**Middle School Number Sense Lesson 57: Circles**

And now back to our study of geometry, although we take a quick break from the polygons of lessons past. (Or is a circle a polygon with an infinite number of sides? ... I'll let you decide.)

On last year's tests, there were

**13 questions**about circles, with the median placement at

**question # 55**.

**Number Dojo Level: 157**

**Let's start with a few definitions:**

**Radius (r)**: the distance from a circle's center to its edge**Diameter (d)**: the greatest distance across a circle (from one edge to the opposite edge)**Circumference (C)**: the distance around a circle--its perimeter**Area (A)**: the amount of space contained within a circle's edge**Pi (π)**: our favorite mathematical constant--irrational but real--equal to the ratio of a circle's circumference to its diameter. Approximately 3.14159265358979323846264338327950288419716939937510...

**And a few formulas:**

**C = πd = 2πr**

*

**A = πr**

^{2}= π(d/2)^{2}*

**d = 2r = C/π**

*

**r = d/2 = C/2π**

* For purposes of

**estimation**problems, π can be estimated as the fraction

**22/7**(or

**3 1/7**).

**Example 1: The area of a circle with a radius of 9 is kπ; k = ___**

^{2}

2. kπ = π(9

^{2}) = 81π. k =

**81**.

**Example 2: The circumference of a circle with a radius of 17 is kπ; k = ___**

2. kπ = 2π(17) = 34π. k =

**34**.

**Example 3: What is the diameter of a circle with a circumference of 14π? ___**

2. d = (14π)/π =

**14**.

**Example 4: What is the radius of a circle with a circumference of 64π? ___**

2. r = (64π)/2π = 64/2 =

**32**.

**Example 5: * The area of a circle with a radius 12 = ___**

**22/7**for π.

2. A = πr

^{2}

3. A = (22/7)(12

^{2}) = (22/7)(144)

4. A = (144/7)(22) = about 20.5 x 22 = (using

**Double/Half**) about 41 x 11 =

**451**.

5. The correct answer is about

**452.389**, and the acceptable (+/- 5%) range is from

**430 to 475**. We are well within range.

**Example 6: * The circumference of a circle with radius 25 is ___**

**22/7**for π.

2. C = 2πr

3. C = 2(22/7)(25) = (22/7)(50) = (50/7)(22)

4. C = about 7 x 22 =

**154**.

5. The correct answer is about

**157.0796**, and the acceptable (+/- 5%) range is from

**150 to 164**. We are within range.

**Here's a free worksheet to help you practice Circles:**

circles.pdf |

**Up Next for Middle School: BaseDiv**