How to cut a circle into 3 equal pieces

Answer

Verified

Hint: We will divide the circle into three parts by dividing it into 3 equal sectors of ${{120}^{\circ }}$ each. We are dividing circle into sectors of ${{120}^{\circ }}$ because a complete circle is of ${{360}^{\circ }}$ and to divide the circle into 3 equal parts, we will divide ${{360}^{\circ }}$ by 3, so we will get ${{120}^{\circ }}$.

Complete step-by-step answer:

It is given in the question that we have to draw a circle and divide it into 3 equal parts.We know that a circle is equal to ${{360}^{\circ }}$. So, on dividing the circle into 3 equal parts, we get 3 equal sectors of ${{120}^{\circ }}$ as $\dfrac{{{360}^{\circ }}}{3}={{120}^{\circ }}$.So, now we will divide a circle into 3 equal sectors of ${{120}^{\circ }}$ each. So, to do the same, we will follow a few steps as shown below.Step1: We will take a random radius and then we will draw the circle.

Step2: We will now draw ${{120}^{\circ }}$ angle. So, for that also, we will follow a few steps. We will use a compass to draw ${{120}^{\circ }}$.So, first, taking any radius on the compass, we will draw an arc with the center as O.

Then, with the same distance on the compass, we will place the compass at point P and then we will cut an arc on the existing arc. So, we get,

Now, with the same distance on the compass, we will place the compass at point Q, and then we will cut an arc on the existing arc. So, we get,

The point R obtained measures ${{120}^{\circ }}$ from point P in the anti-clockwise direction, so we get the angle ${{120}^{\circ }}$.

Step3: We will now draw another ${{120}^{\circ }}$, but this time, we will take the arc downwards. And we will follow step 2 again to draw ${{120}^{\circ }}$. So, we will get,

Step4: Since we have already measured two ${{120}^{\circ }}$, that is we get 2 equal sectors, so the angle in the reaming sector will also be ${{120}^{\circ }}$, so we will get,

Thus, we have divided the circle into three equal parts, that is, three equal sectors of ${{120}^{\circ }}$ each.

Note: Many students divide the circle into three parts as any one of the follows.

But this is not a correct method as the division by this method does not divide the circle equally. Also, there are high chances of errors, as it is a circular shaped figure. Thus, it is recommended that the students follow the correct method to solve this question. Also, the students must remember that the circle has the measure of $360˚\circ$ so to divide the circle to ‘n’ equal number of parts, we will write it as, $\dfrac{{{360}^{\circ }}}{n}$.

Too long for comments

If you want to solve $$\theta+\sin(\theta)=\frac\pi3$$ what you can do is to expand the lhs as a Taylor series and use series reversion. Using ths simple $$y=\theta+\sin(\theta)=2 \theta -\frac{\theta ^3}{6}+\frac{\theta ^5}{120}-\frac{\theta^7}{5040}+O\left(\theta ^9\right)$$ this would give $$\theta=\frac{y}{2}+\frac{y^3}{96}+\frac{y^5}{1920}+\frac{43 y^7}{1290240}+O\left(y^9\right)$$ Making $y=\frac \pi 3$ and computing $$\theta=0.53626300$$

You could also use the $1,400$ years old approximation $$\sin(\theta) \simeq \frac{16 (\pi -\theta)\theta}{5 \pi ^2-4 (\pi -\theta) \theta}\qquad (0\leq \theta\leq\pi)$$ and solve the cubic $$-\frac{5 \pi ^3}{3}+\pi\left(16 +\frac{19 \pi }{3}\right) \theta-16\left(1+\frac{\pi }{3}\right) \theta^2+4 \theta^3=0$$ which shows only one real root (not very nice formal expression) which is $0.53631167$.

Since, by inspection, you know that the solution is close to $\frac \pi 6$, you could perform one single iteration of Newton-like method and have explicit approximations which will be better and better increasing the order $n$ of the method. For example, Newton method would give $$\theta_{(2)}=\frac \pi 6+\frac{1}{3} \left(2-\sqrt{3}\right) (\pi -3)$$ Halley method would give $$\theta_{(3)}=\frac \pi 6+\frac{2 \left(2+\sqrt{3}\right) (\pi -3)}{45+24 \sqrt{3}-\pi }$$ As a function of $n$, the results would be $$\left( \begin{array}{ccc} n & \text{estimate} & \text{method} \\ 2 & 0.536245321326154 & \text{Newton} \\ 3 & 0.536266784935255 & \text{Halley} \\ 4 & 0.536266978676557 & \text{Householder} \\ 5 & 0.536266978987702 & \text{no name} \\ 6 & 0.536266978988890 & \text{no name} \\ 7 & 0.536266978988891 & \text{no name} \end{array} \right)$$

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If you’ve ever wanted to divide a circle into three, you can do it very easily with just a ruler and a compass. I’m working on a sculptural piece of furniture that has three blades radiating from a cylinder, and as many of you know, the Shakers have an iconic candle table which also has three legs radiating off of a cylinder, so this little trick could come in handy.

So if a circle has 360° and we want to divide it into three, how do we find the 120° of each section without a protractor? It’s really very easy. I’m going to use whole numbers to keep everything simple. My circle is 4″ in diameter. Mark the centerpoint and draw a line from one side of the circle to the other through the center.

The radius is 2″. You need to take half of the radius, in this case 1″, and mark the halfway point from the center of the circle on the centerline. Take your ruler and draw a perpendicular line from this point to the outside diameter of the circle. This will give you two points. Connect these two points to the center point and you have the circle divided into three.

If you are interested in building the Shaker Candle Table, we have a collection on sale called The Ultimate Shake Project Collection. It includes three books on CD. Click here to buy it today.

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Updated May 11, 2018

By Kevin Beck

Circles are everywhere in nature, art and the sciences. The sun and the moon, through spherical, form circles in the sky and travel in roughly circular orbits; the hands of a clock and the wheels on automobiles trace out circular paths; philosophically-minded observers speak of a the "circle of life."

Circles in plain terms are mathematical constructs. You may need to know, using math, how to separate a complete circle into equal portions for pie, land or artistic purposes. If you have a pencil, along with a protractor, a compass or both, dividing a circle into three equal parts is straightforward and instructive.

A circle encloses 360 degrees of an arc, so for this exercise you need to create a "pie" with three equal 120° angles at the center.

Use your straightedge (ruler or protractor) to draw a diameter or line through the middle of the circle that reaches both edges. This of course divides your circle in half.

If the center of the circle is not marked, you will find it in this step because the diameter of any circle is the longest distance across the circle. Simply divide the value of the diameter by 2 and place a point halfway along the line from one edge to indicate the center.

Use your ruler or protractor to find a point exactly halfway between the center and one edge, or equivalently, one-fourth of the diameter or half of the radius. Label this point A.

Use your protractor, or if necessary the short edge of your ruler, to draw a line through point A. Extend this line to the edges of the circle. Label the points at which this line intersects the edge of the circle B and C.

Using your straightedge, create lines connecting the center of the circle to points B and C. These lines represent radii of the circle, which have a value of half of the diameter.

You now have two right triangles inscribed within the circle. Because the short leg of each of these is one-half the distance of the hypotenuse of the circle, which is the same as a radius, you may recognize that these right triangles are "30-60-90" triangles, which have the property of the shortest side being half the length of the longest.

Because of this, you can conclude that the interior angles of the circle you have created between the two hypotenuses, and the hypotenuse and the diameter on the opposite side of the circle, are each 120°. You thus have a circle divided into three equal parts.