Eli5: Finding the area inside of a circle but outside of an inscribed equilateral triangle?

Question

Can someone walk me through how to do this? I’m in a Pre-Calculus class and generally am really comfortable with math and mathematical concepts, but this is twisting my brain apart and I need help lol

In: Mathematics

Answer 1

you know the triangle is equalateral, so all its sides are the same. you know the radius of the circle it is inscribed in.

the radius is the length of a line from the circles edge to the center. the circles edge is where all 3 points of the triangle are (meaning of inscribed) so draw a line from 2 of the triangle points to the center. you now have an Isosceles triangle with sides r,r, and x. you know the angles as well, you cut 2 60 degree angles in half, ao thats 30, 30, and whatever adds to make 180 (120). now you can use law of sins to find x.

alternativly, you can further cut the isoscelese trianglemin to 2 right triangles with hypotenuse r, angles 60 30 90, one side x/2, and the other unknown. for a 30 60 90 triangle, the sides are hypotenuse 2x, side opposite 30 x, other side x√3, so you now know that the triangle you just found has sides r, r/2, and x/2=r√3/2. you couks find the area of this triangle and multiply by 6, or you now know x is r√3, and the triangle height is 3r/2 (r/2+r),

now all you have to do is apply triangle area formula base*height/2

but what you want is the area of the circle bits. so subtract that area from the circle (πr^2 )