A regular hexagon with sides of 3" is inscribed in a circle. Find the area of a segment formed by a side of the hexagon and the circle
just look up the formula for the area of a segment of radius r and angle θ:
a = 1/2 r^2 (θ-sinθ)
Now plug in r=3 and θ=π/3
To find the area of the segment formed by a side of the hexagon and the circle, we first need to find the area of the sector formed by the hexagon and the circle. Then, we will subtract the area of the triangle formed by the side of the hexagon to get the area of the segment.
Let's break down the solution step by step:
Step 1: Find the radius of the circle
In a regular hexagon, the distance from the center of the hexagon to any vertex is equal to the radius of the circle. Since the side length of the hexagon is given as 3", the distance from the center of the hexagon to any vertex is half the length of a side, which is 1.5".
Step 2: Find the area of the sector
The formula for the area of a sector is given by A = (θ/360) * π * r^2, where θ is the central angle of the sector in degrees, and r is the radius of the circle.
In a regular hexagon, each central angle is 360/6 = 60 degrees. Therefore, the central angle of the sector is 60 degrees.
Using the formula, the area of the sector is A = (60/360) * π * (1.5)^2.
Step 3: Find the area of the triangle
The area of a triangle can be found using the formula A = (1/2) * base * height.
Since the base of the triangle is equal to the length of a side of the hexagon, which is 3", and the height of the triangle is equal to the radius of the circle, which is 1.5", the area of the triangle is A = (1/2) * 3 * 1.5.
Step 4: Find the area of the segment
To find the area of the segment, we subtract the area of the triangle from the area of the sector: Segment area = Sector area - Triangle area.
Substituting the values from the previous steps, we have Segment area = [(60/360) * π * (1.5)^2] - [(1/2) * 3 * 1.5].
Calculating this expression, we get the area of the segment formed by a side of the hexagon and the circle.