# Know the formulas for the area and circumference of a circle, and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

ACOS 2009 Correlation
• 9 - Solve problems involving circumference and area of circles.
• 9.1 - Identifying pi as an irrational number.
• 9.2 - Verifying formulas for determining circumference and area of circles.
• 9.3 - Using the appropriate customary or metric unit for determining the circumference, diameter, radius, and area of circles.

CMP2 Correlation
 Book Investigation Objective Covering & Surrounding 5 7-G4 Common Core Inv 4 7-G4

My Humble Opinion
Knowing the formulas is new. I don't understand the significance of the relationship between the circumference and area of a circle. Surely I am reading that the wrong way. I would expect them to understand the derivation of the FORMULAS for circumference and area.

2003 ACOS
Solve problems involving circumference and area of circles.
ARMT Possible Points = 4 (MC, GR)

• Word problems/real-life situations may be used.
• The drawing of a circle may be included.
• The value of "pi" will be 3.14.
• Any representation of a rational number may be used for the dimension of the circle.
• The formulas will be given on the reference page.

Sample problems from Item Specs

Pi Day (March 14 or 3/14) is a great day to study circles.

The 6th grade AMSTI unit Covering and Surrounding has an investigation of these topics that is wonderful.

Discovering the Value of Pi
The students use a Java Applet1 to discover the fact that the ratio of the circumference to the diameter is a constant that applies to all circles. In other words, it teaches them the concept of pi and how it is derived. The lesson also includes a sheet with interesting facts about pi.

4 2 explore Pi page
Collection of links activities concerning Pi.

Illuminations Circle Tool
How do the area and circumference of a circle compare to its radius and diameter? This activity allows you to investigate these relationships in the Intro and Investigation sections and then hone your skills in the Problems section.

Resources

North Carolina's Lesson for Learning
Slicing Pi - 7G4

• I can describe the relationship between radius and diameter of a circle.
• I can find the area of a circle.
• I can use the area of a circle to approximate the radius or diameter.
• I can relate the area of a circle to its circumference.

## Hands-On Activities

### Buffon's Needle Experiment

Mathematics Teaching in the Middle School, August 2006, p. 5.

In this activity, students will work in teams to measure a predetermined distance with a cylindrical or circular object. In order to do this, they will first calculate the circumference of their object and then count how many "rolls" of the object it takes to cover the distance. Finally, as they work through this measurement process, they will determine mean distance.
Teamwork Test Prep 7, p. 50

## Cool Problems

### Any Luck in Limbo?

You have stretched a piece of string around the world at the equator and have fastened it tight so that it touches the surface of the earth at every point. You cut a gap in the string and attach a second piece of string, one meter in length, so that the whole length of string is now one meter longer than it was when it was touching the surface of the earth.
Now you stretch out the string so that it is equidistant from the surface of the earth at every point. Which of the following is the closest estimate of how high the string will rise above the surface of the earth? The circumference of the earth at the equator is approximately forty million meters.
(a) One ten-millionth of a meter
(b) Ten centimeters
(c) One million meters
For the answer, see The Daily Spark: Critical Thinking Warm-up Activities, p. 6.

### Eratosthenes in the Round

Eratosthenes of Cyrene was a Greek scholar and mathematician of the third century B.C.E. who, among other things, made a remarkably accurate estimate of the circumference of the earth.
His method involved placing a stick in the ground a few hundred miles north of the Tropic of Cancer on the summer solstice and measuring the shadow cast by the stick.
How could he estimate the circumference of the earth based on the shadow cast by this stick?
Answer: The Daily Spark Critical Thinking, p. 19.

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