**High School Number Sense Lesson 70: Estimating with Powers of Pi (π)**

Some of the most interesting estimation questions on number sense tests involve the irrational (but real) number Pi (better known as π), which is approximately 3.14159265358979323846264338327950288419716939937510... This concept appeared

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

**question # 70**.

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When I was younger, I found it entertaining to try to memorize dozens of digits of π. I don't think I ever got very far, and now, if somebody catches me off guard, I can only recite the first 20 decimal places. (I think my daughter can rattle off about twice that many). One of the many cool things about

**π**is that when you square it, you get about

**10**. (You actually get about

**9.8696044**, but for number sense estimation questions, that's close enough!).

**How close is close enough?**

As you well know, as long as your answer is

**within (+/-) 5%**of the exact answer for estimation questions, it is considered correct. For every two factors of π in a question, you can be safe substituting 10 in the answer...up to a certain point. Assuming you use 3.14 for the

*odd*(or leftover) factors of π, here would be your estimations:

**π is about 3.14**(actually 3.141592653...); 3.14 is

**within 0.06%**of π

**π**(actually 9.869604401...); 10 is

^{2}is about 10**within 1.33%**of π

^{2}

**π**(actually 31.006276680...); 31.4 is

^{3}is about 31.4**within 1.27%**of π

^{3}

**π**(actually 97.409091034...); 100 is

^{4}is about 100**within 2.66%**of π

^{4}

**π**(actually 306.019684785...); 314 is

^{5}is about 314**within 2.61%**of π

^{5}

**π**(actually 961.389193575...); 1000 is

^{6}is about 1000**within 4.02%**of π

^{6}

**π**(actually 3020.2932277767...); 3140 is

^{7}is about 3140**within 3.97%**of π

^{7}

**π**(actually 9488.531016070...); 10000 is

^{8}is about 10000**> 5.39% more than**π

^{8}.

**7 or**

**less**, estimating (π x π) as 10 is fine. As soon as the exponent reaches 8, however, you are out of range.

**Example 1:**

*** π**

^{4}= ___1. π

^{4}= π

^{2}x π

^{2}. Substitute 10 for each π

^{2}.

2. Your answer is 10 x 10 =

**100**.

3. The actual answer is about

**97.409**, and the (+/- 5%) range is

**from 93 to 102**. You are within range.

**Example 2:**

*** (3π)**

^{4}= ___1. (3π)

^{4}= 3

^{4}x π

^{2}x π

^{2}. Substitute 10 for each π

^{2}.

2. Your answer is 81 x 10 x 10 =

**8100**.

3. The actual answer is about

**7890.136**, and the (+/- 5%) range is

**from 7496 to 8284**. You are within range.

**Example 3:**

*** (2π) x (3π) x (2π) = ___**

1. (2π) x (3π) x (2π) = (2 x 3 x 2) x π

^{3}= 12 π x π

^{2}

2. 12 x 3.14 x 10 is about 37 x 10 =

**370**.

3. The actual answer is about

**372.075**, and the (+/- 5%) range is

**from 354 to 390**. You are within range.

**Example 4:**

*** 2π**

^{5}= ___1. 2π

^{5}= 2 x π x π

^{2}x π

^{2}

2. 2 x 3.14 x 10 x 10 is about 6.28 x 100 =

**628**

3. The actual answer is about

**612.039**, and the (+/- 5%) range is

**from 582 to 642**. You are within range.

**Example 5:**

*** (2π)**

^{5}= ___1. (2π)

^{5}= 2

^{5}x π x π

^{2}x π

^{2}

2. 32 x 3.14 x 10 x 10 is about 101 x 100 =

**10100**

3. The actual answer is about

**9792.63**, and the (+/- 5%) range is

**from 9303 to 10282**. You are within range.

**Check back soon for a free worksheet to help you practice EstPiPower.**

**Up Next for High School: SquareRootMem**