Tag Archives: rectangle

Three Little Geometry Problems

Here are three variations of geometry problems I got from @jamestanton on Twitter.

  1. The numerical measure of a rectangle’s area and perimeter are equal (P=A) (obviously the units are different).  If the rectangle’s sides have integer lengths, what are the dimensions of the rectangle?
  2. The numerical measure of a box’s surface area and volume are equal (V=SA).  If the box’s sides have integer lengths, what are its dimensions?
  3. The numerical measure of a right triangle’s area and perimeter are equal (P=A).  If the triangle’s sides have integer lengths, what are its dimensions?

To give a complete solution to a math problem, remember that you must

  1. show that your proffered solution(s) is (are) correct, and
  2. shows that no other solutions exist.

Find convincing arguments that you have found all of the solutions for each.  While my solutions are shown below, I eagerly welcome suggestions for any other approaches.

STOP!!!
SOLUTION ALERT!!
DO NOT READ ANY FURTHER IF YOU WANT TO SOLVE THESE PROBLEMS ON YOUR OWN.

Problem 1:  P=A for a rectangle
Let the a=length and b=width.  Without loss of generality, assume a\le b.  Then, a\cdot b=2(a+b) \Longrightarrow \frac{1}{2}=\frac{1}{a}+\frac{1}{b} which implies 2<a\le 4.

If a=3, b=6.  If a=4, b=4.  Thus, the only rectangles for which P=A are a 3×6 and a square with side 4.

Problem 2:  P=SA for a box
Let the a=length, b=width, and c=height.  Without loss of generality, assume a\le b\le c.  Then, a\cdot b\cdot c=2(a\cdot b+a\cdot c+b\cdot c) \Longrightarrow \frac{1}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} which implies 2<a\le 6.

It wasn’t worth it to find all these by hand, so I wrote a quick spreadsheet to find all c values for given a and b values under the condition a\le b\le c.


Therefore, there are 8 such boxes with integer dimensions: 3x7x42, 3x8x24, 3x9x18, 3x10x15, 4x5x20, 4x6x12, 5x5x10, and 6x6x6.

Problem 3:  P=A for a right triangle
Let the legs be a and b and the hypotenuse be c.  Perhaps there is a way to employ the technique I used on the first two problems, but my first successful solution invoked a variation on Euclid’s formula:  For some integer values of k, m, and n with m\le n, a=k\cdot (m^2-n^2), b=2k\cdot m\cdot n, and c=k\cdot (m^2+n^2) will form all Pythagorean triples (although not uniquely).

Because \frac{1}{2} a\cdot b=a+b+c, Euclid’s formula gives

\begin{tabular}{ r c l }  \(2\) & \(=\) & \(\frac{\displaystyle a\cdot b}{\displaystyle a+b+c}\) \\  & \(=\) & \(\frac{\displaystyle [k\cdot (m^2-n^2)]\cdot [2k\cdot m\cdot n]}{\displaystyle [k\cdot (m^2-n^2)]+[2k\cdot m\cdot n]+[k\cdot (m^2+n^2)]}\) \\  & \(=\) & \(\frac{\displaystyle 2k^2mn(m^2-n^2)}{\displaystyle 2km(m+n)}\) \\  & \(=\) & \(kn(m-n)\)  \end{tabular}

What started out feeling like a difficult search for unknown side lengths has been dramatically simplified.  2=kn(m-n) can only happen if

  1. k=2, n=1, m-n=1 \rightarrow m=2 which Euclid’s formula converts to a=6, b=8, c=10 .
  2. k=1, n=2, m-n=1 \rightarrow m=3 which Euclid’s formula converts to a=5, b=12, c=13 .
  3. k=1, n=1, m-n=2 \rightarrow m=3, but this leads to a repeat of the first solution.

Therefore, there are only two right triangles with the property P=A:  the 6-8-10 and the 5-12-13 right triangles.

Again, any other solution approaches are encouraged and will be posted.

Elementary Multiplication

One of my daughters is now in 2nd grade and I’ve always been interested in keeping her curiosity piqued–whether in math or any other discipline. I never want to push her to memorize anything or accelerate her learning beyond what she’s ready to engage.  But she has always enjoyed games and has been intensely interested in art.  Following are some ideas I’ve been playing with my daughter during our recent conversations.  Perhaps some of parents out there can benefit from my ideas or others can give me some additional leads on other good ideas

I always play number games with my daughter.  A few years ago I asked her how many apples (or dolls, or crackers, or whatever was in front of her at the time) she would have if she had 2 and I gave her 2 more.  There were many variation on this theme.  Eventually the numbers grew larger and then I asked her how many I would need to give her if she had 2 apples now and would have 5 after my donation.  It was my attempt at introducing subtraction without needing to name a new concept.  From my end, this has worked well.  My daughter likes playing with numbers and I keep pushing the window of what she can handle.  I make it clear that she can always ask for hints and that I’m never disappointed if she can’t handle a question I give her so long as she tries.  It’s a delicate balancing act, reading my daughter’s readiness and trying not to overburden her.  When I misjudge, her blank face tells me to go in another direction.

I’ve been seeding the idea of commutativity lately.  When I ask her something like 10+2, I always follow with a 2+10 and ask her if she notices anything about her last two answers.  At first she didn’t notice, then she saw that the answers were the same, and recently she has been been telling me that you can “flop the numbers” in addition and get the same answer.  I knew the idea had begun to sink in when I asked her 4+8 and she asked if it was OK by me if she added 8+4–it was easier for her to add on 4 to 8 than 8 to 4.

Today, she mentioned negative numbers and I jumped on her exploration of commutativity.  She told me that she knew “somehow” that a subtraction gave a negative result if the “second number was bigger.”  I told her that the only difference between 10-13 and the 13-10 that she already knew was that 10-13 gave a negative answer.  Further details can happen later, but for now, I jumped on a moment of interest and continued a game that we’ve been playing for months. Her face lit up when she realized that negative numbers really aren’t that hard!  It’s never about memorizing facts and I’m always ready to back off.  My mantra:  Keep reading your audience and keep it fun.

Here’s another set-up I started a week ago.  I’ve never seen multiplication started from this angle (but I’ve not been trained as an elementary teacher either).  Nevertheless, I was thinking about how to introduce the concept of multiplication without making it a chore or making it a new idea, so I tried tapping into her art interest.  Two weeks ago, I asked her how many ways she could arrange 6 dots into rectangles.  Grabbing some paper, she quickly made an arrangement of 2 rows and 3 columns and a short time later, 3 rows and 2 columns.  It took some prompting to get her to see a line of dots as a rectangle 1 unit high (or 1 unit wide), but the hook was set.  What follows is a sampling from a journal she keeps for playing around with shapes or math ideas.  I had asked her to try this rectangle arrangement of dots for every number from 1 to 20 using what she had learned from arranging 6 dots.  I asked her to list beside each arrangement the dimensions of the rectangles she could find.  She missed a few, but I’m a pretty proud dad right now.

While she’s just scratching the surface of multiplication right now, I’m pretty psyched that she has written multiplication while thinking that she was just describing the dimensions of rectangles.  What I really think is cool is that she is learning multiplication in reverse–starting with the products and learning the factors.  Eventually, we’ll do this in the “normal” order, but for  now, the art connection has her totally hooked.

Also, she already knows about even numbers and odd numbers.  For now, I have future plans to introduce prime numbers as those that can form exactly 2 rectangles.  We’ll also explore commutativity further once she gets more comfortable with multiplication.  Down the road, I see showing her that 1 is a special number because it is the only number that has only 1 rectangle.  Also, perfect squares (the square numbers that we’ve also discussed) are the only numbers with an odd number of possible rectangles–another consequence of commutativity.

I’d love any feedback on these rambling musings.