Honors Geometry

Lateral Area, Surface Area and Volume

Prisms - definitionsA PRISM is any object with two parallel congruent bases with lateral sides that are parallelograms.

In this case, the two blue sides are the bases. The bases are not always on the top and bottom! In the next figure, the bases are the front and the back,because they are theparallel congruent sides.

In this case, the two blue sides are the bases. In this case, the two blue sides are the bases.

In this case, the two pink sides are the bases. In this case, the two yellowyellow sides are the bases.

Naming PrismsPrisms are named by the shape of their bases.

SquarePrism

RectangularPrism

TriangularPrism

Hexagonal

Prism

Right and Oblique PrismsA right PRISM has the sides perpendicular to the bases.

An oblique PRISM has sides that are NOT perpendicular to the bases.

RightPrism

ObliquePrism

Cylinders

A CYLINDERCYLINDER is a special case of a prism where the two parallel sides (bases) are circles.

A right cylinder has its side perpendicular to its bases.

A cylinder can be thought of as a circular prism.

An oblique cylinder has its side NOTNOT perpendicular to its bases.

Lateral Area of a Prism

xy

zz

x xy y

Lateral Area (LA) is the area of the side(s) of the object not including the bases.

In this case, the LA = (x + y + x + y) z, but x + y + x + y = 2x + 2y. This happens to be the perimeter of the base (P). So, the LA of a right prism is given by LA = Ph.

Lateral Area of a CylinderI like to think of the LA of a cylinder as measuring the area of the label of a soup can. If

you cut down the dotted line and peel the label off and lay it out flat, it is a rectangle. The base of the rectangle is the circumference of the base and the height of the rectangle is the height of the cylinder.

h

r

Circumference

Height

Therefore, LA of a cylinder is given by LA = Ch.Since circumference is to a circle as perimeter is to a polygon, this just a variation of LA = Ph.

Surface (Total) Area of a Prism

The Surface area of a Prism is the sum of the area of ALL of the sides. It is also the LA plus the area of the two bases. Since, however the two bases are congruent, the surface area of a right prism (if B is the area of a base) is given by SA = LA + 2BSA = LA + 2B

Example 112cm10cm8cm Find the Surface

AreaLA = Ph; SA = LA + 2B

LA = Ph; P = 2(12cm) + 2(10cm) = 44cm; h = 8cm

LA = (44cm)(8cm) = 352cm2

B = (12cm)(10cm) = 120cm2; so 2B = 2(120cm2) = 240cm2

SA = LA + 2B = 352cm2 + 240cm2 = 592cm2

Example 215 in.9 in.12 in.13 in. Find the Surface

AreaLA = Ph; SA = LA + 2B

LA = Ph; P = 15 in. + 12 in. + 9 in. = 36 in.; h = 13 in.

LA = (36 in.)(13 in.) = 468 in.2

B = 1/2(12 in.)(9 in.) = 54 in.2; so 2B = 2(54 in.2) = 108 in.2

SA = LA + 2B = 468 in.2 + 108 in.2 = 576 in.2

Surface Area of a Cylinder

The Surface Area of a Cylinder is much like that of a Prism. It the LA + the area of the 2 bases or (since the bases are congruent), LA

+ twice the area of a base (B).

SA = Ch + 2B

Example 312 ft.8 ft. Find the Surface

AreaLA = Ch; SA = LA + 2B

LA = Ch; C = 2π(8 ft.) = 16π ft. = 50.27 ft.; h = 12 ft.

LA = (50.27 ft.)(12 ft.) = 603.24 ft.2

B = π(8ft.)2 = 64π ft.2 = 201.06 ft.2; so 2B = 2(201.06 ft.2) = 402.12 ft.2

SA = LA + 2B = 603.24 ft.2 + 402.12 ft.2 = 1005.36 ft.2

Assignment

Lateral Area and Surface Areaall

Dog - PRISM

Volume of a PrismUnit Cube

Base

Volume of a Prism is merely the area of a base (B) times the

height.V = Bh

Example 412cm10cm8cm

Find the Volume

B = (12cm)(10cm) = 120cm2

V = Bh

V = (120cm2)(8cm) = 960cm3

h = 8cm

Example 515 in.9 in.12 in.13 in.

Find the Volume

V = Bh

B = 1/2(12in)(9in) = 54in2

V = (54in2)(13in) = 702in3

h = 13in.

Volume of a CylinderLike a prism, the volume of a

Cylinder is the area of a base times its height.

V = Bh

Example 6

12 ft.8 ft. Find the Volume

V = Bh

V = (64π ft2)(12 ft) = 768π ft2 = 2412.74ft2

B = π(8ft)2 = 64π ft2

h = 12 ft

Assignment

Volume - all

Pyramids and Cones

Square orRectangular

Pyramid

TriangularPyramid

Cone

For pyramids and cones V = 1/3Bh

Example 7

6 mm

6 mm

6 mm

Find the Volume

V = 1/3Bh

B = (6mm)2 = 36mm2

V = 1/3 (36mm2) (6mm) = 72mm3

Example 8

5 cm

8 cm

6 cm

Find the Volume

V = 1/3Bh

B = 1/2(5cm)(8cm) = 20cm2

V = 1/3 (20cm2)(6cm) = 40cm3

h = 6cm

Example 9

15 cm

9 cm

Find the Volume

V = 1/3(81π)15

V = 405π cm3

Assignment

Pyramids and Cones - all

Spheres

r SA = 4πr2

V = 4/3πr3

Example 10

7 cm

Find the Surface Area

SA = 4πr2

SA = 4π(7cm)2

SA = 196πcm2

SA = 615.75cm2

Example 11

12 cm

V = 4/3πr3

Find the Volume

V = 4/3π(12cm)3

V = 2304πcm3

V = 7238.23cm3

Assignment

Spheres - all