Circumference of circle around equilateral triangle?

Question

I need to put up deer fencing around individual fruit trees. How much fencing do I need to buy? I plan to put up three posts in an equilateral triangle around each tree. If the posts are 4 feet apart, and I string chicken wire in a circle around them, what length of fence do I need for each tree? That is, if I make an equilateral triangle with sides of length 4, what is the circumference of the circle that I would draw around the triangle?

Suppose I need to change the lenght of the sides to 3 feet or 5 feet? What is the general procedure/formula for calculating the circumference of such a circle? How do I calculate the diameter of the circle, based on the length of the sides of the equilateral triangle?

Answer 1

This will be easier for you if you sketch the triangle and the circle, so start with that.

Next, divide the triangle into three triangles by drawing three lines, each from the corner of the triangle to the center. Now, you have three congruent triangles, each measure 120º x 30º x 30º. You'll notice that the lines you drew each equal the radius of the circle.

Next, take one of the 3 smaller triangles, and draw a line that separates the triangle into two equal triangles. Now, you'll notice that each of the two smaller triangles measures 30º x 60º x 90º, and the hypotenuse of the smaller triangles is the radius of the circle, and the longer of the legs is half the length of the side of the original equilateral triangle.

Using trig, cos30 = 2/r --> r = 2/cos30

C = π d = 2 π r = 14.51.

In general, if the posts are x feet apart, then C = π x / cos30.

Answer 2

This sounds more like homework to me!

Anyway the if the length between posts in an equilateral triangle is L, then the radius of the circle will be

R*cos(30) = L/2

from geometry. cos(30) = sqrt(3)/2 so

R = L/sqrt(3) then the perimeter is (per circle)

P = 2*pi*R = 2*pi*L/sqrt(3)

if L = 4, P = 14.5104
if L = 3, P = 10.8828
if L = 5, P = 18.1380

Answer 3

if u want to put up a circle you will need the diameter to be the longest length in the triangle which would be 4. so to figure that out you should use the circumference equation for a circle which woul be:

C=3.14xd
where c=circumference and the d=diameter

C=3.14x4

C=12.57

u can use the same equation if u want to find the circumference if u change the length to 3 or 5.
Ex:
for 3 C=3.14x3
for 5 C=3.14x5

Answer 4

The radius of the circle is

(2/3) * 4 * cos 30° = (2/3) * 4 * (3^0.5) / 2

continue ... it's easy

Answer 5

If R ft is the radius of the circle, then:
4 / sin(60deg) = 2R
R = 2 / sin(60deg)
= 2 / (sqrt(3) / 2)
= 4sqrt(3) / 3

The circumfrerence is:
2pi*R
= 8*pi*sqrt(3) / 3
= 14.51ft.

Just over 14ft 6in. per tree, or 43ft. 6in. for three.

Answer 6

14.51ft

x=(a)sqrt(3)/3= radius where a is the distance between posts

circumference=2pi*r