although, here's a baking tip for the problem: after cutting out the cookies, take the leftover dough, mush it back together into one ball and roll it out again. Then you can keep cutting out cookies and you won't be wasting any dough! ðŸ˜„

]]>$$ \frac{\text{area of circular wedges}}{\text{area of equilateral triangle}} $$

To find the area of the equilateral triangle, we recall that we know the ratios of the height of the equilateral triangle to its leg. In fact, that was covered in this forum post.

M2W1D1-30-60-90-triangle-proportions.png

So altitude of an equilateral triangle is equal to \( \sqrt{3} \times \frac{\text{side}}{2} \), which means the area of the equilateral triangle is equal to

$$\begin{aligned} \text{ area of triangle } &= \frac{1}{2} \times \text{ base} \times \text{ height} \\ &= \frac{1}{2} \times 2 \times \sqrt{3} \\ &= \sqrt{3} \\ \end{aligned} $$Inside the equilateral triangle, there are three wedges of cookie, each with an angle of \(60^{\circ}.\)

M2W3D9-y-forum-question-why-divide-sqrt-3-whole-triangle.png

Put together, the wedges make up half of a circle with radius \(1!\)

M2W3D9-y-forum-question-why-divide-sqrt-3-half-circle.png

Since the area of a circle with radius \(1\) is \(\pi ,\) the cookie area is \(\frac{\pi}{2}.\)

And since we know the area of the cookie and the area of the triangle, we can find the answer, which is the ratio of two:

$$ \frac{\text{cookie area}}{\text{triangle area}} = \frac{\frac{\pi}{2}}{\sqrt{3}}$$

I hope this helps! ðŸ™‚

]]>These are great questions! There are actually many ways to go about solving this question. In particular, there are multiple ways to "dissect" that whole sheet of cookies into simpler shapes.

The idea behind "dissecting" this cookie sheet was to find a single shape that can cover the whole sheet, and so that the "pattern" inside every shape is the same. Professor Loh decided to use equilateral triangles by connecting the centers of the circles, and each triangle had this same pattern inside:

58f7648c-b32e-49ae-8d94-ab623e57edf0-image.png

If each triangle has the same "pattern" inside, then each triangle has the same amount of cookie inside! So by just looking at one triangle, you can analyze the whole sheet of cookies because the rest of the triangles are the same!

There are more ways to do it though! For example, maybe we can use hexagons:

40902bc0-76f4-465a-9182-93a69dd0d444-image.png

This also works! Can you try to work it out?

I think the first dissection using triangles is a little easier. One important reason is this: When circles are tangent to each other, it is usually very useful to draw a line connecting their centers! It's almost always a very helpful first step in any geometry problem. For example, how would you find the length of this blue line segment if the radii of the circles are 1 and 2?

eb205dcd-becc-4203-a7db-fbacc0ebe5cc-image.png

(Try connecting the centers!)

As for your other question: Professor Loh divides because he wants to find the fraction of the triangle that is taken up by the circles (the "cookies"). By doing this division, he can actually figure out what fraction of the ENTIRE sheet is covered by circles! If he only subtracted, he'd only find the area within one triangle taken up by cookies, and that's not enough.

Let me know if you have any other questions!

]]>