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Circle and its chords

Circle and its chords


Circle

From Latin: circus – “ring, a round arena”
A line forming a closed loop, every point on which is a fixed distance from a center point.
Try this Drag an orange dot. The circle can be moved by dragging the center point and resized by dragging the point on the circle.

 

A circle is a type of line. Imagine a straight line segment that is bent around until its ends join. Then arrange that loop until it is exactly circular – that is, all points along that line are the same distance from a center point.

There is a difference between a circle and a disk. A circle is a line, and so, for example, has no area – just as a line has no area. A disk however is a round portion of a planewhich has a circular outline. If you draw a circle on paper and cut it out, the round piece is a disk.

Properties of a circle

Center A point inside the circle. All points on the circle are equidistant (same distance) from the center point.
Radius The radius is the distance from the center to any point on the circle. It is half the diameter. See Radius of a circle.
Diameter The distance across the circle. The length of any chord passing through the center. It is twice the radius. See Diameter of a circle.
Circumference The circumference is the distance around the circle. SeeCircumference of a Circle.
Area Strictly speaking a circle is a line, and so has no area. What is usually meant is the area of the region enclosed by the circle. See Area enclosed by a circle .
Chord A line segment linking any two points on a circle. See Chord definition
Tangent A line passing a circle and touching it at just one point. See Tangent definition
Secant A line that intersects a circle at two points. See Secant definition

 

Pi

In any circle, if you divide the circumference (distance around the circle) by its diameter (distance across the circle), you always get the same number. This number is called Pi and is approximately 3.142. See Definition of pi.

Relation to ellipse

A circle is actually a special case of an ellipse. In an ellipse, if you make the major and minor axis the same length, the result is a circle, with both foci at the center.

Circle as a conic section

You can define a circle as the shape created when a plane cuts through a cone at right angles to the cone’s axis.

Circle as a locus

A circle is the locus of all points a fixed distance from a given (center) point. This definition assumes the plane is composed of an infinite number of points and we select only those that are a fixed distance from the center.

 

Parts of a circle

Semicircle

From Latin: semi – “half”
Half a circle. A closed shape consisting of half a circle and a diameter of that circle*.

Semicircle
A semicircle is a half circle, formed by cutting a whole circle along a diameter line, as shown above. Any diameter of a circle cuts it into two equal semicircles.

*  An alternative definition is that it is an open arc. See note at end of page.

Area of a semicircle

The area of a semicircle is half the area area of the circle from which it is made. Recall that the area of a circle is πR2, where R is the radius. (See Area of a circle).

So, the formula for the area of a semicircle is:

Area=πR2

where:
R  is the radius of the semicircle
π  is Pi, approximately 3.142

Perimeter of a semicircle

The perimeter of a semicircle is not half the perimeter of a circle*. From the figure above, you can see that the perimeter is the curved part, which is half the circle, plus the diameter line across the bottom.

Recall that the perimeter of a circle is 2πR, (See Perimeter of a circle).
So the curved part is half that, or πR, and the base line is twice the radius or 2R.

So, the formula for the perimeter of a semicircle is:

Perimeter=πR+2R

where:
R  is the radius of the semicircle
π  is Pi, approximately 3.142By factoring out R, this simplifies slightly to

R(π+2)

Angle inscribed in a semicircle

The angle inscribed in a semicircle is always 90°.

 

Alternative definition*

An alternative definition of a semicircle is that it is simply anarc – a curved line that is half the circumference of a circle, without the straight line linking its ends. This means it is not a closed figure, and so:

  1. Has no area
  2. Has no perimeter. Its length is the length of the arc, or πR.

To avoid confusion, it is best to refer to this as a “semicircular arc”.

 

Chord

From Greek: khorde “gut, string.”
A line that links two points on a circle or curve. (pronounced “cord”)
Try this Drag either orange dot. The blue line will always remain a chord to the circle.

 

The blue line in the figure above is called a “chord of the circle c”. A chord is a lot like a secant, but where the secant is a line stretching to infinity in both directions, a chord is a line segment that only covers the part inside the circle. A chord that passes through the center of the circle is also a diameter of the circle.

Calculating the length of a chord

Two formulae are given below for the length of the chord,. Choose one based on what you are given to start.

1. Given the radius and central angle

Below is a formula for the length of a chord if you know the radius and central angle.

Chord length=2r sin(c/2)

where
r  is the radius of the circle
c  is the angle subtended at the center by the chord
sin  is the sine function

2. Given the radius and distance to center

Below is a formula for the length of a chord if you know the radius and the perpendicular distance from the chord to the circle center. This is a simple application of Pythagoras’ Theorem.

Chord length=2√(r^2−d^2)

where
r  is the radius of the circle
d  is the perpendicular distance from the chord to the circle center

Finding the center

The perpendicular bisector of a chord always passes through the center of the circle. In the figure at the top of the page, click “Show Right Bisector”. Then move one of the points P,Q around and see that this is always so. This can be used to find the center of a circle: draw one chord and its right bisector. The center must be somewhere along this line. Repeat this and the two bisectors will meet at the center of the circle.

Intersecting Chords

If two chords of a circle intersect, the intersection creates four line segments that have an interesting relationship.

Intersecting Chord Theorem
When two chords intersect each other inside a circle, the products of their segments are equal.

A.B = C.D

It is a little easier to see this in the diagram on the right. Each chord is cut into two segments at the point of where they intersect. One chord is cut into two line segments A and B. The other into the segments C and D.

This theorem states that A×B is always equal to C×D no matter where the chords are.

In the figure below, drag the orange dots around to reposition the chords. As long as they intersect inside the circle, you can see from the calculations that the theorem is always true. The two products are always the same.

(Note: Because the lengths are rounded off for clarity, the calculations will be slightly off if you enter the displayed values into your calculator).

 

 

A Practical use

When making doors or windows with curved tops we need to find the radius of the arch so we can lay them out with compasses.

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