Lesson Explainer: Circles Mathematics

In this explainer, we will learn how to identify the parts of a circle, such as the radius, chord, and diameter, and use their properties to solve problems.

We can start by defining what a circle is.

Definition: Circle

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 โ€œa circle of radius 3 cm.โ€

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 ๐‘€ and ๐‘ are 2 cm and 6 cm respectively, determine the length of ๐‘€๐‘.

Answer

We can begin answering this problem by observing that the two given circles meet at a point, which we can designate ๐‘ƒ.

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 ๐‘€๐‘ƒ and ๐‘๐‘ƒ 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 ๐‘€ is 2 cm, then the radius of circle ๐‘€, the length ๐‘€๐‘ƒ, is 2รท2=1cm.

Similarly, given that the diameter of the circle with center ๐‘ is 6 cm, then its radius, the length ๐‘๐‘ƒ, is 6รท2=3cm.

We can then calculate the length of ๐‘€๐‘ by adding the two radii: lengthoflengthoflengthofcm๐‘€๐‘=๐‘€๐‘ƒ+๐‘๐‘ƒ=1+3=4.

Therefore, we can give the answer that the length of ๐‘€๐‘ 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 ๐‘ is 23, and ๐ต๐‘=5. Find the length of ๐ด๐ต.

Answer

We are given the length of the diameter of the circle with center ๐‘ 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 ๐‘; 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 ๐ด๐‘ is a radius of the circle with center ๐‘. The radius of a circle is half of its diameter. Therefore, ๐ด๐‘=23รท2=11.5.

We can add this length to the diagram, along with the information that ๐ต๐‘=5, to help us visualize the problem.

We are required to find the length of ๐ด๐ต. This can be found by subtracting ๐ต๐‘ from ๐ด๐‘; therefore, ๐ด๐ต=๐ด๐‘โˆ’๐ต๐‘=11.5โˆ’5=6.5.lengthunits

Hence, we can give the answer that the length of ๐ด๐ต 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?

Answer

In the figure, we can observe that there are 3 different lines in the circle: ๐ด, ๐ต, and ๐ถ.

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 ๐ด.

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 ๐ด 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 360โˆ˜ 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 ๐‘ and two shorter sides ๐‘Ž and ๐‘, the Pythagorean theorem states that ๐‘Ž+๐‘=๐‘.๏Šจ๏Šจ๏Šจ

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 ๐ด๐ต=20, find the length of ๐‘‚๐ด.

Answer

In this problem, we are given the length of one side in the right triangle ๐‘‚๐ด๐ต. 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 ๐‘‚๐ด, as we are only given the length of one side, ๐ด๐ต. However, we can use the properties of the circle that contains triangle ๐‘‚๐ด๐ต. ๐‘‚๐ด and ๐‘‚๐ต 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 ๐‘‚๐ด=๐‘‚๐ต. We can denote these lengths as ๐‘ฅ and write the lengths on the diagram as below.

Applying the Pythagorean theorem, we can set up an equation and simplify, given that ๐ด๐ต=20. This gives us (๐ด๐ต)=๐‘ฅ+๐‘ฅ20=2๐‘ฅ400=2๐‘ฅ200=๐‘ฅ.๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ

We can now take the square root of each side of the equation, noting that ๐‘ฅ must be a positive value since it is a length. Thus, we have โˆš200=๐‘ฅโˆš100ร—2=๐‘ฅ10โˆš2=๐‘ฅ.

Hence, we can give the answer that the radius of this circle, the length of ๐‘‚๐ด, is 10โˆš2.

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 ๐‘šโˆ ๐‘€๐ด๐ต?

Answer

We can begin this question by recalling that the angles about a point sum to 360โˆ˜. Thus, we can calculate ๐‘šโˆ ๐ด๐‘€๐ต as ๐‘šโˆ ๐ด๐‘€๐ต=360โˆ’306=54.โˆ˜โˆ˜โˆ˜

Next, we observe that ๐‘€๐ด and ๐‘€๐ต are both radii of the circle. This means that they will be of the same length and, furthermore, โ–ณ๐‘€๐ด๐ต must be an isosceles triangle as it has two sides with the same length. Hence, we have two congruent angles: ๐‘šโˆ ๐‘€๐ด๐ต=๐‘šโˆ ๐‘€๐ต๐ด.

We can define both of these angles to be ๐‘ฆโˆ˜.

To find the missing angle, we remember that the internal angles in a triangle sum to 180โˆ˜. Therefore, we have ๐‘šโˆ ๐ด๐‘€๐ต+๐‘šโˆ ๐‘€๐ด๐ต+๐‘šโˆ ๐‘€๐ต๐ด=180.โˆ˜

Substituting the values for the angles, we have ๐‘šโˆ ๐ด๐‘€๐ต+๐‘ฆ+๐‘ฆ=18054+2๐‘ฆ=1802๐‘ฆ=180โˆ’542๐‘ฆ=126๐‘ฆ=63.โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜

As we defined ๐‘šโˆ ๐‘€๐ด๐ต as ๐‘ฆโˆ˜, we can give the answer that ๐‘šโˆ ๐‘€๐ด๐ต is 63โˆ˜.

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 ๐‘€ and ๐‘ are congruent and ๐ด๐ต=๐ถ๐ท.

  1. Find the value of ๐‘ฅ.
  2. Find the value of ๐‘ฆ.

Answer

Part 1

In the circle with center ๐‘€, since ๐‘€๐ถ and ๐‘€๐ท are both radii of the circle, they will be of equal length. Hence, โ–ณ๐‘€๐ถ๐ท is an isosceles triangle. An isosceles triangle has two sides of equal length and two angles of equal measure. Therefore, ๐‘šโˆ ๐‘€๐ท๐ถ=๐‘šโˆ ๐‘€๐ถ๐ท=25.โˆ˜

Next, using the fact that the angle measures in a triangle sum to 180โˆ˜, we have ๐‘šโˆ ๐‘€๐ท๐ถ+๐‘šโˆ ๐‘€๐ถ๐ท+๐‘šโˆ ๐ถ๐‘€๐ท=18025+25+๐‘ฅ=18050+๐‘ฅ=180๐‘ฅ=130.โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜โˆ˜

Thus, we have that ๐‘ฅ=130โˆ˜.

Part 2

Given that the circles with centers ๐‘€ and ๐‘ are congruent, then the radii of both will be congruent. We are given that ๐ด๐ต=๐ถ๐ท; hence, โ–ณ๐‘€๐ถ๐ทโ‰…โ–ณ๐‘๐ต๐ด, by the side-side-side (SSS) congruency criterion. Note that since these are isosceles triangles, we could also write that โ–ณ๐‘€๐ถ๐ทโ‰…โ–ณ๐‘๐ด๐ต.

Therefore, ๐‘šโˆ ๐‘๐ต๐ด=๐‘šโˆ ๐‘€๐ถ๐ท๐‘ฆ=25.โˆ˜

Hence, we have that ๐‘ฆ=25โˆ˜.

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.

Key Points

  • 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.

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