Area of major sector

Segment and Area of a Segment Of Circle Segment and area of a segment of the circle: A segment is a part of a circle basically the region between the chord and an arc. A circle is a path traced by a point that is equidistant from a unique point on the plane, this point is called the centre of the circle and the constant distance is called the radius of the circle. A segment is an interior region of a circle. The area enclosed by a segment and the angle subtended by a segment is called a sector . The area of the segment of a circle is determined by subtracting the triangle formed inside the sector from the sector which has the segment. In this article, we shall discuss in detail the segment and area of a segment of a circle and all related theorems with proof. Table of Contents: • • • • • • • • • • • • Segment of a Circle Definition A segment of a circle can be defined as a region bounded by a chord and a corresponding arc lying between the chord’s endpoints. In other words, a circular segment is a region of a circle which is created by breaking apart from the rest of the circle through a secant or a chord. We can also define segments as the parts that are divided by the circle’s arc and connected through a chord by the endpoints of the arc. It is to be noted that the segments do not contain the center point. Types of Segments in a Circle According to the definition, the part of the circular region which is enclosed between a chord and corresponding arc is known as a segme...

Sector of a Circle is one of the components of a circle like a segment which students learn in their academic years as it is one of the important geometric shapes. From a slice of pizza to a region between two fan blades, we can see sectors of the circle in our daily life everywhere. In this article, we will explore this geometric shape of the sector which is derived from the circle in detail including its areas, perimeter, and all the formulas related to it. Other than that, we will also learn how to solve problems related to it using all the formulas we will learn in this same article. So, let’s fill our brains with the knowledge of the concept with the name “Sector of a Circle”. Sector of a Circle Definition A sector of a circle is a portion of a circle that is enclosed by two radii and the arc that they form. In other words, a sector of a circle is a pie-shaped section of a circle formed by the arc and its two radii and it is produced when a section of the circle’s circumference (also known as an arc) and two radii meet at both extremities of the arc. A semi-circle, which represents half of a circle, is the most frequent sector of a circle. • Major Sector: The sector with a larger arc length is called the major sector. • Minor Sector: The sector with a smaller arc length is called the minor sector. Sector Angle The angle subtended by the arc at the centre of the circle is known as the sector angle or central angle of the sector. In the above diagram, we can see that, t...

Which can be simplified to: θ 2 × r 2 Area of Sector = θ 2 × r 2 (when θ is in radians) Area of Sector = θ × π 360 × r 2 (when θ is in degrees) Area of Segment The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here). There is a lengthy reason, but the result is a slight modification of the Sector formula:

• Afrikaans • العربية • Azərbaycanca • Беларуская • Català • Чӑвашла • Čeština • ChiShona • Dansk • Deutsch • Eesti • Español • Esperanto • Euskara • فارسی • Français • 한국어 • Հայերեն • Italiano • Latviešu • Lëtzebuergesch • Lietuvių • Magyar • Македонски • മലയാളം • Nederlands • 日本語 • Norsk bokmål • Norsk nynorsk • Oʻzbekcha / ўзбекча • ភាសាខ្មែរ • Piemontèis • Polski • Português • Română • Русский • Slovenščina • کوردی • Suomi • Svenska • தமிழ் • ไทย • Українська • Tiếng Việt • 吴语 • 粵語 • 中文 A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a minor sector and the larger being the major sector. θ is the r is the arc length of the minor sector. The angle formed by connecting the endpoints of the arc to any point on the circumference that is not in the sector is equal to half the central angle. Types [ ] A sector with the central angle of 180° is called a quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. Confusingly, the Compass [ ] See also: The total area of a circle is πr 2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2 π (because the area of the sector is directly proportional to its angle, and 2 π is the angle for the whole circle, in radians): A = ∫ 0 θ ∫ 0 r d S = ∫ 0 θ ∫ 0 r r ~ d r ~ d θ ~ = ∫ 0 θ 1 2 r 2 d θ ~ = r 2 θ 2 Converting the central angle in...

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 𝐴 𝐵 in the circle with center 𝑂. The angle between the two radii connecting the endpoints of chord 𝐴 𝐵 to the center of the circle is known as the central angle, and we will denote this angle as 𝜃. 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 𝐴 𝐵 𝑂 (outlined in green) and the area of the triangle 𝐴 𝐵 𝑂 (outlined in pink). We recall that the area of a triangle with side lengths 𝑎 and 𝑏 and the included angle 𝐶 is given by 1 2 𝑎 𝑏 𝐶 s i n. Applying this formula to our diagram, we have a r e a o f t r i a n g l e s i n 𝐴 𝐵 𝑂 = 1 2 𝑟 𝜃 .  ∘ We also recall that the formula for the area of a sector with a central angle in degrees is a r e a o f s e c t o r = 𝜋 𝑟 𝜃 3 6 0 .  We therefore have a r e a o f s e g m e n t a r e a o f s e c t o r a r e a o f t r i a n g l e s i n = 𝐴 𝐵 𝑂 − 𝐴 𝐵 𝑂 = 𝜋 𝑟 𝜃 3 6 0 − 1 2 𝑟 𝜃 .   ∘ We can factor by 1 2 𝑟  to give a r e a o f s e g m e n t ...

Paul Mazzola • • • • Parts of a circle A circle is the set of all points equidistant from a given point on a plane. It encloses a measurable area. You can divide circles into sectors or pieces. Pieces cut off by connecting any two points on the circle are segments. Since both sectors and segments are part of a circle's interior, both have area. We can calculate their area using formulas. Pies, cakes, pizzas; so many foods we eat neatly lend themselves to mathematics, because they are models of circles. The area of a circle is always calculated using the known relationship of π \pi π between a circle's radius, r, (or diameter, d) and its circumference: A = π ( d 2 ) 2 A=\pi (\frac A chord is a line created by connecting any two points on the circle, without worrying about the center. A chord creates an area called a segment. The areas of both segments and sectors can be calculated in square units of whatever linear measurement you are given. Area of a sector For Pi Day, your Math Club is celebrating with pies. You make or buy an 8" blueberry pie. You can quickly calculate the area of the pie, using either formula: A = 50.265482 i n 2 A=50.265482 i A = 16 1 ​ × π × r 2 But you can also use a general formula, based on the central angle of the slice, represented by n° or θ \theta θ: A = 1.968894 i n 2 A=1.968894 i A = 1.968894 i n 2 What would a 10° slice cost, at $0.31 a square inch? Only $0.61. Such a deal! Area of segment of a circle To calculate areas of segments, you firs...

Circular Sector A circle sector, is the portion of a circle enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. A sector with the central angle of 180° is often called a half-disk and is bounded by a diameter and a semicircle. Sectors with other central angles are given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one 4th, 6th or 8th part of a full circle, respectively. In the diagram \(\theta\) is the central angle, \(r\) is the radius of the circle, \(s\) is the arc length of the sector, and \(c\) is the chord connecting the endpoints of the arc. The sector arc length can be expressed using the following quite obvious formula: $$s = r \cdot \theta,$$ where \(\theta\) is the angle in radians. The length of the chord is given by the formula: $$c = 2r \cdot sin\frac .$$ These formulas are used in our Sector Area Calculator. With this calculator you can easily find all the parameters of a circle sector (\(r\), \(\theta\), \(c\), \(s\), \(A\)) if any two of these parameters are known, except for the following parameter pairs: chord, arc and chord, area. The reason for the latter is that knowledge of these pairs of parameters does not allow, in the general case, to unambiguously find the remaining parameters. The angle \(\theta\) can be specified both in degrees and in radians. Please note that angles greater than 360 degrees (\(2\pi\)) are ta...

\( \newcommand ~. \nonumber \] Solving for \(A \) in the above equation, we get the following formula: Example 4.8 Find the area of a sector whose angle is \(\frac ~. \nonumber \] Example 4.9 Find the area of a sector whose angle is \(117^\circ \) in a circle of radius \(3.5 \) m. Solution: As with arc length, we have to make sure that the angle is measured in radians or else the answer will be way off. So converting \(\theta=117^\circ \) to radians and using \(r=3.5 \) in Equation \ref \nonumber \] Note: The central angle \(\theta \) that intercepts an arc is sometimes called the angle subtended by the arc. Example 4.10 Find the area of a sector whose arc is \(6 \) cm in a circle of radius \(9 \) cm. Solution Using \(s=6 \) and \(r=9 \) in Equation \ref \) rad. Example 4.11 Find the area \(K \) inside the belt pulley system from Example 4.7 in Section 4.2. Solution: Recall that the belt pulleys have radii of \(5 \) cm and \(8 \) cm, and their centers are \(15 \) cm apart. We showed in Example 4.7 that \(EF=AC=6\,\sqrt we must have \(\;\theta > \sin\;\theta\) for \(0 < \theta \le \pi \) (measured in radians), since the area of a segment is positive for those angles. Solution Figure 4.3.5 shows the segment formed by a chord of length \(3 \) in a circle of radius \(r=2 \). We can use the Law of Cosines to find the subtended central angle \(\theta\): \[ \cos\;\theta ~=~ \frac \nonumber \] Example 4.13 The centers of two circles are \(7 \) cm apart, with one circle having a ra...