A plot of land is bounded by a 140-degree circular arc and two 80-foot radii of the same circle.

Question

Find the perimeter of the plot, as well as its area.

Answer 1

Hi Moxa,

 

The first thing to do here is to draw a picture of the plot. Imagine a circle of radius R, but only consider the upper hemisphere, i.e. the semicircle above the x axis. Draw a radial at 20 degrees above the positive x axis. Draw a second radial at 20 degrees above the negative x axis. These radials form the straight edges of a circular sector. The section of the circular circumference between the ends of these radials is an arclength, and the length of an arclength s is equal to the product of the radius R and the radian measure θ.

 

The perimeter is the distance around a figure. The distance along the straight edges is 2R, which is an R for each radial. The arclength is Rθ, where θ is in radians.

 

What is the radian measure of 140 deg? θ = (140 deg)(π radians / 180 deg) = 7π/9

 

- So, the perimeter is 2R + 7πR/9 = R(2 + 7π/9) = (80)(2 + 7π/9) = 515.48 feet

 

- Regarding area, we know the area of a circle of radius R is πR². So, the area of a semicircle is (1/2)(πR²). But, this circular sector is 7/9 of a semicircle. So, its area is (7/9)(1/2)(πR²) = (7/18)(πR²) =(7/18)(π)(80)² = 7819 feet².