Surface Areas

8.7 Surface Area

Objective

• Apply the surface area formula to various

3-dimensional figures in order to find the area

8.7 Surface Area

A polyhedron is a three-dimensional figure whose surfaces are polygons.

Each polygon is a face of the polyhedron.

An edge is a segment that is formed by the intersection of two faces.

A vertex is a point where three or more edges intersect.

These faces are rectangles.

Each edge is a segment. A vertex is a corner.

8.7 Surface Area

How many faces does this polyhedron have?

How many edges?

How many vertices?

8.7 Surface Area

Platonic Solids

There are five regular polyhedrons, called Platonic Solids (named after Plato).

Tetrahedron Hexahedron Octahedron

Dodecahedron Icosahedron

8.7 Surface Area

A prism is a polyhedron with exactly two congruent, parallel faces, called bases.

Other faces are called lateral faces.

Bases(pentagons)

Lateral faces(rectangles)

How many faces does this polyhedron have?8.7 Surface Area

Rectangular PrismShape of bases?Number of lateral faces?

Pentagonal PrismShape of bases?Number of lateral faces?

Triangular PrismShape of bases?Number of lateral faces?

8.7 Surface Area

An altitude of a prism is a perpendicular segment that joins the planes of the bases.

The height h of the prism is the length of an altitude.

Altitude or height

8.7 Surface Area

The lateral area of a prism is the sum of the areas of the lateral faces.

How many lateral faces?

Area of one lateral face?

Area of all lateral faces?

8.7 Surface Area

The surface area is the sum of the lateral area and the area of the two bases.

What shape is one base?

What is the area of one base?

What is the area of two bases?

What is the surface area? 8.7 Surface Area

Theorem 10-1 Lateral and Surface Areas of a Prism

The lateral area of a right prism is the product of the perimeter of the base and the height.

L.A. = ph

The surface area of a right prism is the sum of the lateral area and the areas of the two bases.

S.A. = L.A. + 2B8.7 Surface Area

Find the lateral area and surface area.

L.A. = ph

S.A. = L.A. + 2B

8.7 Surface Area

Find the lateral area and surface area.

L.A. = ph

S.A. = L.A. + 2B

8.7 Surface Area

Find the lateral area and surface area.

L.A. = ph

S.A. = L.A. + 2B

8.7 Surface Area

A cylinder has two congruent parallel bases.

The bases of a cylinder are circles.

An altitude of a cylinder is a perpendicular segment that joins the planes of the bases.

The height h of a cylinder is the length of an altitude.

8.7 Surface Area

How do you find the lateral area?

Think about a soup can.

What happens if you “unroll” the label?

To find the lateral area, find the area of the resulting rectangle: 2πrh

Surface Area = Lateral Area + Area of Bases

8.7 Surface Area

Surface Area = Lateral Area + Area of Bases

SA = L.A. + 2B

SA = 2πrh + 2πr2

8.7 Surface Area

Find the lateral area and the surface area:

8.7 Surface Area

Some oatmeal comes in a cardboard box in the shape of a cylinder. The entire box, top and bottom, is made of cardboard. If the diameter of the bottom of the box is 6 inches and the height is 11 inches, find the lateral area and the surface area of the cardboard.

Now suppose the top is made of plastic, not cardboard. Find the lateral area and the surface area of the cardboard.

8.7 Surface Area

A soup can is made of metal and covered with a paper label. If the radius of the bottom of the soup can is 4 inches and the height is 5 inches, find the following:

The surface area of metal used in an unopened can.

The area of paper used in the label.

The area of metal remaining once the lid has been thrown away.

8.7 Surface Area

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8.7 Surface Area