history of circumference

history of circumference

Proving that the formula for the circumference of a circle is always true is harder than just figuring out that it is true. Now we need to show that C=2πr is always true for every possible circle.

Here is how Archimedes proved it. Draw any circle. Make a point anywhere on the circumference of the purple circle. Use that point as the center of a blue circle with the same radius as the purple circle. The edge of the blue circle should touch the center of the purple circle.

Draw the line segment connecting the centers of the two circles. That's the radius of the both circles. Now draw the line connecting the center of the blue circle to where it crosses the purple circle on both sides, and complete the triangles. You should have two equilateral triangles whose sides are equal to the radius of the purple circle.

Now extend all of the radius lines so they become diameter lines, all the way across the circle, and finish drawing all of the triangles to connect them. You've got six equilateral triangles now, that make an orange hexagon. So the perimeter of your hexagon is the same as six times the radius of your circle. But your circumference is a little bigger than the perimeter of your hexagon, because the shortest distance between two points is always a straight line. This shows you that the circumference of the purple circle has to be more than 6r, so if C=2πr then π (pi) has to be a little bigger than 3, which it is.

Now let's try to get a little closer to the real value of π. Suppose we make our triangles narrower, so that instead of drawing a hexagon, we draw a dodecagon - a shape with twelve sides? We can do that by drawing more points halfway between the points of the hexagon. If we do that, we'll see that that the perimeter of the dodecagon is a little bigger, and closer to being a circle. We have twelve congruent isosceles triangles. Each triangle has two sides that are as long as the radius of the circle, and a third side that we want to know the length of, in order to figure out the perimeter of the dodecagon.

It's pretty hard to figure out the perimeter of the dodecagon. Start by drawing a red line from the center of the purple circle to one of the points of the dodecagon (B). We know that the line AC is the same as the radius of the circle - let's say the radius is 4. It's also the hypotenuse of the bluish right triangle ADC. We also know that the line AD is the same as half the radius of the circle (remember the hexagon was made of equilateral triangles), so AD = 2. Because the bluish triangle is a right triangle, we can use the Pythagorean Theorem to tell us that the third side, CD, is the square root of 12. (4 squared = 2 squared + 12).

We also know that the red line CB is the same as the radius of the circle, so that's also 4. So the distance from point B to point D must be the radius (4) minus the length of CD, or the square root of 12. BD = 4 - the square root of 12. Now look at the little orange triangle ABD. That's also a right triangle, and now we know two of its sides. AD = 2, and BD = 4 - the square root of 12. We can use the Pythagorean Theorem again to calculate that the green line AB must be 2.07, so the whole perimeter is 12 x 2.07 = 24.84. The circumference of the purple circle has to be more than 24.72, or more than 6.21 r. If C = 2πr, then π has to be a little bigger than 3.1.

The more sides we draw on our polygon, the closer we will get to the real value of pi (3.14159 etc.). Using a polygon with 96 sides, Archimedes was able to calculate that π was a little bigger than 3.1408, which is pretty close.

sector and segment of circle

Circle Sector and Segment

A circular sector or circle sector (symbol: ⌔), is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, the radius of the circle, and is the arc length of the minor sector.
A sector with the central angle of 180° is called a semicircle. Sectors with other central angles are sometimes given special names, these include quadrants (90°), sextants (60°) and octants (45°).
The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle.

n geometry, a circular segment (symbol: ⌓) is an area of a circle informally defined as an area which is "cut off" from the rest of the circle by a secant or a chord. The circle segment constitutes the part between the secant and an arc, excluding of the circle's center.

Contents

Slices There are two main "slices" of a circle: The "pizza" slice is called a Sector. And the slice made by a chord is called a Segment.

Try Them!

Sector	Segment

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

	Quarter of a circle is called a Quadrant. Half a circle is called a Semicircle.

 

Area of a Sector You can work out the Area of a Sector by comparing its angle to the angle of a full circle. Note: I am using radians for the angles.
This is the reasoning: A circle has an angle of 2π and an Area of: πr2 So a Sector with an angle of θ (instead of 2π) must have an area of: (θ/2π) × πr2 Which can be simplified to: (θ/2) × r2 Area of Sector = ½ × θ × r2   (when θ is in radians) Area of Sector = ½ × (θ × π/180) × r2   (when θ is in degrees)

Arc Length By the same reasoning, the arc length (of a Sector or Segment) is: L = θ × r   (when θ is in radians) L = (θ × π/180) × r   (when θ is in degrees)

Area of Segment The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here). There is a lengthy reason, but the result is a slight modification of the Sector formula:
Area of Segment = ½ × (θ - sin θ) × r2   (when θ is in radians) Area of Segment = ½ × ( (θ × π/180) - sin θ) × r2   (when θ is in degrees)

The line segment from the center of a regular polygon to the midpoint of a side, or the length of this segment. Same as the inradius; that is, the radius of a regular polygon's inscribed circle.

Note: Apothem is pronounced with the emphasis on the first syllable with the a pronounced as in apple (A-puh-thum).

Apothem

Given a circle, the apothem is the perpendicular distance from the midpoint of a chord to the circle's center. It is also equal to the radius minus the sagitta ,

For a regular polygon, the apothem simply is the distance from the center to a side, i.e., the inradius of the polygon.
The apothem is also the radius of the incircle of the polygon. For a polygon of n sides, there are n possible apothems, all the same length of course. The word apothem can refer to the line itself, or the length of that line. So you can correctly say 'draw the apothem' and 'the apothem is 4cm'.
Each formula below shows how to find the length of the apothem of a regular polygon. Use the formula that uses the facts you are given to start.

Apothem given the length of a side.

By definition, all sides of a regular polygon are equal in length. If you know the length of one of the sides, the apothem length is given by the formula:

Calculator

where
s  is the length of any side
n  is the number of sides
tan  is the tangent function calculated in degrees (see Trigonometry Overview)

Apothem given the radius (circumradius)

If you know the radius (distance from the center to a vertex):

Calculator

where
r  is the radius (circumradius) of the polygon
n  is the number of sides
cos  is the cosine function calculated in degrees (see Trigonometry Overview)

Irregular Polygons

Since irregular polygons have no center, they have no apothem. In the figure above, uncheck the "regular" checkbox and note how there can be no center or apothem.

Example : 
How can I find the apothem of a regular pentagon given the only the side lengths? 
The only 2 information given is that the pentagon is a regular polygon and that the side lengths are 5. Can someone not only give me the answer, but also explain it?

Answer : 1) Draw a straight horizontal line. It doesn't have to be too long. This line will represent one of the sides of your polygon. In your case, it's one of the 5 sides of the regular polygon.

2) Now from the middle of that line, draw a line straight up, so that the two lines are perpendicular. This second line drawn is the "apothem." This is the segment's length your trying to find. Let's call it "b".

3) Finally draw one more line...connecting the top of the apothem (which is the center of your polygon) to either end of the first line you drew (it doesn't matter which end - left or right). We don't know this length either. Let's call it "c".

Now that you have a triangle picture, let's call 1/2 of the first line you drew "a", and its length = 2.5, since it is half the length of each side of the polygon.

Using the fact that this pentagon is "regular" (convex and equilateral), you can find the angle that is formed between sides "a" and "c". To do this, first determine what each interior angle in this regular pentagon equals. You've probably already learned that a 5-sided convex polygon's interior angles add up to 3 x 180 degrees = 540 degrees. So, since the pentagon is regular: 540 / 5 = 108 degrees. That's how large each interior angles is. But, the angle we're interested in (the one between "c" and "a") is exactly half of 108 degrees or 54 degrees.

Ok, so here's your picture:

|\
| \ ` c
|__\
` a

In this picture "b" is the vertical line on the left.

Remember, side "b" is the "apothem". You know the length of "a" and the angle between "a" and "c".

Finally, using trigonometry, specifically tangent, you can find the length of "b".

tan 54 = b / 2.5

I'm getting b = 3.44 (rounded to two decimal places)