Lesson Explainer: Areas of Circular Segments Mathematics

In this explainer, we will learn how to find the area of a circular segment.

Definition: A Circular Segment

A circular segment is a region of a circle bounded by an arc and a chord passing through the endpoints of the arc.

In fact, each chord divides a circle into two segments: a minor segment, which is smaller than a semicircle, and a major segment, which is larger than a semicircle.

Let us consider how to derive the formula for the area of a minor circular segment. Consider the minor circular segment formed by the chord 𝐴𝐡 in the circle with center 𝑂. The angle between the two radii connecting the endpoints of chord 𝐴𝐡 to the center of the circle is known as the central angle, and we will denote this angle as πœƒ. The formula we derive will depend on the unit of measurement used for this angle. We will consider the formula in degrees first and then see how it can be adapted if the central angle is measured in radians.

From the figure below, we see that the area of the minor circular segment (shaded in orange) is the difference between the area of the sector 𝐴𝐡𝑂 (outlined in green) and the area of the triangle 𝐴𝐡𝑂 (outlined in pink).

We recall that the area of a triangle with side lengths π‘Ž and 𝑏 and the included angle 𝐢 is given by 12π‘Žπ‘πΆsin. Applying this formula to our diagram, we have areaoftrianglesin𝐴𝐡𝑂=12π‘Ÿπœƒ.∘

We also recall that the formula for the area of a sector with a central angle in degrees is areaofsector=πœ‹π‘Ÿπœƒ360.

We therefore have areaofsegmentareaofsectorareaoftrianglesin=π΄π΅π‘‚βˆ’π΄π΅π‘‚=πœ‹π‘Ÿπœƒ360βˆ’12π‘Ÿπœƒ.∘

We can factor by 12π‘ŸοŠ¨ to give areaofsegmentsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰.∘

Formula: Area of a Circular Segment Where the Central Angle Is Measured in Degrees

The area of a circular segment of radius π‘Ÿ units and central angle πœƒ measured in degrees is areaofcircularsegmentsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰.∘

Now let us consider a variant of this formula when the central angle is measured in radians. We recall that the formula for the area of a sector with a central angle of πœƒ radians is areaofsector=12π‘Ÿπœƒ.

Hence, using the same formula for the area of triangle 𝐴𝐡𝑂, the area of the segment is areaofsegmentsin=12π‘Ÿπœƒβˆ’12π‘Ÿπœƒ.

We can factor by 12π‘ŸοŠ¨ to obtain the simplified formula for the area of a segment in radians: areaofsegmentsin=12π‘Ÿ(πœƒβˆ’πœƒ).

Formula: Area of a Circular Segment Where the Central Angle Is Measured in Radians

The area of a circular segment of radius π‘Ÿ units and central angle πœƒ measured in radians is areaofcircularsegmentsin=12π‘Ÿ(πœƒβˆ’πœƒ).

In each problem, we must check carefully whether the central angle is given in degrees or radians and select the appropriate formula to apply. We must also ensure that our scientific calculator is in the correct mode for the unit of measure used for the central angle. If the central angle is not given, we can choose which formula to use. We will consider later the validity of these formulas for calculating the area of major segments as well as minor segments.

In our first example, we are given the central angle in radians and so we will apply the second formula to calculate the area of a circular segment.

Example 1: Finding the Area of a Circular Segment given the Diameter of the Circle and the Central Angle

The diameter of a circle is 14 cm and the central angle is 5.79 rad. Find the area of the circular segment giving the answer to two decimal places.

Answer

The formula for calculating the area of a circular segment, when the central angle is measured in radians, is areaofcircularsegmentsin=12π‘Ÿ(πœƒβˆ’πœƒ).

We are given that the central angle is 5.79 radians, and we can calculate the radius of the circle by halving the diameter: π‘Ÿ=142=7.cm

Substituting π‘Ÿ=7 and πœƒ=5.79 into the area formula, and ensuring our calculator is in radian mode, we have areaofcircularsegmentsin=12Γ—7Γ—(5.79βˆ’5.79)=153.454β€¦β‰ˆ153.45.

The area of the circular segment, to two decimal places, is 153.45 cm2.

In other problems, we may need to relate the area of a circular segment to the length of the arc connecting the endpoints of the chord. We therefore need to recall the formulas for calculating an arc length, which again depend on the unit of measure used for the central angle.

If the central angle is measured in degrees, the arc length 𝑠 is given by 𝑠=πœƒ360Γ—2πœ‹π‘Ÿ, and if the central angle is measured in radians, the arc length is given by 𝑠=π‘Ÿπœƒ.

In our next example, we will use the second formula above to find the central angle in radians given an arc length and radius. Then, we will use this central angle to compute the area of a circular segment.

Example 2: Finding the Area of a Circular Segment given Its Circle’s Radius Length and Its Arc Length

The radius of a circle is 40 cm and the arc length of a segment is 18 cm. Find the area of the segment giving the answer to two decimal places.

Answer

In this problem, we are not given the measure of the central angle, so we can choose whether to work in radians or degrees. As the formulas for arc length and area of a circular segment are simpler in radians, this would be the easier choice.

Recall that the arc length 𝑠 is given by 𝑠=π‘Ÿπœƒ. Substituting the given arc length and radius, we have 18=40πœƒ.

Rearranging the equation, we obtain πœƒ=1840=920.

We can now substitute the central angle of 920 radians and radius of 40 cm into the formula for the area of a circular segment and evaluate, ensuring our calculator is in radian mode: areaofcircularsegmentsinsinsin=12π‘Ÿ(πœƒβˆ’πœƒ)=12Γ—40ο€Ό920βˆ’ο€Ό920=800ο€Ό920βˆ’ο€Ό920=12.027β€¦β‰ˆ12.03.

The area of the segment, to two decimal places, is 12.03 cm2.

It is also possible to work backward from knowing the area of a circular segment and either its radius or central angle to calculating the other. Let us now consider such an example.

Example 3: Finding the Radius of a Circle given the Area of a Circular Segment and the Central Angle

The area of a circular segment is 34 cm2 and the central angle is 75∘. Find the radius of the circle giving the answer to the nearest centimetre.

Answer

Recall the formula for the area of a circular segment when the central angle is measured in degrees: areaofcircularsegmentsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰.∘

Substituting the given area and central angle, we have 34=π‘Ÿ2ο€Ό75πœ‹180βˆ’75.∘sin

We can now solve this equation to determine the radius of the circle. We must ensure that our scientific calculator is in degree mode: 34=π‘Ÿ2Γ—0.343β€¦π‘Ÿ=34Γ—20.343…=198.209….

Taking the positive square root, we obtain the radius: π‘Ÿ=14.078β€¦β‰ˆ14.

The radius of the circle, to the nearest centimetre, is 14 cm.

In the next example, we will demonstrate how to find the area of a segment given the radius of a circle and the length of the chord which bounds the segment.

Example 4: Finding the Area of the Minor Circular Segment given the Radius of the Circle and the Length of the Chord

A chord and radius of a circle are each 24 cm. Find the area of the minor circular segment giving the answer to two decimal places.

Answer

We begin by producing a sketch of this circle.

Our sketch emphasizes the fact that both the radius of the circle and the chord are of length 24 cm, and so the triangle formed by the chord and the two radii is equilateral. Hence the central angle πœƒ is 60∘, or πœ‹3 radians. Since the formula for the area of a circular segment is simpler when the central angle is measured in radians, we define πœƒ=πœ‹3radians.

We recall the formula for the area of a segment when the central angle is measured in radians: areaofcircularsegmentsin=12π‘Ÿ(πœƒβˆ’πœƒ).

We substitute π‘Ÿ=24 and πœƒ=πœ‹3 into this formula and evaluate. Our calculator must be in radian mode to ensure the correct answer: areaofcircularsegmentsinsin=12Γ—24Γ—ο€»πœ‹3βˆ’ο€»πœ‹3=288ο€»πœ‹3βˆ’ο€»πœ‹3=52.177β€¦β‰ˆ52.18.

The area of the minor circular segment, to two decimal places, is 52.18 cm2.

When we derived the formula for the area of a segment, we assumed that the segment was a minor segment, as we calculated its area as the difference between the areas of a minor sector and a triangle. In fact, the same formula applies when calculating the area of a major segment, even though its area cannot be geometrically broken down in the same way. Let us consider the validity of the formula only in degrees, since a very similar method can be followed to demonstrate its validity in radians.

Geometrically, the area of a major segment with central angle πœƒβˆ˜ is the sum of the area of the major sector, shown in blue on the figure below, and a triangle, shown in orange.

Note that the angle of the major sector is πœƒβˆ˜, but the angle in the triangle is (360βˆ’πœƒ)∘. The area of the major sector is given by areaofmajorsector=πœƒπœ‹π‘Ÿ360, and the area of the triangle is given by areaoftrianglesin=12π‘Ÿ(360βˆ’πœƒ).∘∘

Therefore, the area of the major segment is given by areaofmajorsegmentareaofmajorsectorareaoftrianglesin=+=πœƒπœ‹π‘Ÿ360+12π‘Ÿ(360βˆ’πœƒ).∘∘

We can factor by 12π‘ŸοŠ¨ to obtain areaofmajorsegmentsin=12π‘Ÿο€½πœƒπœ‹180+(360βˆ’πœƒ).∘∘

To achieve the same formula we have already seen for the area of a minor segment, some further manipulation is required. We recall two properties of the sine function: firstly, the sine function has a period of 360∘, so adding or subtracting any integer multiple of 360∘ to the argument does not change its value. Therefore, sinsinsin(360βˆ’πœƒ)=(360βˆ’πœƒβˆ’360)=(βˆ’πœƒ).∘∘∘∘∘∘

Secondly, we recall that sine is an odd function, and hence sinsin(βˆ’πœƒ)=βˆ’(πœƒ).∘∘

Substituting this into our formula gives areaofmajorsegmentsin=12π‘Ÿο€½πœƒπœ‹180βˆ’(πœƒ),∘ which agrees with the formula we have already found for the area of a minor segment. Thus, even though the derivation is different, the same formula is valid for both major and minor segments.

In the next example, we will demonstrate how to calculate the area of a major segment given its central angle and the length of the chord.

Example 5: Finding the Area of the Major Circular Segment given the Central Angle and the Length of the Chord

𝐴𝐡 is a chord of length 17 cm with a central angle of 155∘. Find the area of the major circular segment giving the answer to the nearest square centimetre.

Answer

We begin by sketching the circle.

Note that we are calculating the area of the major circular segment. This is the larger of the two segments, whose central angle is larger than 180∘. We can calculate the measure of the reflex angle 𝐴𝑂𝐡 by subtracting the known angle from 360∘: π‘šβˆ π΄π‘‚π΅=360βˆ’155=205.∘∘∘

We are given the length of the chord but not the radius of the circle. In order to calculate the radius, let us examine triangle 𝐴𝐡𝑂. As 𝑂𝐴 and 𝑂𝐡 are radii of the circle, this triangle is isosceles, so it can be divided into two congruent right triangles by drawing a line from 𝑂 to the midpoint of 𝐴𝐡.

This divides 𝐴𝐡 into two equal lengths of 8.5 cm and divides the central angle of 155∘ into two congruent angles measuring 77.5∘. We can now apply right triangle trigonometry in these triangles to calculate the length of the radius. The radius 𝑂𝐴 is the hypotenuse of triangle 𝐴𝑀𝑂, and in relation to the angle of 77.5∘, the side with length 8.5 cm is the opposite side. Applying the sine ratio, we find sinoppositehypotenuse77.5==8.5π‘Ÿ.∘

Rearranging this equation and evaluating, ensuring our calculator is in degree mode, gives π‘Ÿ=8.577.5=8.706….sincm∘

We are finally able to calculate the area of the major circular segment, using the central angle of 205∘ and the radius of 8.706… cm: areaofmajorsegmentsinsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰=(8.706…)2ο€Ό205πœ‹180βˆ’205=151.622β€¦β‰ˆ152.∘∘

The area of the major circular segment, to the nearest square centimetre, is 152 cm2.

The formulas we have encountered in this explainer can also be applied to problems in real-world contexts. In such problems, sketching a diagram will often be a helpful first step. Let us consider one final example where we apply these concepts in a real-world context.

Example 6: Finding the Area of Minor Circular Segments in Context

A circular flower bed is divided into four parts by an equilateral triangle inscribed in the circle. The radius of the flower bed is 9 m. Find the area of each minor circular segment giving the answer to two decimal places.

Answer

We begin by sketching the flower bed.

We also sketch in the radii from each vertex of triangle 𝐴𝐡𝐢 to the center of the circle. The area we are looking to calculate is shaded in blue on the figure below.

Recall the formula for the area of a circular segment whose central angle is measured in degrees: areaofsegmentsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰.∘

We are given the radius of the circle, but we need to calculate the central angle. As triangle 𝐴𝐡𝐢 is equilateral, the figure has rotational symmetry of order 3 and so the three circular sectors in the diagram are congruent. Hence, the areas of the three minor circular segments are the same and they all have the same central angle of 3603=120∘∘.

We can now substitute π‘Ÿ=9 and πœƒ=120∘ into the formula for the area of a circular segment: areaofminorsegmentsin=12Γ—9Γ—ο€Ό120πœ‹180βˆ’120.∘

Evaluating, with our calculator in degree mode, gives areaofminorsegment=12Γ—81Γ—(1.228…)=49.748β€¦β‰ˆ49.75.

The area of each minor circular segment, to two decimal places, is 49.75 m2.

Let us finish by recapping some key points.

Key Points

  • A circular segment is a region of a circle bounded by an arc and a chord passing through the endpoints of the arc.
  • The area of a minor circular segment is the difference between the area of a circular sector and a triangle, while the area of a major circular segment is the sum of these two areas. The same formula is valid in both cases.
  • The area of a circular segment with radius π‘Ÿ and central angle πœƒ measured in degrees is areaofsegmentsin=π‘Ÿ2ο€½πœ‹πœƒ180βˆ’πœƒο‰.∘
  • The area of a circular segment with radius π‘Ÿ and central angle πœƒ measured in radians is areaofsegmentsin=12π‘Ÿ(πœƒβˆ’πœƒ).

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