Assignment

• P. 806-9: 2-20 even, 21, 24, 25, 28, 30

• P. 814-7: 2, 3-21 odd, 22-25, 30

• Challenge Problems: 3-5, 8, 9

In Glorious 3-D!

Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.

Polyhedron

A solid formed by polygons that enclose a single region of space is called a polyhedron.

Separate your Geosolids into 2 groups: Polyhedra and others.

Parts of Polyhedrons

• Polygonal region = face• Intersection of 2 faces = edge• Intersection of 3+ edges = vertex

face vertexedge

Warm-Up

Separate your Geosolid polyhedra into two groups where each of the groups have similar characteristics.

What are the names of these groups?

1. Prisms2. Pyramids

Polyhedra:

12.2 & 12.3: Surface Area of Prisms, Cylinders, Pyramids, and Cones

Objectives:

1. To find and use formulas for the lateral and total surface area of prisms, cylinders, pyramids, and cones

Prism

A polyhedron is a prism iff it has two congruent parallel bases and its lateral faces are parallelograms.

Classification of Prisms

Prisms are classified by their bases.

Right & Oblique Prisms

Prisms can be right or oblique. What differentiates the two?

Right & Oblique Prisms

In a right prism, the lateral edges are perpendicular to the base.

Pyramid

A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex.

Classification of Pyramids

Pyramids are also classified by their bases.

Pyramid

A regular pyramid is one whose base is a regular polygon.

Pyramid

A regular pyramid is one whose base is a regular polygon.

• The slant height is the height of one of the congruent lateral faces.

Solids of Revolution

The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution.

Cylinder

A cylinder is a 3-D figure with two congruent and parallel circular bases.

• Radius = radius of base

Cone

A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base.

• Altitude = perpendicular segment connecting vertex to the plane containing the base (length = height)

Cone

A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base.

• Slant height = segment connecting vertex to the circular edge of the base

Right vs. Oblique

What is the difference between a right and an oblique cone?

Right vs. Oblique

In a right cone, the segment connecting the vertex to the center of the base is perpendicular to the base.

Nets

Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.

An unfolded pizza box is a net!

Nets

Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.

Activity: Red, Rubbery Nets

Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?

Activity: Red, Rubbery Nets

Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?

A sphere doesn’t have a true net; it can only be approximated.

Exercise 1

There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net?

Surface Area

The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid.

• Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it.

Lateral Surface Area

The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid.

• Think of the lateral surface area as the size of a label that you could put on the figure.

Exercise 2

What solid corresponds to the net below?

How could you find the lateral and total surface area?

Exercise 3

Draw a net for the rectangular prism below.

To find the lateral surface area, you could:

• Add up the areas of the lateral rectangles

A B C D

Exercise 3

Draw a net for the rectangular prism below.

To find the lateral surface area, you could:

• Find the area of the lateral surface as one, big rectangle

Hei

gh

t o

f P

rism

Perimeter of the Base

Exercise 3

Draw a net for the rectangular prism below.

To find the total surface area, you could:

• Find the lateral surface area then add the two bases

Hei

gh

t o

f P

rism

Perimeter of the Base

Surface Area of a Prism

Lateral Surface Area of a Prism:

• P = perimeter of the base• h = height of the prism

Total Surface Area of a Prism:

• B = area of the base

S Ph 2S Ph B

Exercise 4

Find the lateral and total surface area.

Exercise 5

Draw a net for the cylinder.

Notice that the lateral surface of a cylinder is also a rectangle. Its height is the height of the cylinder, and the base is the circumference of the base.

Exercise 6

Write formulas for the lateral and total surface area of a cylinder.

Surface Area of a Cylinder

Lateral Surface Area of a Cylinder:

• C = circumference of base• r = radius of base• h = height of the cylinder

Total Surface Area of a Cylinder:

S Ch2S Ph B

2S rh

S Ph22 2S rh r

2S r h r

Exercise 7

The net can be folded to form a cylinder.

What is the approximate lateral and total surface area of the cylinder?

Height vs. Slant Height

By convention, h represents height and l represents slant height.

Height vs. Slant Height

By convention, h represents height and l represents slant height.

Exercise 8

Draw a net for the square pyramid below.

To find the lateral surface area:

• Find the area of one triangle, then multiply by 4

12 40 25A

124 40 25S

12 4 40 25S

Exercise 8

Draw a net for the square pyramid below.

To find the lateral surface area:

• Find the area of one triangle, then multiply by 4

12 40 25A

124 40 25S

12 4 40 25S

12 4S s l

12S Pl

Exercise 8

Draw a net for the square pyramid below.

To find the total surface area:

• Just add the area of the base to the lateral area

12 40 25A

124 40 25S

12 4 40 25S

12 4S s l

12S Pl12S Pl B

Surface Area of a Pyramid

Lateral Surface Area of a Pyramid:

• P = perimeter of the base• l = slant height of the

pyramid

Total Surface Area of a Prism:

• B = area of the base

12S Pl 1

2S Pl B

Exercise 9

Find the lateral and total surface area.

Exercise 10

You may have realized that the formula for the lateral area for a prism and a cylinder are basically the same. The same is true for the formulas for a pyramid and a cone. Derive a formula for the lateral area of a cone.

12S Pl

Lateral area of a Pyramid: Lateral area of a Cone:12S Cl

12 2S r l

S rl

Surface Area of a Cone

Lateral Surface Area of a Cone:

• r = radius of the base• l = slant height of the

cone

Total Surface Area of a Cone:

S rl 2S rl r

S r l r

Exercise 11

A traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. To the nearest tenth of a square inch, find the approximate lateral area of the traffic cone.

Tons of Formulas?

Really there’s just two formulas, one for prisms/cylinders and one for pyramids/cones.

Assignment

• P. 806-9: 2-20 even, 21, 24, 25, 28, 30

• P. 814-7: 2, 3-21 odd, 22-25, 30

• Challenge Problems: 3-5, 8, 9