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Circle : Radius | Diameter | Area | Sector | Segment
Radius (of a circle)
2. The distance from the center of a circle to a point on the circle.
The radius of a circle is the length of the line from the center to any point on its edge. The plural form is radii (pronounced “ray-dee-eye”). In the figure above, drag the orange dot around and see that the radius is always constant at any point on the circle.
Sometimes the word ‘radius’ is used to refer to the line itself. In that sense you may see “draw a radius of the circle”. In the more recent sense, it is the length of the line, and so is referred to as “the radius of the circle is 1.7 centimeters”
If you know the diameter
Given the diameter of a circle, the radius is simply half the diameter:
where:
D is the diameter of the circle
If you know the circumference
If you know the circumference of a circle, the radius can be found using the formula
where:
C is the circumference of the circle
π is Pi, approximately 3.142
If you know the area
If you know the area of a circle, the radius can be found using the formula
where:
A is the area of the circle
π is Pi, approximately 3.142
Circumference, Perimeter of a circle
You sometimes see the word ‘circumference’ to mean the curved line that goes around the circle. Other times it means the length of that line, as in “the circumference is 2.11cm”.
The word ‘perimeter’ is also sometimes used, although this usually refers to the distance around polygons, figures made up of straight line segments.
If you know the radius
Given the radius of a circle, the circumference can be calculated using the formula
where:
R is the radius of the circle
π is Pi, approximately 3.142
If you know the diameter
If you know the diameter of a circle, the circumference can be found using the formula
where:
D is the diameter of the circle
π is Pi, approximately 3.142
If you know the area
If you know the area of a circle, the circumference can be found using the formula
where:
A is the area of the circle
π is Pi, approximately 3.142
Diameter (of a circle)
The diameter of a circle is the length of the line through the center and touching two points on its edge. In the figure above, drag the orange dots around and see that the diameter never changes.
Sometimes the word ‘diameter’ is used to refer to the line itself. In that sense you may see “draw a diameter of the circle”. In the more recent sense, it is the length of the line, and so is referred to as “the diameter of the circle is 3.4 centimeters”
The diameter is also a chord. A chord is a line that joins any two points on a circle. A diameter is a chord that runs through the center point of the circle. It is the longest possible chord of any circle.
The center of a circle is the midpoint of its diameter. That is, it divides it into two equal parts, each of which is a radius of the circle. The radius is half the diameter.
If you know the radius
Given the radius of a circle, the diameter can be calculated using the formula
where:
R is the radius of the circle
If you know the circumference
If you know the circumference of a circle, the diameter can be found using the formula
where:
C is the circumference of the circle
π is Pi, approximately 3.142
If you know the area
If you know the area of a circle, the diameter can be found using the formula
where:
A is the area of the circle
π is Pi, approximately 3.142
Thales’ Theorem
Put another way: If a triangle has, as one side, the diameter of a circle, and the third vertexof the triangle is any point on the circumference of the circle, then the triangle will always be a right triangle.
In the figure above, no matter how you move the points P,Q and R, the triangle PQR is always a right triangle, and the angle ∠PRQ is always a right angle.
A practical application – finding the center of a circle
The converse of Thales Theorem is useful when you are trying to find the center of a circle. In the figure on the right, a right angle whose vertex is on the circle always“cuts off” a diameter of the circle. That is, the points P and Q are always the ends of a diameter line.
Since the diameter passes through the center, by drawing two such diameters the center is found at the point where the diameters intersect.
An easy way to find the center of a circle using any right-angled object. Here we use a 45-45-90 drafting triangle, but anything that has a 90° corner will do, such as the corner of a sheet of paper.
Click on ‘Next’ to go through the construction one step at a time, or click on ‘Run’ to let it run without stopping.
This page shows how to find the center of a circle using any right-angled object. This method works as a result of using Thales Theorem in reverse. The diameter of a circle subtends a right angle to any point on the circle. By placing the 90° corner of an object on the circle, we can find a diameter. By finding two diameters we establish the center where they intersect.
Area enclosed by a circle
A circle is actually a line, one that connects back to itself making a loop. Imagine the circle to be a loop of string. The string itself has no area, but the space inside the loop does. So strictly speaking a circle has no area.
However, when we say “the area of a circle” we really mean the area of the space inside the circle. If you were to cut a circular disk from a sheet of paper, the disk would have an area, and that is what we mean here.
If you know the radius
Given the radius of a circle, the area inside it can be calculated using the formula
where:
R is the radius of the circle
π is Pi, approximately 3.142
If you know the diameter
If you know the diameter of a circle, the area inside it can be found using the formula
where:
D is the diameter of the circle
π is Pi, approximately 3.142
If you know the circumference
If you know the circumference of a circle, the area inside it can be found using the formula
where:
C is the circumference of the circle
π is Pi, approximately 3.142
Concentric Circles
Concentric circles are simply circles that all have the same center. They fit inside each other and are the same distance apart all the way around. In the figure above, resize either circle by dragging an orange dot and see that they both always have a common center point.
Annulus
An annulus is a flat ring shaped object, much like the throw-ring shown on the right. One way to think of it is a circular disk with a circular hole in it. The outer and inner circles that define the ring are concentric (share a common center point).
The dimensions of an annulus are defined by the two radii R2, R1 in the figure above, which are the radii of the outer ring and the inner ‘hole’ respectively.
The area of the annulus can be found by subtracting the area of the ‘hole’ from the area of the overall disk.
The adjective form is ‘annular’. So for example the ring on the right could be called an ‘annular plastic ring’.
Area of an annulus
Area formula
The area of the annulus is the area of the ring-shaped space between the two circles that define it. In the figure above it is the number of gray square units it takes to fill the annulus. This area is the area of the entire disk, minus the area of the ‘hole’ in the middle. ( See Area of a Circle):
Where:
R | is the radius of the outer circle |
H | is the radius of the inner ‘hole’ |
π | is Pi, approximately 3.142 |
This simplifies a little to:
Sector
As you can see from the figure above, a sector is a pie-shaped part of a circle. It has two straight sides (the two radius lines), the curved edge defined by the arc, and touches the center of the circle.
Properties
Radius | The radius of the circle of which the sector is a part | |
Central Angle | The angle subtended by the sector to the center of the circle. | |
Arc length | The length around the curved arc that defines the sector (shown in red here). |
Area
The area of a sector can be found if you know the radius and the central angle or arc length.
Sector area
What the formulae are doing is taking the area of the whole circle, and then taking a fraction of that depending on what fraction of the circle the sector fills. So for example, if the central angle was 90°, then the sector would have an area equal to one quarter of the whole circle.
If you know the central angle
where:
C is the central angle in degrees
r is the radius of the circle of which the sector is part.
π is Pi, approximately 3.142This is the method used in the animation above.
If you know the arc length
where:
L is the arc length.
R is the radius of the circle of which the sector is part.
Sector area is proportional to arc length
The area enclosed by a sector is proportional to the arc length of the sector. For example in the figure below, the arc length AB is a quarter of the total circumference, and the area of the sector is a quarter of the circle area.Similarly below, the arc length is half the circumference, and the area id half the total circle.
Segment of a Circle
The chord AB in the figure above defines one side of the segment. As you drag the points you will notice that the segment is always the smaller part of the circle. This is a definition of a segment. Its Central Angle is always less than 180°
In fact, if the chord divides the circle exactly in half (becoming a diameter) neither of the two halves are segments. They are semicircles. If you careful with the mouse, you can create this situation in the figure above. Move A or B so that the line AB passes through the center of the circle. No segment is then present.
Attributes
A segment is defined by the arc and chord that form its outer boundary. The key properties a segment inherits from them are shown below. In the figure above click on “show details” to see these items.
Arc Length | The length of the curved arc line. See Arc Length page for more. |
Radius | The radius of the circle of which the segment is a part. |
Central Angle | The angle subtended by the segment to the center of the circle of which it is a part. |
Area of a Circle Segment Given the Central Angle
The formula to find the area of the segment is given below. It can also be found by calculating the area of the whole pie-shaped sector and subtracting the area of theisosceles triangle △ACB.
Where:
C | is the central angle in DEGREES |
R | is the radius of the circle of which the segment is a part. |
π | is Pi, approximately 3.142 |
sin | is the trigonometry Sine function. |
If you know the radius of the circle and the height of the segment, you can find the segment area from the formula below. The result will vary from zero when the height is zero, to the full area of the circle when the height is equal to the diameter.
area= r^2 cos^-1 {(r-h)/r} -(r-h) √{(2rh)-h^2}
where:
r is the radius of the circle of which the segment is a part.
h is the height of the segment.
Note: The result of the cos^{-1} function in the formula is in radians.
Applications
This calculation is useful as part of the calculation of the volume of liquid in a partially-filled cylindrical tank. For more on this see Volume of a horizontal cylindrical segment.
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