Can you calculate the shaded area?

[u/MathorionMath]

[u/stools_in_your_blood]

Join the centres of the three small circles and you get an equilateral triangle with side length 2.

The non-blue area is therefore the area of this triangle + the area of the three circles, each with a 60 degree slice removed.

So the blue area is area of big circle - 3 * 5/6 * area of small circle - area of triangle.

Fiddle around with the trig and this comes to something like (4/sqrt(3) - 1/6) * pi - sqrt(3), although I did that in my head so it's probably wrong.

[u/Icy_Sector3183 → u/stools_in_your_blood]

I missed the part about getting at the shaded area, so I went for the area of the equilateral triangle of sides 2, minus half the area of a circle with radius 1 (three 60 degree arcs make one 180 arc, half a circle)

Area of equilateral triangle with sides = 2 is sqrt(3)

Area of circle with r = 1 is pi.

Wrong solution is sqrt(3) - 1/2 pi

:)

[u/BrotherInJah → u/stools_in_your_blood]

Yea.. that triangle cuts of area of half circle.

[u/PiasaChimera]

it will be the area of the blue circle minus the area of 3 partial circles of 300deg (5/6ths), radius 1 then minus the remaining equilateral triangle of side length 2. the center of that triangle is the center of the blue circle. the distance of the center to any corner is 2/(sqrt(3)), making the radius of the blue circle 1+2/sqrt(3).

this gives the blue circle area as ~14.5855. the area of the partial circles as ~7.8539, and the center triangle as ~1.7321, for a shaded area that looks to be 5.

[u/White_Rabbit_26]

Same as above, the larger circle minus the three small circles minus the area in between the small circles. Start with the equilateral triangle made by connecting radii of the small circles. Area of large circle minus 3 small circles minus internal area = (7+4sqrt(3))/3 x pi - 3 x pi - (2sqrt(3) - pi)/2 =(2sqrt(3) x (4 x pi - 3) - pi)/6 = 4.999547873769695 cm^2 ~ 5 cm^2.

[u/BadJimo]

Illustrated on Desmos

[u/imscreamingeternally]

no :(

[u/Automatic-Fan9297]

1. Connect all the centers of smaller circles to make equilateral triangle of side 2 units. This triangle covers a 60 degree sector from each of the circle. Remaining area not covered by triangle in each circle(300 angle out of total 360) = 5/6 area of circle.

2. Area of unshaded region = Area of triangle + 3(remaining area of circle) = (√3/4)22 + 3(5/6)π1*1 = √3 + 2.5π.

3. Radius of bigger circle = circumcircle radius of equilateral triangle + radius of small circle = 2/√3 + 1

4. Area of shaded part = area of big circle - area of unshaded region = π*(2/√3 + 1)² - (√3 + 2.5π) = π(7/3 + 4/√3) - √3 - 2.5π = π(8√3 - 1)/6 - √3.

[u/Purdude1983]

Connect the centers of the three white circles to make an equilateral triangle with side lengths of 2. It's area is sqrt(3). That leaves 3 circles with a wedge missing that have a pac-man shape. These each have an area of 5*Pi/6. The radius of the outer circle is 1+2/sqrt(3) which can be rewritten as (3+2*sqrt(3))/3 The area of the blue circle is then Pi*(7+4*sqrt(3))/3 The blue shaded area is then Pi*(8*sqrt(3)-1)/6-sqrt(3) which is about 4.99954787376969 So close to 5, but not quite.

[u/Copy_Outrageous]

Both deepseek and Google ai gave the wrong answer for the first time, they did not take into account that the central area was not shaded, only after pointing it out, they calculated it correctly.

So trust the AI after that.