In this explainer, we will learn how to use the properties of a circle and its parts in order to solve problems.
We can start by defining what a circle is.
A circle is a shape consisting of all points in a plane that are an equal distance from a given point, the center.
We recap the key terminology about different parts of a circle.
The circumference of a circle is its perimeter. It is the measurement of the boundary of the circle.
A radius of a circle is a straight line extending from the center of the circle to the circumference. The plural of radius is radii. There are an infinite number of radii in any circle. We commonly use radius as the length; for example, a circle might be described as “a circle of radius 3 cm.”
A diameter of a circle is a line segment passing through the center and joining two points on the circle. As with a radius, it is common to use the diameter to just mean the length of this line segment.
Because we would have two lengths in a diameter that consist of a line from the center to the circumference, there are two radii in one diameter, so the length of the diameter is twice the length of the radius.
We can now consider the terminology arising from sections of the area or circumference of a circle.
A chord of a circle is a line segment joining two distinct points on the circumference of the circle.
There are an infinite number of chords that can be drawn in a circle. The largest chord in a circle is that created by the diameter, the line joining two points on the circumference and passing through the center.
We will now apply our understanding of the parts of a circle, including the relationship between radius and diameter, in the following examples. In the first example, we will see how we can apply the given information about the radii of two circles to find the length of a line segment joining their centers.
Example 1: Solving a Problem Involving the Relationship between the Radius and the Diameter of a Circle
If the diameters of the two given circles, with centers and are 2 cm and 6 cm respectively, determine the length of .
We can begin answering this problem by observing that the two given circles meet at a point, which we can designate .
We are given the diameter of each of the circles. We can recall that the diameter of a circle is the length of the line segment that passes through the center and joins two points on the circumference.
However, we observe that on the diagram, the lengths of and are, in fact, radii of the respective circles. A radius of a circle is a line passing through the center of a circle to its circumference. The radius and diameter of any circle are related, as the radius of a circle is half the diameter.
Thus, given that the diameter of the circle with center is 2 cm, then the radius of circle , the length , is .
Similarly, given that the diameter of the circle with center is 6 cm, then its radius, the length , is .
We can then calculate the length of by adding the two radii:
Therefore, we can give the answer that the length of is 4 cm.
We will now see another example of using the properties of circles to determine a length. As in many geometry questions, adding any given, or calculated, lengths to a diagram can be useful in helping us to solve a problem.
Example 2: Finding an Unknown Distance Using Lengths between the Centers of Two Circles
The diameter of the circle with center is 23, and . Find the length of .
We are given the length of the diameter of the circle with center as 23. We recall that the diameter of a circle is the length of the line segment passing through the center and joining two points on the circumference. We observe in the diagram that there is no line representing the diameter of ; however, there is a line that represents its radius. A radius of a circle is the line passing through the center of the circle to its circumference. This means that is a radius of the circle with center . The radius of a circle is half of its diameter. Therefore,
We can add this length to the diagram, along with the information that , to help us visualize the problem.
We are required to find the length of . This can be found by subtracting from ; therefore,
Hence, we can give the answer that the length of is 6.5.
We can now consider the reflection, or line symmetry of a circle.
Definition: Line Symmetry of a Circle
Any line passing through the center of the circle is an axis of symmetry because it divides the circle into two identical parts.
As there are an infinite number of lines that can pass through the center, a circle has an infinite number of lines of symmetry.
In the next question, we will consider the reflection symmetry of a circle.
Example 3: Identifying the Lines of Symmetry of a Circle
Which of these are lines of symmetry of the circle?
In the figure, we can observe that there are 3 different lines in the circle: , , and .
A line of symmetry is a line that cuts a shape exactly in half. In order to obtain a line of symmetry in a circle, we would need to determine which line passes through the center of the circle. This line divides the circle exactly in half. This can be achieved by folding along line .
In general, any line that passes through the center of a circle is a line of symmetry. A circle has an infinite number of lines of symmetry. In this figure, the line that creates a line of symmetry is line only.
We have seen that a circle has an infinite number of lines of symmetry. Note also the rotational symmetry of a circle. The order of rotational symmetry of a geometric shape is how many times the figure fits onto itself during a full rotation of about its center. A circle has an infinite order of rotational symmetry. This is because a circle will always fit into its original outline, regardless of how much it is rotated.
We are often required to solve a range of geometrical problems involving circles. A key geometry fact that we should recall is the Pythagorean theorem.
Recap: The Pythagorean Theorem
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
For a hypotenuse and two shorter sides and , the Pythagorean theorem states that
We will now see how we can apply the Pythagorean theorem in the following example.
Example 4: Finding an Unknown Length Using the Equality of the Radii of a Circle and the Pythagorean Theorem
Given that , find the length of .
In this problem, we are given the length of one side in the right triangle . We recall that we can find a missing length in a right triangle by using the Pythagorean theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides.
Initially, it may seem like we do not have enough information to find the length of side , as we are only given the length of one side, . However, we can use the properties of the circle that contains triangle . and are radii of the circle: they are each a straight line extending from the center to a point on the circumference. Therefore, we can say that . We can denote these lengths as and write the lengths on the diagram as below.
Applying the Pythagorean theorem, we can set up an equation and simplify, given that . This gives us
We can now take the square root of each side of the equation, noting that must be a positive value since it is a length. Thus, we have
Hence, we can give the answer that the radius of this circle, the length of , is .
In the previous example, we saw an example of a right triangle. Notice that this triangle was also an isosceles triangle, as it had two equal lengths created by radii of the circle. We commonly find isosceles triangles in circle geometry problems, and it is useful to note that any two radii of a circle and the chord that connects them form an isosceles triangle.
We will see an example of this in the following question.
Example 5: Finding an Unknown Angle around the Center of a Circle to Solve a Problem
What is ?
We can begin this question by recalling that the angles about a point sum to . Thus, we can calculate as
Next, we observe that and are both radii of the circle. This means that they will be of the same length and, furthermore, must be an isosceles triangle as it has two sides with the same length. Hence, we have two congruent angles:
We can define both of these angles to be .
To find the missing angle, we remember that the internal angles in a triangle sum to . Therefore, we have
Substituting the values for the angles, we have
As we defined as , we can give the answer that is .
We will now consider the similarity and congruence of circles. Generally, two geometric shapes are similar if they are the same shape, but they may be a different size. As all circles are the same shape, all circles are similar.
Next, two geometric shapes are congruent if they are the same shape and the same size. Although circles are the same shape, they are not always the same size. In order to identify and prove that any two given circles are congruent, we would need to identify that a common measure in each is congruent.
How To: Proving Two Circles Are Congruent
Congruent circles are the same shape and size. Two circles are congruent if any one of the following conditions is satisfied:
- The radii are congruent.
- The diameters are congruent.
- The circumferences are congruent.
In the final example, we will see a problem involving two equal chords in congruent circles.
Example 6: Finding Unknown Angles given Equal Chords in Congruent Circles
Consider that circles and are congruent and .
- Find the value of .
- Find the value of .
In the circle with center , since and are both radii of the circle, they will be of equal length. Hence, is an isosceles triangle. An isosceles triangle has two sides of equal length and two angles of equal measure. Therefore,
Next, using the fact that the angle measures in a triangle sum to , we have
Thus, we have that .
Given that the circles with centers and are congruent, then the radii of both will be congruent. We are given that ; hence, , by the side-side-side (SSS) congruency criterion. Note that since these are isosceles triangles, we could also write that .
Hence, we have that .
In the previous example, we proved that two chords connected by radii are congruent, using the SSS congruency criterion. This rule applies in instances where congruent chords are in the same circle or in a pair of congruent circles. We can define this below.
Definition: Congruent Chords Connecting Two Radii
If chords of the same length connect radii in the same circle, or in congruent circles, then the two isosceles triangles formed are congruent.
We will now summarize some important points from this explainer.
- A circle is a shape consisting of all points in a plane that are an equal distance from a given point, the center.
- The length of the diameter of a circle is twice the length of its radius.
- A diameter of a circle is the longest chord in the circle.
- A circle has an infinite number of lines of symmetry, and all of the axes of symmetry pass through the diameters of the circle.
- A circle has an infinite order of rotational symmetry about its center.
- A triangle formed within a circle that consists of two radii and the chord that connects them is an isosceles triangle.
- All circles are similar: they are the same shape but may be a different size.
- We can prove that two circles are congruent if their radii, diameters, or circumferences are of equal length.
- If chords of the same length connect radii in the same circle, or in congruent circles, then the two isosceles triangles formed are congruent.