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# Perimeter & Circumference

Directions: Using the digits 1-6, at most one time each, fill in the boxes to create the largest and smallest combined perimeter/circumference for the rectangle and circle. ### Hint

Which box number change has the greatest effect on the combined perimeter/circumference? Why?
What part of your calculations will be the same every time?

How many permutations are there of the numbers 1-6 in four positions?
Which permutations matter for the largest sum? Why?
Which permutations matter for the smallest sum? Why?

greatest: (53 · 6) + (2π · 4) = 343.13… units

least: (24 · 1) + (2π · 3) = 42.85… units

Source: Christin Smith

## Sides of a Triangle

Directions: The perimeter of a triangle is 20 units. Using whole numbers, how many sets …

1. The solution given is actually for area of the rectangle and circumference of the circle

• Where do you put the numbers? Like which boxes? Thanks.

• You can choose to use any of the digits 1-6 in each of the boxes so that the rectangle has side lengths and the circle has a radius. You can only use each digit once.

• David, you are right. With the solution (with given numbers) should look like:
greatest: (2*53+2*6)+(2π · 4)=143.13….
least: (2*24+2*1)+(2π · 3)=68.84…..

HOWEVER: Neither are the greatest or smalles that we found.
Greatest: (2*65+2*4)+(2π · 3)=156.84…
Least: (2*13+2*4)+(2π · 2)=46.56…

• Brianna… I believe this is the greatest that we have found
(2*65+2*3) + (2pi * 4) = 161.13

2. Greatest: Rectangle with sides 63 and 4 and circle with radius 5. Answer: 165.4

• This is a solution but the one given in the answers is greater.

• It is, but the solution used for the rectangle is the area, not the perimeter.

3. Least: Rectangle with sides 13 and 4 and circle with radius 2. Answer: 46.56

• Similarly with this solution Eric – it works but is not the smallest possible area.

4. Are we finding the smallest perimeter or area???

5. My students came up with the following for greatest:

2(64+3) + (5*2*pi) = 134 + 31.4 = 165.4

• Us, too. Was super confused by the 313 answer until I really looked at.

• I agree with 165.4 as the greatest total.
Note that you can also get that same total by reversing the 3 and the 4:

2(63+4) + (5 * 2*pi)

6. Richard Buettner

I wrote a program that solves these. The largest combined perimeter/circumference is the one that was already found with:

A 64 x 3 rectangle and a circle of radius 5 = 165.42
A 63 x 4 rectangle and a circle of radius 5 = 165.42

Least:

A 14 x 3 rectangle and a circle of radius 2 = 46.57
A 13 x 4 rectangle and a circle of radius 2 = 46.57

• Richard Buettner

For combined areas:

Greatest:
A 53 x 6 rectangle and a circle of radius 4 = 368.27

Least:
A 34 x 1 rectangle and a circle of radius 2 = 46.57