# Thread: Area of Goat Grazing

1. ## Area of Goat Grazing

A goat is tied to the corner of 7 foot square barn. It has a 20 foot long rope. What is the area of its grazing? I thought it would be 300*pi+169*pi/4-1/4*36*pi. However I got about 1179 and the correct answer rounds to 1176. Did I make a mistake or did I do it wrong. I said 300*pi because 3/4 sector of a 20 foot radius circle, 169*pi/4 because of two quarter sectors of 13 radius circles, and -1/4*36*pi, because the two radius thirteen circles overlap each other by 6 foot radii in a 90 degree circular arc.

2. Originally Posted by empireruler
A goat is tied to the corner of 7 foot square barn. It has a 20 foot long rope. What is the area of its grazing? I thought it would be 300*pi+169*pi/4-1/4*36*pi. However I got about 1179 and the correct answer rounds to 1176. Did I make a mistake or did I do it wrong. I said 300*pi because 3/4 sector of a 20 foot radius circle, 169*pi/4 because of two quarter sectors of 13 radius circles, and -1/4*36*pi, because the two radius thirteen circles overlap each other by 6 foot radii in a 90 degree circular arc.
1. I've attached a drawing.

2. There are 3 different areas to calculate:

Threequarter of a circle with radius r = 20: $a_1=\frac34 \cdot \pi \cdot 20^2$

Two sectors with radius r = 13 and the central angle $\alpha = \arctan \left(\frac{12}5\right) \approx 67.38^\circ$
Thus the area is $a_2 = 2 \cdot \frac{67.38}{360} \cdot \pi \cdot 13^2$

Two triangles with the base length of 7 and the height of 5: $a_3 = 2 \cdot \frac12 \cdot 7 \cdot 5$

The complete area is $a = a_1+a_2+a_3$

I've got $a \approx 1176.222$