×

# Finding Area

I posted a problem "How Much Can a Goat Eat #2". Using a false assumption I calculated the overlapped area incorrectly. I request assistance in this regard. Please calculate the area of the green colored figure in this image. Thanks.

Note by Guiseppi Butel
3 years, 4 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

yup its a good sum

- 3 years, 4 months ago

It looks like there are several ways to go about this, but this is my initial thought. First, we have $$AB = 3\sqrt{3}$$. Thus $$\theta = \angle CAD = \cos^{-1}(\frac{\sqrt{3}}{4}) - \frac{\pi}{6}$$. The area of the green region will then be the area of sector $$CAF$$ minus the area of $$\Delta CAD$$. This comes out to

$$(\frac{1}{2})(6^{2})\theta - (\frac{1}{2})(3)(6\sin(\theta)) = 18\theta - 9\sin(\theta)$$.

This "simplifies" to

$$18\cos^{-1}(\frac{\sqrt{3}}{4}) - 3\pi - (\frac{9}{8})\sqrt{3}(\sqrt{13} - 1)$$,

which equals $$5.7115$$ to $$4$$ decimal places.

- 3 years, 4 months ago

Thanks, Brian. I obtained the same answer by using a different method. In the sector ACF I calculated A by Law of Sines to be 34.34109373 degrees, altitude from AC to D to be 1.692355197.

Area of the green = Area of the sector - Area of triangle ADC

Area of the green = (34.34.../360 * Pi * 6^2) - (1/2 * 6* 1.69..)

Area of the green = 10.788573 - 5.0770656 = 5.7115074

This amount was taken to the problem "How Much Can a Goat Eat #2" and a corrected answer was obtained.

- 3 years, 4 months ago

Glad we came to the same conclusion. I enjoyed the goat problem; thanks for posting it. :)

- 3 years, 4 months ago

as i tried to solve it i got 4.9 as area of shaded region

- 3 years, 3 months ago

what is the correct answer of this problem ????

- 3 years, 3 months ago

Look back at what Brian and I posted on Oct. 20.

- 3 years, 3 months ago

5.748

- 3 years, 4 months ago

5.7115 Where did you err?

- 3 years, 4 months ago

it can be solved by using integration in a very easy manner .just shift the whole diagram on a xy plane and do the integration by writing the equation .

- 3 years, 4 months ago

The area is 8.58

- 3 years, 4 months ago

How did you get that answer?

- 3 years, 4 months ago

4.92 square units

- 3 years, 4 months ago

49

- 3 years, 4 months ago

Yup got 5.711

- 3 years, 4 months ago

5.711

- 3 years, 4 months ago

wow

- 3 years, 4 months ago