Connecting Geometric and Algebraic Representations

Cheryl Olsen

Shippensburg University

October 11, 2005

Objectives• Geometric figures:

– Understand the concepts of length, area, volume & surface area

– Recognize/represent 3-dimensional figures– Understand formulas for area and volumes of

2-dimensional & 3-dimensional figures

• Measurement– Solve problems involving length, area, &

volume of geometric objects

Question 1: Imagine a cone inscribed in a cylinder of the same size, so that the base of the cone is the same as the base of the cylinder and the top of the cone touches the top of the cylinder. Imagine also a sphere inscribed in a cylinder so that the sphere touches the cylinder at the north and south pole and all the way around the equator. What is the ratio of the volumes of a cone, sphere and cylinder?

Question 2:

Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

Area of Rectangle

Area of rectangle = base * height

h

b

Area of Parallelogram

b

h

b

h

b

h

Area of parallelogram = base * height

Area of Triangle

h

b

Area of triangle = ½ * area of parallelogram

Area of triangle = ½ * base * height

Area of Trapezoid

b1

b2

h

b2

b1

Area of trapezoid = ½ * area of parallelogram

Area of trapezoid = ½ * (b1+b2) * h

base of parallelogram = b1+b2

Area of a Circle

Cut apart your pizza and rearrange the slices so that it is a figure that we know how to find the area.

Area of a Circle

base = ½ of circumference of circle

height =

radius of circle

Area of Circle = ½ * circumference * radius

Area of Circle = ½ * ( 2 * Pi * radius ) * radius

Area of Circle = Pi * radius2

Area of a Circle

Surface Area & Volume of 3-dimensional shapes

• Cylinder

• Cone

• Sphere

Activities A,B,C

• Groups of 4

• 3 groups start with Activity A, 3 groups start with Activity B, & 3 groups start with Activity C

• After 15 minutes we’ll rotate

Activity AActivity ARelationship between Volume of Cylinder and Volume of Sphere

• Volume of sphere is 2/3 of the volume of the cylinder.

Activity BActivity BRelationship between Volume of

Cylinder and Volume of Cone

• Volume of cone is 1/3 of the volume of the cylinder.

Activity CActivity CSurface Area of a Sphere

• Surface Area = 4*Pi*radius2

Volume of Cylinder(in which a cone & sphere fit inside it)

Volume =

heightradius *)( 2

)*2(*)( 2 radiusradius

3)(3

6radius

3)(*2 radius

Volume of Sphere(which fits inside previous cylinder)

Volume =heightradius *)( 2

)*2(*)( 2 radiusradius3)(

3

4radius

2/3 * volume of cylinder

= 2/3 *

= 2/3 *

Volume of Cone(which fits inside previous cylinder)

Volume =heightradius *)( 2

)*2(*)( 2 radiusradius3)(

3

2radius

1/3 * volume of cylinder

= 1/3 *

= 1/3 *

Volume Cylinder vs. Volume Cone vs. Volume Sphere

• Volume of Cylinder

• Volume of Sphere

• Volume of Cone

3)(3

6radius

3)(3

2radius

3)(3

4radius

Ratios of the volumes are

(Cylinder : Sphere : Cone )6:4:2 OR 3:2:1

Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

can of Volume

ball tennisof Volume*3)*6(*)(

)(3

4*3

2

3

radiusradius

radius

3

3

)(6

)(4

radius

radius

%673

2

6

4

Case Study Video on Volume of Cylinder

Cylinder

Surface Area– What does a net of a cylinder look like? How

does this help determine the surface area of a cylinder?

– Surface Area = heightradiusradius *)(2)(*2 2

Milk Tanker

• A stainless steel milk tanker in the shape of a right circular cylinder is 38 feet long and 5 feet in diameter.

• Determine the amount of stainless steel material needed to construct the tanker.

• Assume that 12% of the material you start with will be wasted in the construction process.

Milk Tanker

Circumference = Pi * diameter

5 ft

circumference

38 ft

Surface Area = (38)(circumference) + 2(Pi * radius2)

= (38)(Pi * 5) + 2(Pi * 2.52)

= (190*Pi) + (50*Pi/4)

= 202.5 Pi square feet

Milk Tanker

Surface Area = 202.5 Pi square feet ~ 636.17 square feet

• Assume that 12% of the material you start with will be wasted in the construction process.

Since 12% of the original material will be wasted we can think of 88% of the original material = 636.17 sq ft

.88 * original material = 202.5 Pi ~636.17 sq ft

original material = 636.17 sq ft ~ 723 sq ft

.88

Name that Common Solid

1. Side view and front view are triangles. Top view is a circle.

2. Side view and front view are rectangles. Top view is a circle.

3. Side view and front view are triangles. Top view is a square.

Cone

Cylinder

Square pyramid

Name that Common Solid

4. Side view and front view are triangles. Top view is a rectangle.

5. Side view and front view are rectangles. Top view is a rectangle.

Rectangular Prism

Rectangular pyramid

Name that Common Solid

6. Side view, front view, and top view are all congruent squares.

7. Side view, front view, and top view are all congruent, and all triangles.

Triangular Pyramid

Cube