Find The Area

The blue and white circles combined have half the area as the outside circle. What is the area of the red part relative to the area to the whole circle?

Note: The parts are formed by lines tangent to the white circle and when the borders between colors are extended, it forms an equilateral triangle.

#Geometry

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Is it not 1/3*1/2=1/6? Or you mean full white circle + full blue circle?

Saya Suka - 4 years, 6 months ago

Prove it

alex wang - 4 years, 6 months ago

The red part, green part, and yellow part must all have an equal area. This is simply because of symmetry and someone who is not as lazy as I am can prove it in full. The total area of these consists of half of the full logo. This means that it is 1/3 of 1/2 or 1/6 as said by Saya Suka.

Ayush Kumar - 3 years, 8 months ago

You are right but

alex wang - 4 years, 6 months ago

If that had been the case then I'm pretty the sure that there would be multiple answers.

Ayush Kumar - 3 years, 8 months ago

yeah its just as simple as half the area of the big circle divided by three because each of those coloured sections out side the white and blue circle combined are congruent figures

Krishna Karthik - 2 years, 8 months ago

nice job and they are congruent

alex wang - 2 years, 7 months ago

not similar

alex wang - 2 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$