How to Find The Area of a Segment of Circle

A segment in geometry refers to a part of a circle, specifically the space between a chord and an arc. To understand this better, let’s first grasp what a circle is. A circle is the path created by a point that is an equal distance away from a particular point in a flat plane.

This central point is called the circle’s center, and the constant distance from the center to any point on the circle is known as the radius.

Now, back to segments. A segment is simply the area inside a circle. To define it more precisely, it’s the space enclosed by a section of the circle and the angle formed by that section. We also call this enclosed area a sector. To find the area of a circle segment, we subtract the triangular part inside the sector from the sector itself.

In this article, we’ll dive deep into segments, their areas, and all the related theorems, complete with proofs.

Segment of Circle

A circle segment is a part of a circle defined by a straight line (chord) and the curved part between the chord’s ends. Simply put, it’s like cutting out a slice from a pizza. These segments are created by a line that either touches or crosses the circle.

In simpler terms, segments are the portions separated by the curved part of a circle, with the connecting line being the chord between the two ends of the curved part. One important thing to remember is that these segments don’t include the center of the circle.

Types of Segment of Circle

In geometry, a circle can be divided into two primary segments: the major segment and the minor segment. These segments are determined by the relationship between a chord and its corresponding arc. The major segment, which encompasses a larger area, is the portion of the circle enclosed by the chord and the arc.

On the other hand, the minor segment, with a smaller area, is the region bounded by the same chord and arc. This division of a circle into major and minor segments is fundamental to understanding its properties and applications in various mathematical and practical contexts.

Area of a Segment of Circle Formula

You can find the area of a segment in two ways: by using radians or degrees. The formulas for calculating the area of a segment in a circle are as follows:

How to Find the Area of Segment of Circle?

If you know the angle θ (in degrees) in a circle, you can find the area of a sector called “AOBC” using this formula:

Area of AOBC = (θ/360°) × πr²

Now, let’s find the area of a part of the circle called “ABC”:

Area of ABC = Area of AOBC – Area of ΔAOB

Area of ABC = (θ/360°) × πr² – AΔAOB

To calculate the area of ΔAOB, follow these two steps:

Step 1: Find the height OP of ΔAOB using the Pythagoras theorem. You can use one of two methods:

– If you know the length of AB, use this formula:

OP = √(r² – (AB/2)²)

– If you know θ (in degrees), use this formula:

OP = r * cos(θ/2)

Step 2: Calculate the area of ΔAOB using the formula for the area of a triangle:

Area of ΔAOB = ½ × base × height = ½ × AB × OP

Once you’ve found the height and the area of ΔAOB, you can substitute these values into the formula for the area of segment ABC to calculate its area.

Segment of Circle Theorems

There are three key theorems related to segments within a circle:

1. Alternate Segment Theorem
2. Angle in the Same Segment Theorem
3. Alternate Angle Theorem

Theorem 1: Inscribed Angle Theorem

The inscribed angle theorem simplifies a relationship between angles in a circle. It states that when you have an angle (let’s call it θ) inscribed within a circle, this angle is exactly half the size of the central angle (which we’ll call 2θ) that covers the same arc on the circle.

In other words, if we label the inscribed angle as ∠AOC and the central angle as ∠ABC:

∠AOC = 2∠ABC

Now, let’s prove this theorem:

Imagine we have a circle with a center point O, and three points A, B, and C lying on its circumference. If we draw lines from O to A (OA) and from O to C (OC), we create a triangle AOC. Inside this triangle, we have an angle ∠ACD (let’s call it xz, which is equal to angle ABC. This situation occurs when DC is a tangent to the circle.

Since a tangent to a circle is always at a 90-degree angle to the circle’s radius, we can conclude that the sum of angles x and y is 90 degrees:

x + y = 90° ——————————(i)

Now, if we bisect (cut in half) triangle AOC from point O, we create a right-angled triangle with an angle z. So, angles ∠AOE and ∠COE are both equal to z. Since the sum of angles in a triangle is always 180 degrees, we can say:

y + z + 90° = 180°

This equation simplifies to:

y + z = 90° —————————(ii)

Equating equations (i) and (ii), we find that:

x = z

And because we’ve already established that ∠ABC = ∠ACD = x, we can conclude that:

∠AOC = 2z

and

∠ABC = x

This ultimately leads us to the result that:

∠AOC = 2∠ABC

Theorem 2: Angle in the Same Segment Theorem

In the same segment of a circle, angles are always equal. This means that when you have two angles formed by the same arc on the circle’s circumference, they will have the same measurement.

For instance, if you have a triangle ABC and another triangle ADC, with both ∠ABC and ∠ADC in the major part of the circle, you can be sure that these two angles are of equal size.

To prove this equality, you can start by connecting points A and C with the center of the circle, O.

Now, if you call the angle ∠AOC ‘x’, you can use a theorem to show that:

x = 2∠ABC (Equation i)

And at the same time:

x = 2∠ADC (Equation ii)

By comparing these two equations, you can conclude that:

∠ABC is indeed equal to ∠ADC.

Theorem 3: Alternate Angle Theorem

The alternate segment theorem, also known as the tangent-chord theorem, tells us that in a circle, the angle formed between a chord and a tangent line from one of the chord’s endpoints is equal to the angle in the alternate segment.

So, if we have ∠ACD = ∠ABC = x in a circle, this theorem is at play.

Area of Segment of Circle Example

Question:

You have a circle with a radius of 8 cm. Find the area of the segment corresponding to an arc subtending an angle of 120° at the center.

Solution:

To determine the area of this circle segment, we can divide it into two parts:

The area of the sector AOB (the shaded region plus the unshaded region):

Area = (θ/360°) × πr² = (120°/360°) × π × 8² = 32π square cm.

The area of triangle AOB (ΔAOB):

To calculate this, we need to find the values of OC and AB.

OC = 8 × cos 60° = 8 × ½ = 4 cm,

AB = 2 × 8 × sin 60° = 2 × 8 × (√3/2) = 8√3 cm.

Now, we can find the area of ΔAOB:

Area = ½ × OC × AB = ½ × 4 × 8√3 = 16√3 square cm.

Now, the area of the segment AB is determined by subtracting the area of ΔAOB from the area of the sector AOB:

Area of segment AB = 32π square cm – 16√3 square cm.

Area of a Segment of Circle FAQs

What is a circle segment in geometry?

A circle segment is a portion of a circle defined by a chord (a straight line) and the curved part between the ends of that chord. It's like cutting out a slice from a pizza. These segments are created by lines that either touch or cross the circle.

How can I divide a circle segment into major and minor segments?

In geometry, you can divide a circle into two primary segments: the major segment and the minor segment. The major segment encompasses a larger area and is the region enclosed by the chord and the arc. The minor segment, with a smaller area, is the part bounded by the same chord and arc. This division is based on the relationship between a chord and its corresponding arc.

What are the key theorems related to circle segments?

There are three important theorems related to circle segments: the Alternate Segment Theorem, the Angle in the Same Segment Theorem, and the Alternate Angle Theorem. These theorems help in understanding and solving problems involving circle segments and angles within them.