Chapter 12Surface Area and Volume

• Identify the parts of prisms, pyramids, cylinders and cones.

• Find the lateral areas, total areas and volumes of right prisms, regular pyramids, cones and cylinders.

12-1: Prisms

Objectives

• Learn and apply the area and volume formula for a prism.

Anatomy of a Prism

BaseBase Edge

Lateral edge

Lateral Face

Altitude

Oblique vs. Right

The lateral edges of a right prism are perpendicular to the base.

The name of a prism comes from its base.

Names of Prisms

Rectangular Oblique Prism Triangular Right Prism

Surface Area

The total surface area of a prism is the sum of all the lateral faces and the two bases.

Lateral Area

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

LA = p x h

Volume

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

V = B x h

Total Area

• TA=LA+2B

White Board Practice

A right triangular prism has base edges of 5, 12, and 13. It has a volume of 450. Find the height of the prism.

White Board Practice

The base of a triangular prism is an isosceles right triangle with legs of 3 cm. The height of the prism is 10 cm. Find the lateral area, total area, and volume of the prism

12-2:Pyramids

Objectives

• Learn and apply the area and volume formula for a pyramid.

Anatomy of a Pyramid

Base

Base Edge

Lateral edge

Lateral Face

Altitude

Slant Height

Vertex

Rotate the pyramid…..this is hard

Oblique vs. Right

The altitude of a right pyramid passes through the center of the base.

The name of a pyramid comes from its base.

Names of Pyramids

Triangular Oblique Pyramid Square Right Pyramid

Lateral Area

The lateral area of a right pyramid is one half the product of the perimeter of the base and the slant height.

p

l

1

2LA pl

Total AreaThe total surface area of a pyramid is the sum of all the lateral

faces and the base or it is the total are plus one base area

TA = LA + B

Volume

The volume of a right pyramid is one third the product of the area of the base and the height.

h

B

1

3V Bh

White Board Practice

• A regular square pyramid had base edge 6m and lateral edge 5m.

a) Find the length of a slant height

b) Find the Lateral Area

c) Find the Base Area

d) Find the Total Area

e) Find the Length of the altitude

f) Find the Volume

12-3: Cylinders and Cones

Objectives

• Find the area and volume for a cylinder and a cone.

Cylinder

A cylinder is a prism with a circular base.

r

h

Base

LA and TA

The lateral area of a cylinder is the product of the perimeter(?) of the base and the height.

r

h

2TA LA B

LA = ph

LA = (2Пr) h

TA = LA + 2 Пr2

Volume

The volume of a cylinder is the product of the area of the base and the height.

V Bh

r

h

Cone

A cone is a pyramid with a circular base.

r

h

Base

LA and TA

The lateral area of a cone is one half the product of the circumference of the base and the slant height.

TA LA B r

h

LA = ½ pl

LA = ½ (2Пr) h

LA = Пrh

Volume

The volume of a cone is one third the product of the area of the base and the height.

1

3V Bh

r

hV = 1/3 Пr2h

White Board Practice

• Find the lateral area, total area and volume of a cone with height and radius 3 cm.cm33

White Board Practice

• Find the lateral area, total area and volume of a cone with height and radius 3 cm.

LA = 18 cm2

TA = 27 cm2

V = 9 cm3

cm33

3

12-4: Spheres

Objectives

• Determine the area and volume of a sphere.

Sphere

r

A sphere is the locus (set) of points in space equidistant from a given point.

Volume of a Sphere

r

V = 4/3 Пr3

Area of a Sphere

The area of a sphere is four times the area of the circle with the same radius.

r

24A r

White Board Practice

• If the surface area of a sphere is 16, find the diameter and the volume

White Board Practice

• If the surface area of a sphere is 16, find the diameter and the volume

d = 4

V = 3

32

White Board Practice

• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.

White Board Practice

• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.

144

White Board Practice

• Find the area of the circle formed when a plane passes 9 cm from the center of a sphere with a radius of 15 cm.

15 9

White Board Practice

• Betty made two wax candles, one in the shape of a sphere with radius 5cm and another in the shape of a cylinder with radius 5cm and height 6 cm. Which candle required more wax?

White Board Practice

• Betty made two wax candles, one in the shape of a sphere with radius 5cm and another in the shape of a cylinder with radius 5cm and height 6 cm. Which candle required more wax?

Volume of Sphere = 166 2/3 cm3

Volume of Cylinder = 150 cm3

12-5: Areas and Volumes of Similar Solids

Objectives

• Determine the ratios of the areas and volumes of solids.

Similar SolidsFor two solids to be similar, all angles must be congruent and

all corresponding measurements must be proportional.

rh rh

Remember

• Scale Factor a:b

• Ratio of perimeters a:b

• Ratio of areas a2:b2

Theorem 12-11If the scale factor of two similar solids is a:b, then

• the ratio of their perimeters is a:b

• the ratio of their areas is a2:b2

• the ratio of their volumes is a3:b3

rr

rr

• Scale Factor a:b

• Ratio of (anything that is not area of volume) a:b– Radii, diameter, heights, circumference,

perimeter, etc….

• Ratio of areas a2:b2

• Ratio of volumes a3:b3

Ex 1

• Two regular pyramids have equilateral triangular bases with sides 4 and 6. Their heights are 6 and 9 respectively. Are the two pyramids similar?

Ex 2

• Two similar cones have bases with area ratios of 4:9. Find the ratios of the following:

a) Radii

b) Heights

c) Total areas

d) Volumes

Ex 3

• The volumes of two spheres have a ratio of 27:64. Find the area of the larger sphere if the area of the smaller sphere is 18.

Ex 4

• The radii of two similar cylinders are 2 and 5. Find the ratios of their volumes and of their lateral areas.

Ex 5

• The volumes of two similar rectangular solids are 125 and 64. Find the ratio of their base perimeters.