# How can i Prove that the gray area is the same as white area? [duplicate]

This question already has an answer here:

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area?

I just want any hint for solving this question. how can I prove this?

## marked as duplicate by Martin R, Xander Henderson, Hayk, Blue geometry StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 22 at 13:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• The area of a triangle with respect to given two sides $a,b$ and angle between the sides is $0.5 a b \sin \theta$ s.t $\theta$ is the angle between the two sides can you proceed ? – Ameryr May 22 at 7:56
• This is the Pizza Theorem. See here: en.wikipedia.org/wiki/Pizza_theorem – Christian Blatter May 22 at 8:26
• – Martin R May 22 at 11:13

## 2 Answers

Carter and Wagon's proof without words:

(Image author: Christian Lawson-Perfect, source)

• Thanks for Help. – Pankaj Solanki May 22 at 8:53

This theorem is known as the Pizza theorem (see https://en.wikipedia.org/wiki/Pizza_theorem for further reading). There exist algebraic proofs, but the most elegant proof for the 8 segment case is below:

In general, alternating areas are equal iff the number of segments is a multiple of 4 at least 8.

• How are segments counted? If this is eight from the concurrence point, then four will not work. – Oscar Lanzi May 22 at 13:44
• @OscarLanzi Sorry about that, I've fixed my answer. – auscrypt May 22 at 13:46