Calculate area of the Green shaded region in between three identical circles | Circle radius is 2

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Calculate area of the Green shaded region in between three identical circles | Circle radius is 2

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[AnonimityAssured]

I think I can see a very simple way of doing this.

Spoiler alert.

The green area will be the area of an equilateral triangle of side-length 4 minus the area of a semicircle of diameter 4.

Area of triangle = 4√3.

Area of semicircle = 2π.

Green area = 4√3 − 2π = 2(2√3 − π) square units.

Numerical equivalent ≈ 0.645 square units.

[alster]

Thanks for bringing my memory of the Sector Area back

[bobeyay]

I've been watching your content for some while now and finally for the first time in a long time, I manage to solve one! Thanks for making my interest in Math come back again, it was one of my favorite subjects back in my days in school

[vidyadharjoshi]

ht of triangle ABC = sqrt(16-4). Area ABC = 0.5*4*3.46 = 6.93. Green Area = = Area of triangle ABC - 3*60 deg part of circle. = 6.93 - 2pi = 0.645

[batavuskoga]

My first thought was drawing a square around the circles for calculating. Then I saw I realised it wasn't possible to find the solution this way.
Suddenly I saw it. I've drawn a triangle like in the video and find the solution as shown in the video.
And of course I found the solution myself and then I checked if I've done it the right way.

[muttleycrew]

Enjoyed solving this one. Solving these geometric problems is especially satisfying for whatever reason.

[jan-willemreens]

... Good day, Coincidentally I derived the general formula of the area in between 3 identical circles with radius R (lol) ... so Area = [ SQRT(3) - pi/2 ] * R^2 ... finally applying to your problem, where R = 2 ... A(Green shaded region) = 4*SQRT(3) - 2*pi ... thanks for your presentation .... best regards, Jan-W

[lindafromcalifornia]

That was so beautifully explained. Splendid job Pre-Math.

[alexundre]

Good Morning MASTER
Thanks Sir
A Hug from Rio de Janeiro

[erautome]

1)calculer la hauteur du triangle avec Pythagore. 2) calculer la surface d'un cercle. 3) calculer la surface du morceau d'un cercle inclus dans le triangle. Le triangle étant équilatéral, les angles sont de 60° ou 1/6 de l'aire du cercle. On multiplie l'aire des trois quartier de cercle par 3 et on soustrait de l'aire du triangle.

[nilsalmgren]

That one is easy to see how it is done. I figured it out in about 30 seconds, including the sectors of the three circles making half a circle.

[HappyFamilyOnline]

Awesome👍
Thank you so much for sharing.😊

[MrPaulc]

Without a calculator or putting pen to paper I calculated the area of the equilateral and the 1/6th circles and ended up with 4root3 - 2pi. The calculator gives that as 0.645(3dp) sq units. I'm fairly ancient and trying these puzzles helps keep my mind alert and active.

[sorourhashemi]

Thanks. I got 0.648 square units. For getting area I used Heroin formula= rad.48

[misterenter-izrz]

The answer is the difference of area of the triangle and the semicircle=(4x4 root 3/2)/2-pix2^2=0.645 approximately.

[williamwingo]

No peeking:
Consider triangle ABC. Invoking Pythagoras, altitude = 2 sqrt(3); base = 4; and area = (1/2)(2)(sqrt(3))(4) = 4 sqrt(3).
The three 60-degree circle segments inside the triangle add up to 180 degrees, which is half the area of one of the circles, or (1/2)(4)(pi) = 2 pi.
The green shaded area is the difference, or 4 sqrt(3) -- 2 pi.
Staying sharp, one problem at a time.... 🤠

[piersonm]

Haven't seen it yet, my guess is you construct an equilateral triangle ABC, then subtract the area of the three segments? The angle is 60°, convert into radians and get the three areas, subtract from ABC's area

[rudychan]

Angles Are the Same.
No Need Phytagoras there, Pal. 😓
Area = (1/2)•4•4•sin60° =4√3
Sectors = 3×(60°/360°)×π×2" = 2π
Green = 4√3 - 2π = 0.645 units"

Anyway, I give U #161st thumb's Up 🙂 👍

[bentels]

Solved it the same way (without expounding on the height of the triangle, obviously 😅).

[wackojacko]

The approximation is what's killing me here! Because I like exact! So I'm thinking because Pi and 3^⅓ are so Fractal so too speak... I don't know, just throw the resulting 0.645 square units out the curved space window. I'm just trying too be funny...I absolutely love your channel.