# The smallest circle that two parts of a semi-circle can fit into?

• MHB
• TrevorE
In summary, the speaker made a large circular tortilla and ate half of it before putting the remaining half in the fridge. They then considered cutting the remaining semi-circle into two pieces but wondered if there was a mathematical solution for the most optimal cut. They proposed three possible solutions, including one where the cut is a straight line from the center to the edge of the original circle. They also asked if anyone could provide a demonstration and received a suggestion for a different method of cutting the tortilla in the future.
TrevorE
So, true story:

I made a large circular tortilla.

Ate half of it. Then decided to put the rest into the fridge on a smaller plate.I raised the knife to cut the remaining semi-circle in two, and then went : "Hmmmmmmmm...".

Anyway, it's in the fridge now with an approximate solution, but I'm wondering if anyone knows the mathematical one?

I'm wondering if there are 3 different optimal solutions:

1) In which the straight-line cut necessarily is from the center of the original circle to the edge. I.e. a radius.
2) In which the straight-line cut may divide the semi-circle in any possible way.
3) In which the cut is not necessarily a straight line.

Thanks in advance if anyone can offer a demonstration.View attachment 9517

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Next time, don't cut it that way. Instead, draw a circle on your pizza with radius $\frac r {\sqrt 2}$ and eat the annulus for your first half. You will be left with the smallest possible plate size.

## 1. What is the smallest circle that two parts of a semi-circle can fit into?

The smallest circle that two parts of a semi-circle can fit into is known as the "inscribed circle". It is the largest circle that can fit inside a shape, touching all sides without crossing any of them.

## 2. How is the radius of the inscribed circle calculated?

The radius of the inscribed circle can be calculated using the formula r = (a * b) / (a + b), where a and b are the radii of the two semi-circles.

## 3. What is the significance of the inscribed circle in geometry?

The inscribed circle is significant in geometry as it is a key component in finding the center and radius of a circle inscribed in a polygon. It is also used in various geometric constructions and proofs.

## 4. Can the inscribed circle fit into any shape?

No, the inscribed circle can only fit into certain shapes, such as polygons and circles. It cannot fit into irregular or curved shapes.

## 5. How is the inscribed circle related to the circumcircle?

The inscribed circle and circumcircle are both types of circles that can be drawn inside or around a shape. The inscribed circle is the largest circle inside a shape, while the circumcircle is the smallest circle that can enclose a shape, touching all vertices. In some cases, the inscribed circle and circumcircle may be tangent to each other.

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