1.7: Various Formulas1.7: Various Formulas

Objectives:1.To use formulas to find the perimeter,

circumference, and area of squares, rectangles, triangles, and circles

2.To identify various polyhedra and solids of revolution

3.To use formulas to find the volume of prisms, pyramids, cylinders, cones, and spheres

VocabularyVocabulary

In your notebook, define each of these without your book. Draw a picture for each word and leave a bit of space for additions and revisions.

Area Perimeter

Volume Polyhedron

Prism Pyramid

Cylinder Cone

Sphere

Units vs. Square UnitsUnits vs. Square Units

The lengthlength of something (a segment, a room, a board) is measured in units: inches, feet, centimeters, meters, etc. It’s a one-dimensional measurement.

1

Length: 1 unit

PerimeterPerimeter

The perimeterperimeter of a polygon is the sum of lengths of all the segments that make up the polygon. It is basically the distance around the shape.

5.35 cm

4.82 cm

6.04 cm

4.30 cm

3.72 cm

Perimeter = 5.35 + 4.82 + 6.04 + 4.30 +3.72 = 24.23 cm

Let’s Have Some Let’s Have Some ππ

The distance around a circle is its circumferencecircumference.

DiameterCircumferenceCircumference

Units vs. Square UnitsUnits vs. Square Units

The areaarea of something (a polygon, a floor, a wall) is measured in square units: in2, ft2, cm2, m2, etc. It’s a two-dimensional measurement.

1

1

Area: 1 square unit

AreaArea

The areaarea of a plane figure is the measure, in square units, of the region enclosed by the figure.– This is simply the number of unit squares that

can be arranged to completely cover the figure.

– If two polygons are congruent, what do you suppose is true about their areas?

Formula City Formula City (MEMORIZE THESE)(MEMORIZE THESE)

Example 2Example 2

Find the area and perimeter (or circumference) of each figure.

P=37.4A=74.1

P=6.4A=2.56

C=4π OR 12.6A= 4π OR 12.6

Example 3Example 3

The recently up and running Large Hadron Collider is a circular tunnel that spans a 27 km circumference. What is the diameter of the LHC ring?

27 OR 2.2π

Example 4Example 4

If the irrigated circular field shown has an area of 804.25 m2, what is the length of the rotating sprinkler (the radius of the circle)?

16

Example 5Example 5

Find the area of the blue figure shown.

642.5

Example 6Example 6

Given any of the previous formulas, what would it mean to solve for a particular variable?

To solve for a variablesolve for a variable in an equation or formula means to isolate that variable on only one side of the equation:

variable = everything else

Example 7Example 7

1. Solve C = 2πr for r. Then find the radius of a circle with a circumference of 44 in.

2. Solve V = (4/3)πr3 for r. Then find the radius of a sphere with a volume of 36 cm3.

C = r2π

44 = r2π

7 = r

r= 3V 4π √

3r= 2.05 r= 3(36)

4π √3 r= 27 π √

3r= 8.59 √

3

Example 8Example 8

Sometimes the variable you are solving for is part of an expression.

1. Solve P = 2l + 2w for l. Then find the length of the rectangle whose perimeter is 30 in. and whose width is 7 in.

2. Solve A = (1/2)(b1 + b2)h for b1.

l = P-2w 2

l = 30-2(7) 2

l = 8

2A - b2 = b1

h

Example 9Example 9

Write a formula for the area of a circle in terms of its circumference. Then find the area of a circle with a circumference of 12.5 cm.

C=π2r A=πr2

r = C A= π (C)2 2π (2π)2

A= C2

4π

A= C2

4πA= (12.5)2

4πA = 12.43

In Glorious 3-D!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.

PolyhedronPolyhedron

A solid formed by polygonspolygons that enclose a single region of space is called a polyhedronpolyhedron.

Parts of PolyhedronsParts of Polyhedrons

• Polygonal region (flat surface) = faceface

• Intersection of 2 faces = edgeedge

• Intersection of 3+ edges = vertexvertex

faceface vertexvertexedgeedge

Example 10Example 10

a. Name all of the faces

b. Name all of the edges

c. Name all of the vertices

Classifying PolyhedraClassifying Polyhedra

Polyhedrons are classified by the # of faces:

[Insert Greek prefix for # of faces]-hedron[Insert Greek prefix for # of faces]-hedron

Regular PolyhedronRegular Polyhedron

A regular regular polyhedronpolyhedron is a polyhedron whose faces are regular congruent polygons.

EarthHexahedron

WaterIcosahedron

AirOctahedron Fire

Tetrahedron

CosmosDodecahedron

Regular PolyhedronRegular Polyhedron

Regular polyhedra are commonly called Platonic Platonic solidssolids.

Classification of PrismsClassification of Prisms

Prisms are classified by their bases.

Classification of PyramidsClassification of Pyramids

Pyramids are also classified by their bases.

Example 11Example 11

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

Solids of RevolutionSolids of Revolution

Cylinder

Cone

VolumeVolume

VolumeVolume is the measure of the amount of space contained in a solid, measured in cubic units.– This is simply the

number of unit cubes that can be arranged to completely fill the space within a figure.

Exercise 12Exercise 12

Find the volume of the given figure in cubic units.

20

More Formulas!More Formulas!

The volume of a solid is also easily computed with a formula.

What does the B represent?

B = area of base

Exercise 13Exercise 13

The previous formulas for volume all contained B, the area of the base. Rewrite the following formulas without B.

1.Rectangular Prism:

2.Cylinder:

3.Cone:

V = (l x w) h

V= πr2h

V = 1/3 (πr2h)

Exercise 14Exercise 14

Find the volume of each solid.

189

36π

126

Exercise 15Exercise 15

The triangle shown can be rotated around the y-axis or the x-axis to make two different solids of revolution. Which solid would have the greater volume?

Calculate both and compare

The one with the longest radius: rotated around the y-axis.

Example 16Example 16

A pipe in the shape of a cylinder with a 30-inch diameter is to go through a passageway shaped like a rectangular prism. The passageway is 3 ft high, 4 ft wide, and 6 ft long. The space around the pipe is to be filled with insulating material.

Example 16Example 16

What is the volume of the insulating material?

Vprism= 72Vcylinder = 29.5Vmaterial = 42.5 ft.3