Calculate area of the Green shaded region | 3 identical intersecting circles | Circle radius is 7

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Calculate area of the Green shaded region | 3 identical intersecting circles | Circle radius is 7

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[MenaDemian]

I am sorry but when you assumed the circles intercest in the middle, when you said its not to the scale. What if the center of the middle circle is below the line you drew.

[keithwood]

Cut each green area lengthwise with a chord. You now have four identical areas. Distribute these around the inside of the circle. Interior to them they define a square with diagonal 14. Area of the green regions equals area of the circle minus area of the enclosed square, which turns out to be 49pi-98.

[tylerwright]

What an awesome problem and an elegant solution! Blessings to you!

[wackojacko]

It's so cool when dropped perpendiculars and similarities between shapes are revealed too simplify solutions. I absolutely love it.!🙂

[j.r.]

The given conditions should have included a statement that the middle circle intersects the tangent point of the two lower circles. Just eyeballing it as such isn't really enough.

[EnnioPiovesan]

Very good solution. You could also consider that, if yuo call T the intersection of the three circles (the midpoint of segment AB) the green area is four times that of the circular segment subtended by the chord DT (or CT). But the area of the circular segment is that of the quarter of circle DAT minus that of the triangle DAT. So, called As the area of the segment:
As = (1/4)(π·49) - (1/2)·49=(49/4)·(π-2).
So the green area is 49(π-2).

[williamwingo]

Here's a different and possibly simpler method:
Consider the left-hand green area. Draw a chord cutting it in half.
The portion of the left-hand green area below and to the left of the chord is equal to one-fourth the area of the upper circle minus the area of an isosceles right triangle with legs of length seven. That is,
49π/4 – (1/2)(7)(7) = 49π/4 – 49/2 = 49(π/4 – 1/2);
But the total green area is four times this, so we have
(4)(49)(π/4 – 1/2) = 49(π – 2) = 49π – 98.
Cheers. 🤠

[montynorth]

Joining points A, B, and O.
AB = 14.
The height of O above line AB = 7.
Therefore, angle ABO = 45 degrees.
Therefore, the green areas each subtend an angle of 90 degrees with respect to their centres.
Then calculating the area of a 1/4 circle, subtracting the area of a 7 x 7 right angled triangle, then multiplying by 4.
[ ( Pi x 49 /4 ) - (1/2 x 7 x 7 ) ] x 4 = 55.94.

[theoyanto]

Another masterpiece... I did it the other way, but more than one way is good to see🤓👍🏻

[neelaramramesh]

May I know, who inspired you to enlighten in scopes of math

[mohanramachandran]

Quarter. Circle. - Triangle. =1/4 Green Area
( 1/4. * π * 7 * 7 ) - ( 7 * 7 ÷ 2 ) =
1/4 Green Area
38.4845. - 24.5 = 13.9845
13.9845 * 4 = 55.938
Ans = 55.938 square units

[AnujMaurya]

Thanks professor!!
Mensuration also cover so help for exam .. ❤

[solimana-soli]

Let's call H the intersection point of the three circles, then DH segment, which divides the intersection of the circle with center O and the circle with center A into two equal parts, can be calculated with the Pythagorean theorem
DH =sqrt(DO^2+ OH^2)= 7sqrt(2).
DH is also the chord of the circle with center A and we can calculate the area of the relative circular segment by subtracting the area of the right triangle BOA (7*7*0.5=49/2)
from the area of the circular sector BOA (7^2)/ 4*pi = 49/4*pi
and multiply the result obtained by 4
4*(49/4*pi - 49/2)
This leads to 49pi - 98

[trombone_pasha]

so the general formula for this case is πr^2-2r^2

[ybodoN]

As a formula: the green shaded area is _r² (π - 2)_ where _r_ is the radius of the circle.

[yvessioui]

There ia a point missing in the demo, you have first to justify/demonstrate that both top 2 intersection points and O are colinear. No given data can tell us that withouth some kind of reasoning. That point is important to justify talking about a semi-circle. As coarllary, A, B and the bottom intersection point are also colinear and therefore both lines are parallel.

[zsoltszigeti]

1/4 of the green area is 1/4 circle area (49pi/4) minus 7x7 right-angled triangle area (49/2) => green area is (49pi/4-49/2)*4 = 49pi-98

[vidyadharjoshi]

4*(Area of arc - Area of triangle) = 4*49*(pi/4 - 1/2) = 55.94

[santiagoarosam]

(14²/2) - (14² - 7²Pi) =7²(Pi - 2) =55.94
Gracias y saludos.

[alinayfeh]

Green Area = semicircle Area- Brown, radius (r)=7, Area circle =76.969, 112-76.969=21.03 Brown Area, 49π-98=55.938