A circle is a round plane figure with a boundary (called the circumference) that is equidistant from its center. It is a fundamental object studied in geometry.
In order to describe the shape of an object, we give the object appropriate dimensions. For example, a rectangle can be described with its height and width.
It is harder to describe the shape of a triangle, since we would require all the lengths of the three edges ( and ).
In case of a circle, it is much easier since we only need its radius or diameter to describe its geometry.
Then, what are the radius and diameter of a circle? Their concepts are very important in geometry of a circular shape, so let's review the terminologies.
The radius of a circle is the distance from the center of the circle to any point on its circumference.
The diameter of a circle is the length of a line that starts at one point on the circle, passes through the center and ends on another point on the circle's opposite side. It's also referred as the longest possible chord in the circle.
The radius and the diameter are interrelated as
The formula for the circumference of a circle is
where and is the mathematical constant, "pi."
The first digits of are , but any finite list of digits is can only be an approximation of . Furthermore, , (written "pi" and pronounced as "pie"), is an irrational number. (Meaning it cannot be described by any ratio of whole numbers, .)
For more information about the constant, , check out the wiki page.
This is proven by dividing a circle into even parts
The area of a circle with radius is .
and then rearranging them into a crooked parallelogram.
Observe that the area of a parallelogram is Then the "base" of the rearranged circle is half of the circumference, which is equal to and the "height" is equal to the radius itself.
Therefore, the area of the circle is equal to
What is the area of a circle if the circle has a radius of
The area of a circle with radius is
What is the area of a circle if the circle has a diameter of
The radius of the circle is . Therefore, the area is
What is the area of a circle if the circle has a circumference of
The circumference of the circle with radius is For this problem, so Therefore, the area of the circle is
One half of a circle is called a semicircle. What is the area of a semicircle if the whole circle has a radius of
The area of the circle with radius is Therefore, the area of the semicircle is
One half of a circle is called a semicircle. What is the area of a semicircle if the whole circle has a diameter of
The area of the circle with diameter is Therefore, the area of the semicircle is
One half of a circle is called a semicircle. What is the area of a semicircle if has a circumference of
The circumference of the semicircle is the diameter and one half of the circumference of the circle. If the radius of the circle is the circumference of the semicircle is so
Thus, the area of the semicircle with radius is
Arc length of a circle is the length of the curved part. Arc length of a full circle is its circumference, but what about the arc length of sectors (pieces of circles)? They are calculated by a formula where is the arc length, is the radius of the circle, and is the angle of the sector.
NOTE: The angle should be in radians.
In order to measure an arc of a circle we use the size of the central angle that forms the arc.
A central angle of a circle is an angle where its vertex is the center of the circle and its sides are radius of the circle.
For example, In the diagram above, is a central angle that forms arc . If we call .
Two equal arcs are formed by two equal central angles.
Among two arcs, the longer one is formed by a larger central angle.
If the central angle is , the arc formed by this angle is half the circumference. If the central angle is , the arc formed by this angle is the whole circumference.
An inscribed angle is an angle where its vertex is a point on the circumference of the circle and its sides are chords of the circle that passes through the vertex.
In the diagram, and are both inscribed angles that form arc .
The size of an inscribed angle is half the size of the central angle that forms the same arc.
Two inscribed angles forming the same arc or forming two equal arcs are equal.
If two inscribed angles are equal, the arc(s) that they form are equal.
Among two inscribed angles, the larger one forms the longer arc.
The figure above shows a circle with two chords intersecting. The two chords are each cut into two segments by the point of intersection. One chord is cut into two line segments each of lengths and and the other into two segments each of lengths and
The intersecting chord theorem states that the two chords in the figure above satisfy Thus, is always equal to regardless of where the two chords intersect inside the circle.
Find the value of in the figure below.
From the intersecting chord theorem, we have
Find the value of in the figure below.
According to the intersecting chord theorem we have
The lengths of some line segments in the figure below are
If then what is
Since we have
According to the intersecting chord theorem, we have
Now observe that and which implies that the triangles and are in SAS similarity with a ratio of 2:3.
Thus, if we let then
In the figure below, is a chord of a circle of radius 4 centered at If the line segments and bisect each other, what is the length of
Let be the point on the circle where the extension of passes through. Then, since the radius of the circle is 4 and the two chords bisect each other, the length of and are
Now, according to the intersection chord theorem we have
Since we know that
Hence, our answer is
A circle sector is a closed figure bounded by two radii of a circle and the circle's arc. A sector of angle and radius is drawn in the figure below.
If the angle is in degrees, then the area of sector is of the circle. Since the area of the circle is , the area of sector is
If is in radians, then the area of sector is of the circle, and hence the area of sector is
A sector has radius and angle . Find its area.
A sector has area and angle . Find its radius.
The following are what we know about the above diagram:
- is a diameter of the large circle with radius so
- The two medium-sized circles have equal radius and are both tangent to and to the large circle.
- The small circle has radius and is tangent to the other three circles.
Now, how can we express in terms of
Note that and Then in triangle we have
Suppose that we now have an even smaller circle tangent to the large circle, and the upper medium-sized circle, as shown below. Can you prove that the radius of this new circle is