Area and Volume

Areas and Volumes

Common shape areas

We rarely measure areas directly. For example, there is no tape measure that has an area.

Rectangle

Rectangle

Circle

Circle

Ellipse

Ellipse

Any shape

General shape

Areas

• Notice that these all involve the length and the width and a factor

shape	area
rectangle	$l \cdot w$
circle	$0.79\ l \cdot w$
ellipse	$0.79\ l \cdot w$

Area

• Has dimension of length squared

Common Shape Volumes

Basic Volumes

• Cube

• Cylinder

• Pyramid

• Sphere

What is in common?

• A length times a length times a length

• Has dimensions of length cubed

• Volume only differs from a rectangular prism by a factor

Cylinder

Sphere

Unit Conversion

Converting areas

A common source of area is forgetting to apply a linear conversion twice to get an area conversion.

$1 m^2 = 1 \cdot meter \cdot meter$

$1 \cdot meter \cdot meter \cdot \frac{100cm}{meter} \cdot \frac{100cm}{meter} = 10^4 cm^2$

Converting volumes

$1 m^3 \cdot \frac{100cm}{m} \cdot \frac{100cm}{m} \cdot \frac{100cm}{m} = 10^6 cm^3$ $1 m^3 \cdot \left( \frac{100cm}{m} \right)^3 = 1m^3 \cdot \frac{10^6 cm^3}{m^3} = 10^6 cm^3$

Converting volumes

• How many cubic inches in a cubic foot?

Converting volumes

• Convert cubic meters to liters

Roots

Roots answer the question, what is the size of square or cube that I can fit a given quantity in?

Square roots

• If I have a certain area, how do I find the square that contains that

  area?

Square Roots

Cube roots

• If I have a volume, how do I find the cube that contains that volume?

Cube Roots

Examples

GPA

The calculation of a grade point average can be thought of as an area problem.

The average is the height of a rectangle that is 10 units long, or 2.79.

River flow

We can estimate the flow of a creek if we know the rate of rainfall and the land area that drains into the creek.

Average Power

If we have a power that is changing over time, we can interpret the area under the curve as an energy. The average power is the height of a rectangle with the equal width and area.

Area as a way of understanding formulas

Pythagorean theorem

$a^2 + b^2 = c^2$

• We usually interpret as a relation between the sides of a right

  triangle

• We can also interpret as a statement about the areas

Pythagorean theorem

• Strogatz, The Joy of X

Pythagorean theorem

• Strogatz, The Joy of X

Pythagorean theorem

• Strogatz, The Joy of X

Quadratic theorem as an area

• Blends algebra and areas

$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Completing the square

• www.mathisfun.com

Activities

Estimations

• What is the area of skin on the human body?

• Without looking, how much volume in a can of soda?

• How much water in your water bottle?

• What is the volume of the aluminum?

• How much does an empty can weigh?

Exercise

• Estimate the area of our classroom

• Using units of your stride squared

• Using the floor tiles

• Convert between square strides and square feet

ETC Area

• Estimate the volume of our classroom

Question

• I have a coupon for 1000 square feet of carpet.

• What is the largest square room I can cover?

Creek Flow Estimation

• Make a model of the flow

  • What are your assumptions?

  • Is all the water flowing at same speed?

  • What is the basic depth and shape of the creek bed?

• Estimated the flow of the creek in volume per time

• Estimate the linear speed of flow

• Estimate the volume per time

• Where is the water coming from?