Math Help - Finding the area of the shaded region (2)

2. Re: Finding the area of the shaded region (2)

Originally Posted by Mhmh96
You can make an equilateral triangle of side length 4 units (each side is made up of two radii from the circles), which you can use Heron's Formula to evaluate.

The shaded area is then the area of this triangle minus the areas of the three circular sectors inside that triangle.

3. Re: Finding the area of the shaded region (2)

Can you show how to use Heron's Formula here?

4. Re: Finding the area of the shaded region (2)

Originally Posted by Mhmh96
$A=\frac{(2r)^2 \sqrt 3}{4}-3\cdot \frac{r^2 \pi \cdot 360^{\circ}}{60^{\circ}}$

where r=2

5. Re: Finding the area of the shaded region (2)

Herons formula is not required. It provides a method to find the area of a triangle from the three sides but we are dealing here with 30-60-90 triangles where the area is readily calculated using the 2-1- rad3 relationship. prnceps solution has a typo error.Its 60/360 not 360/60 in the area for the three sectors.Google Herons formula to see it

6. Re: Finding the area of the shaded region (2)

This is relatively old post..but i was wondering if there another method to solve it ..other than using what princeps used and Heron's Formula.

7. Re: Finding the area of the shaded region (2)

Originally Posted by Mhmh96
This is relatively old post..but i was wondering if there another method to solve it ..other than using what princeps used and Heron's Formula.
I expect there are many other methods, none of which are as straightforward as the ones posted by myself or Princeps, and all of which would be pointless to try to explain if you are not willing to attempt to master the straightforward methods you have been shown.

8. Re: Finding the area of the shaded region (2)

In fact i solved this problem before just like princeps ,which is drawing an equilateral triangle then the area of the shaded region =Area of triangle-3*(Area of sectors),but i was looking for other way

9. Re: Finding the area of the shaded region (2)

Dear Mhmh96,
Heron's Formula for area of triangle given three sides

A=sq rt of s(s-a)(s-b)(s-c) where s= 1/2 of perimeter and in my opinion is the long way to find the area of an equilateral triangle. Princips uses the property of a 30-60-90 triiangle whch ought to be known by a geometry student but does not describe it that way