I like to think about a sector of a circle as having the same shape as a slice of pizza. This first part of the solution requires that we calculate the area of one slice of a circular pizza with a central angle of 60 degrees.
Area for a sector of a circle = ( central angle / 360 )(π)(r)2 where r is the radius.
= 60/360 (π) (23 inches)2
= 1/6 (π) 529
= 276.8433333 square inches
Area for a triangle is 1/2(b)(h) where b is the base of the triangle and h is the height of the triangle.
1/2(23 inches)(h inches)
To calculate the height of the triangle, we can bisect any of the angles with a line that is perpendicular to the base. Essentially we create two right triangles that when added together equal the whole equilateral triangle. Because we have a right triangle, we can use the pythagorean theorem to calculate the height of the right triangle which also happens to be the height of the equilateral triangle.
a2 + b2 = c2
a2 + (23/2)2 = 232
a2 + 11.52 = 529
a2 + 132.25 = 529
a2 +132.25-132.25 = 529-132.25
a2 = 396.75
a = 19.918584 inches, which is also the height or h
Area of the triangle is 1/2(b)(h)
1/2(23 inches)(19.918584 inches) = 229.0637 square inches
Area of the shaded region in the problem is equal to the Area of the Sector minus the Area of the Triangle.
Ashaded = Asector - Atriangle
= 276.8433333 - 229.0637
= 47.78 square inches