what is the side length of a triangle circumscribing a circle of diameter x?

Question

Circle is inscribed in an equilateral triangle; want to know the relation between the side of triangle and diameter of circle.

Answer 1

Image at http://i127.photobucket.com/albums/p152/penta10au/CircumTriangle.jpg


tan 30° = (x/2)/(a/2)
= x/a

So 1/√3 = x/a

Therefore a = √3 * x

The side length is √3 * diameter of the circle

Answer 2

Draw the picture.
You'll see that the radius of the circle is equal to the distance from the center of the triangle to its base. Since the equilateral triangle is 60 degrees all around, you can dissect it into six 30 60 90 triangles, and solve for x.

In a 30 60 90 triangle, if the short side is x, then the next larger side is sqrt3 x and the longest side is 2x. This is because of the pythagorean relation a^2 + b^2 = c^2 and the fact that the short side of a 30 60 90 triangle is half of the longest side. If you look at the way your dissected triangle sits over the circle, you can see that x is the radius of the circle and so each side of the circumscribing equilateral triangle would have sides length 2x sqrt 3.

This is a lot easier to explain with a picture, but this media doens't draw pictures. So draw the picture and you'll see what's going on here.

Answer 3

Draw a line from the center of the circle to a corner of the circumscribed triangle, and a line from the center to the point where the circle and the triangle intersect. These 2 lines and half of one side of the triangle form a 30-60-90 triangle, and you should know the sides have a ratio 1: √3: 2. Since the short side is the radius of the circle, the middle side is r√3, and so the length of one side of the equilateral triangle is 2r√3, which is d√3, diameter times √3.

Answer 4

the answer:
the triangle circumscribed will be an equilateral triangle,always

so, if the Diameter of circle is X then the radius will be X/2

therefore, Using the Cosine rule for triangle the angle will be 30degree
ie., Cos(30) =half the length of the sides of triangle/(X/2)
therefore,
Total Length = 2.Cos(30).X
= 2 . sqrt of 3/2 .X
Answer = 2.4624X

Answer 5

The triangle has three corners each 60 degrees.

Draw a circle.
Draw a straight line up-down through the center of the circle (extend well outside the circle).
Draw a straight line left right, touching the bottom of the circle (extend well left and right).
Draw a line on the right hand side of the circle, intersecting the horizontal line at 60 degrees and the vertical line at 30 degrees.
Draw a line on the left hand size of the circle, intersecting the horizontal line at 60 degrees and the vertical line at 30 degrees.

This completes the triangle.

The vertical line divides one angle of 60 degrees in two angles of 30 degrees. You can draw two more of such lines, in the two remaining corners.

You will find small 30-60-90 triangles, for example one in the lower left hand corner. One side is half of the horizontal line, one side is half of the diameter of the circle, one side is the long side.

Extend this triangle, multiple all sides by two. You have one side equal to the diameter of the circle, and one other side is one of the sides of the large triangle.

In a 30-60-90 triangle, sides are 1:root(3):2 so if the diameter is 1, the large triangle's side is root(3). In other words, multiply the diameter by root(3).

Answer 6

Assuming that the triangle is an equilateral triangle, all of the angles are 60° angles. If you draw a small triangle formed by one corner of the triangle, the center of the circle and the tangent point of one of the sides, you have a triangle with a right angle (at the center of the circle, a 60° angle (at the tangent point) and a 30° angle (at the corner point of the triangle, the angle included in the small triangle is half of the 60° angle at that point. The short side of this triangle is equal to the radius of the circle, or x/2. The long side of the little triangle triangle is the hypotenuse. The short side, equal to x/2 or x times 0.5, is equal to the hypotenuse times the sin of 30°. Sin 30° = 0.5. So Hypotenuse times 0.5 = x times 0.5; so the hypotenuse of the little triangle is equal to x, the diameter of the circle. The hypotenuse of the small triangle is one half of a side of the big circumscribing triangle, so the side of the big triangle equals 2 times x.

Answer 7

The triangle's heigh is Lsqrt3/2, where L is the side of the triangle.

The center of the triangle is the barycenter. The barycenter is the crosspoint of the lines who join the vertix and the middle point of the other side. The barycenter divides this line so that 2 parts of3 are next to the vertix.

The distance between the triangle's center and the vertix is (Lsqrt3/2) (2/3), and the diameter is the double of this distance

2L sqrt3/3 is the answer or 2L/sqrt3

Ana

Answer 8

A perpendicular raised on each side of the triangle will intersect the other perpendiculars at the centre of the circle. This perpendicular happens to be the radius of the circle and is also located at the centre of the side of the triangle. You can look up the exact book and theorem numbers in your geometry text. Trigonometry will let you calculate the length of the line connecting the centre of the circle and the apex of the triangle and also resolve the unknown side of the triangle. I simply don't have that formula to hand. Sorry.

Answer 9

In triangle ABC
let O be the centre of the circle. Draw OP perpendicular on AB where P is a point on AB.

then In Right triangle OBP
tanOBP = OP/BP
angle OBP = 30 deg ( triangle is equilateral)

tan 30 = OP/BP
1/sqrt(3) = OP/BP
BP = OP*sqrt(3)
thus
2*BP =2*OP*sqrt(3)

Side = Diameter * sqrt(3)

Answer 10

Let ABC be the equilateral triangle
Draw a line from C perpendicular AB at Point E.
Draw a line from A perpendicular BC at Point F
Then the line AF and CE intersect at Point O, the center of the
inscribed circle.

Now the triangle AEC similar to triangle OEA
Therefore CE/AE = AE/OE **
But OE is the radius of the incircle = r , and
AE = s/2 , where s =AC = side of equilateral triangle.
CE = (s/2)*sqrt(3)
Substituting the above into ** we get:
[(s/2)*sqrt(3)]/(s/2) = (s/2)/r
Cross multiplying we get:
r*[(s/2)*sqrt(3)] = s^2/4
r = (s^2/4)/[(s/2)*sqrt(3)]
Simplifying we get:
r= (s*sqrt(3))/6
So diameter = 2r = (s*sqrt(3))/3