# Maximise perimeter of triangle in a circle

• twinkle21
In summary, the conversation is about finding the maximum perimeter for an equilateral triangle inscribed in a circle. One person has tried a few methods involving algebra, while another suggests using the sine rule or joining the vertices to the center of the circle to avoid square roots. The conversation ends with a request for pointers and a thank you.
twinkle21
Hey guys, I hope someone can give me some pointers with this because it should be really easy but I am just not getting it!

I want to show that for a triangle inscribedin a circle an equilateral traingle gives the maximal perimeter. I've tried a few things and just get bogged down in algebra and I am sure there should be a clean geometric proof!

For example if you take a unit circle on the origin then I can set one of my points at the north pole (0,1), then in polars assign the other 2 points at B and C. But this gives me the problem of maximising 2sin(C/2) + 2sin(B/2) + sqrt(2-2cos(C-B)) which is very messy... can anyone give me some pointers?

Thank you!

Try joining the vertices to the center of the circle, and find the perimeter in terms of the angles at the center. That won't involve any square roots.

Or start from the sine rule: ##a / \sin A = b / \sin B = c / \sin C = 2R## where ##R## is the radius of the circle.

## What is the "Maximise perimeter of triangle in a circle" problem?

The "Maximise perimeter of triangle in a circle" problem is a mathematical problem that involves finding the largest possible perimeter of a triangle inscribed within a circle. This is typically done by varying the dimensions of the triangle and using mathematical techniques to determine the maximum perimeter.

## Why is this problem important in science?

This problem is important in science because it has practical applications in fields such as engineering and physics. For example, in engineering, this problem can be used to determine the maximum size of a circular foundation for a building. In physics, it can be used to determine the maximum area of a solar panel that can be inscribed within a circular space.

## What are the key steps in solving this problem?

The key steps in solving this problem involve setting up the problem by defining the variables and constraints, finding the derivatives of the equations, and then using techniques such as the derivative test or Lagrange multipliers to find the maximum perimeter. This is typically followed by verifying the solution and providing a proof of the result.

## What are some common challenges in solving this problem?

Some common challenges in solving this problem include finding the appropriate equations to represent the problem, determining the appropriate constraints, and applying the correct mathematical techniques to find the solution. Additionally, the problem can become more complex when dealing with triangles that are not equilateral or when the circle is not centered at the origin.

## What are some real-world applications of this problem?

Some real-world applications of this problem include determining the maximum area of a circular garden or playground, finding the maximum size of a circular window for a building, and optimizing the size of a circular dish for a satellite. It can also be applied to various other engineering and scientific problems that involve circular objects or spaces.

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