

Surface area (S) is the sum of the areas of all its surfaces.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.9 in.

3 in.3 in.4 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.9 in.

3 in.3 in.4 in.4 in.

9 in.9 in.

3 in.3 in.4 in.4 in.

basebase

3 in.3 in.

4 in.4 in.

4 in.4 in.

4 in.4 in.

3 in.3 in.

9 in.9 in.

9 in.9 in.

9 in.9 in.

basebase

4 in.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 andS = L + 2B = 78 + 72 = 150 in.2.

Formula: LateralSurface Area of a Prism

Formula: LateralSurface Area of a PrismL = pH The 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).

L = pH The 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

Formula:Surface Area of a PrismS = L + 2B The 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).

S = L + 2B The 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 1Example 1Find the lateral surface area of the prism.Find the lateral surface area of the prism.

L = pHL = pHH = 8 cmH = 8 cm

5 cm5 cm

8 cm8 cm

3 cm3 cm4 cm4 cm

p = 3 + 4 + 5 = 12p = 3 + 4 + 5 = 12L = 12(8)L = 12(8)

= 96 cm2= 96 cm2

= 108 cm2= 108 cm2

S = L + 2BS = L + 2B= 96 + 2(6)= 96 + 2(6)= 96 + 12= 96 + 12

5 cm5 cm

8 cm8 cm

3 cm3 cm4 cm4 cm

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

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

236 units2236 units2

ExampleExample

Circular CylinderCircular Cylinder

A circular cylinder is a three-dimensional figure with two parallel bases that are congruent circles.

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.

The lateral surface of a circular cylinder is the curved surface.

basesbases

Formula: Lateral Surface Area of a Cylinder

Formula: Lateral Surface Area of a Cylinder

L = cH The lateral surface = 2prH 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.

L = cH The lateral surface = 2prH 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

Formula: Surface Area of a Cylinder

S = L + 2B The surface area= 2prH + 2pr2 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.

S = L + 2B The surface area= 2prH + 2pr2 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.Find the surface area of the cylinder.

L = 2prHL = 2prH= 2p(3)(9)= 2p(3)(9)

9 in.9 in.= 54p= 54pB = pr2B = pr2

= p(32)= p(32)= 9p= 9p

3 in.3 in.

rr

Example 2Example 2

≈ 226.1 in.2≈ 226.1 in.2

S = L + 2BS = L + 2B= 54p + 2(9p)= 54p + 2(9p)= 72p= 72p= 72(3.14)= 72(3.14)

9 in.9 in.

3 in.3 in.

rr

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

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

150.72 units2150.72 units2

ExampleExample

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

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

ExampleExample

It doubles.It doubles.

What happens to the lateral surface area if the height is doubled?

What happens to the lateral surface area if the height is doubled?

ExampleExample

It doubles.It doubles.

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

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

ExampleExample

It quadruples.It quadruples.