Help, Please! Math ...

Question

 is an equilateral triangle with side length 23 in. What is the area of the shaded segment? Round your answer to the nearest hundreth.

Answer 1

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.

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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.

A_shaded = A_sector - A_triangle

= 276.8433333 - 229.0637

= 47.78 square inches

Answer 2

Since cad is an equilateral triangle then angle cad is 60° knowing this and that the radius is 23 in then the area of the sector that is shaded is A= 60πr²/ 360 = 3.141(23²)/6 A= 276.93 in²

Answer 3

The above answer gives the area of the entire sector enclosed byCAD

The area of the very small shaded equals the area of the small sector minus the area of triangle CAD. Angle CAD IS 60 degrees since triangle CAD is an equilateral triangle

So area of shaded portion is (60/360)(pi)rsquared=

(1/6)(pi)(23)sq= 276.99

Now we need to find the area of the triangle. Base is 23

If we drop a perpendicular bisector from a to be we create a pair of 30-60-90 triangles; we can use one of these to solve for the altitude of the triangle (AB)which is the side opposite the 60 degree angle. this equals

(23/2)sq root of 3

So area of the triangle is (1/2)(23)(11.5)sq root of 3= 229.06

276.99-229.06= 47.93 square inches