The complicated area?

Geometry Level 3

Find the total area denoted by the red colour.


The answer is 19.504.

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

My way of solving this question was to subtract areas of other shapes. You will know it when you go through my answer. Hope you enjoy!

Solution part 1: \large \color{magenta}\text{Solution part 1:}

Solution part 2: \large \color{magenta}\text{Solution part 2:}

Here is a simplified diagram of what we are trying to find: \large \color{magenta}\text{Here is a simplified diagram of what we are trying to find:}

This is the most complex part: \large \color{magenta}\text{This is the most complex part:}

A bit of explanation to the previous diagram: We start out by making a new sector of the circle [Please note that the area of the sector is given in radians] which is then divided to form an isosceles triangle and then we calculate the area of the 2 shapes as shown on the right side of the image. \large \color{#3D99F6}\text{A bit of explanation to the previous diagram: We start out by making a new sector of the circle [Please note that the area of the sector is given in radians] which is then divided to form an isosceles triangle and then we calculate the area of the 2 shapes as shown on the right side of the image.}

The Final part: \large \color{magenta}\text{The Final part:}

Hope you understand :))

3 Helpful 0 Interesting 0 Brilliant 0 Confused

i think it will be much simpler to use calculus to find area .

Sabhrant Sachan - 3 years, 8 months ago

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I didn't learn calculus yet. I will learn in a few months time. And I also wanted to show how to solve it using basic shapes. Why don't you give a solution using calculus?

Syed Hamza Khalid - 3 years, 8 months ago

The ratio of the area of the red region in the top right corner to the total area of the top right corner region is the ratio of angles (as in the case of circular sectors) which is arctan(2)/(pi/2) = 1.107..../1.570....... = 0.704.....

There are a total of 8 corner regions with a total area of 200-50 pi (Essentially, the area of the square - area of the 2 circles). So, area of each corner region is : (200-50 pi)/8

Other than the top right corner, there are 3 such regions shaded red

So, the total area of the region shaded red = 3* (200-50 pi)/8 + 0.704..... (200-50*pi)/8

Initially attempted the same approach as the above for finding the area of the curvilinear triangle but ran into problems relating to relation between external and central angles relating to the same arc for a non-unit circle

Sundar R - 3 years, 8 months ago

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