**Middle School Number Sense Lesson 61: Spheres**

A couple of months ago we introduced the topic of 3-dimensional objects in

**Polyhedron**. Today we will

**(or maybe**

*get the ball rolling***?) with**

*round out our discussion***spheres**. This concept appeared

**3 times**last year, with a median placement at

**question # 61**.

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Questions about spheres don't show up very often on number sense tests, but it is important for students to know the properties of spheres. I have noticed that these questions are showing up a little earlier on the tests than they used to. There are two formulas that need to be learned:

**surface area (A)**and

**volume (V)**. About 80% of the questions showing up deal with surface area, and the other 20% deal with volume.

**Formulas:**

*** A (surface area) = 4πr**

* V (volume) = (4/3)πr

^{2}* V (volume) = (4/3)πr

^{3}

Question Varieties:Question Varieties:

**Calculate the surface area, given the radius or diameter (or vice versa)****Calculate the volume, given the radius or diameter (or vice versa)****Estimate**either the volume or the surface area

**Example 1:****The surface area of a sphere with radius 3 ft is ___π ft**

1. Use the formula for surface area: A = 4πr

2. A = 4π(3

^{2}1. Use the formula for surface area: A = 4πr

^{2}2. A = 4π(3

^{2}) = 4π(9) = 36π. The answer is 36.**Example 2:**

**The volume of a sphere with radius 3 inches is ___π cubic inches**

1. Use the formula for volume: V = (4/3)πr

^{3}

2. V = (4/3)π(3

^{3}) = (4/3)(27)π = 4(9)π =

**36π**. The answer is

**36**.

**Example 3:**

**The radius of a sphere with surface area 64π is ___**

1. Use the formula for surface area: A = 4πr

^{2}

2. 64π = 4πr

^{2}. Cancel out 4π from each side.

3. 16 = r

^{2}, so r =

**4**.

**Example 4:**

**If the surface area of a sphere is 900π, then the diameter is ___**

1. Use the formula for surface area: A = 4πr

^{2}

2. 900π = 4πr

^{2}. Cancel out 4π from each side.

3. 225 = r

^{2}, so r =

**15**.

4. Note that the question asked for the diameter, so double the 15 to get

**30**.

**Example 5:**

*** (Estimate): The surface area of a sphere with a diameter of 4 inches is ___ square inches.**

1. Use the formula for surface area: A = 4πr

^{2}. Note that we are given the diameter (4), so the radius is 2.

2. A = 4π(2

^{2}) = 4π(4) = 16π. 3. Use 3.14 or 3 1/7 to approximate π. 16(3 1/7) = 48 + 16/7 = about

**50**.

4. The acceptable (+/- 5%) range is

**from 48 to 52**. We are within range.

**Example 6:**

*** (Estimate): The volume of a sphere with a radius of 6 feet is ___ cubic feet.**

1. Use the formula for volume: V = (4/3)πr

^{3}

2. V = (4/3)π(6

^{3}) = (4/3)(216)π = 4(72)π = 288π

3. You can round 288 up to 300 and round π down to 3. Your answer is

**900**.

4. The acceptable (+/- 5%) range is

**from 860 to 950**. We are within range.

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

sphere.pdf |

**Up Next for Middle School: MultPowers2&5**