Area of sector formula

\( \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...

The angle marked off by the original and final locations of the radius line (that is, the angle at the center of the pie / pizza) is the "subtended" angle of the sector. This angle can also be referred to as the "central" angle of the sector. In the picture above, the central angle is labelled as " θ" (which is pronounced as "THAY-tuh"). What is the area A of the sector subtended by the marked central angle θ? What is the length s of the arc, being the portion of the circumference subtended by this angle? To determine these values, let's first take a closer look at the area and circumference formulas. The area and circumference are for the entire circle, one full revolution of the radius line. The subtended angle for "one full revolution" is 2π. So the formulas for the area and circumference of the whole circle can be restated as: What is the point of splitting the angle value of "once around" the circle? I did this in order to highlight how the angle for the whole circle (being 2π) fits into the formulas for the whole circle. This then allows us to see exactly how and where the subtended angle θ of a sector will fit into the sector formulas. Now we can replace the "once around" angle (that is, the 2π) for an entire circle with the measure of a sector's subtended angle θ, and this will give us the formulas for the area and arc length of that sector: Confession: A big part of the reason that I've explained the relationship between the circle formulas and the sector formulas...

Anderson Gomes Da Silva Anderson holds a Bachelor's and Master's Degrees (both in Mathematics) from the Fluminense Federal University and the Pontifical Catholic University of Rio de Janeiro, respectively. He was a Teaching Assistant at the University of Delaware (UD) for two and a half years, leading discussion and laboratory sessions of Calculus I, II and III. In the Winter of 2021 he was the sole instructor for one of the Calculus I sections at UD. • Instructor Circular Sector Consider a circumference is a diameter and the circle is divided in two regions with equal area. Otherwise, one of the circular sectors will have smaller area and, for that reason, is going to be called the minor sector. The other, as expected, is going to be the major sector. The goal of this lesson is to explain how to find the area of a Sketch of a pizza with 8 slices How to Find the Area of a Circular Sector? We need to know two of the following three items to find the area of a circular sector: central angle ( cm. Note that in the formula for the area of the sector the unknowns are the angle and the radius, not the angle and the arc length. Observe the following scheme. Central angle, arc, and radius The length of an arc of a circle is proportional to the corresponding central angle (. Finding the Area of a Sector: Word Problem In this activity, you will check your knowledge of how to find the area of a sector. Directions For this activity, print or copy this page on a blank piece of paper. ...

Area Of Sector Of A Circle Before knowing about a sector of a circle, let’s know how the area of a circle is calculated. When it comes to the area, it is always related to two dimensions. Anything which is two-dimensional can form a plane. So, any two-dimensional figure will have an area. What about a circle? Definition 1: A circle is the collection of all the points in a plane which are at a fixed distance from a fixed point. The fixed point is known as the center of the circle and the fixed distance is known as the radius of the circle. Definition 2: If all the points which lie inside and on the circle are taken together, the plane constructed is known as a disk. A disk is basically the region bounded by a circle. So, the \(\begin \) Definition 3: The portion of the circle enclosed by two radii and the corresponding arc is known as the sector of a circle. Basically, a sector is the portion of a circle. It would hence be right to say that a semi-circle or a quarter-circle is a sector of the given circle. In fig.1, OPAQ is called the minor sector and OPBQ is called the major sector because of lesser and greater areas. The angles subtended by the arcs PAQ and PBQ are equal to the angle of the sectors OPAQ and OPBQ respectively. When the angle of the sector is equal to 180°, there is no minor or major sector. Area of sector In a circle with radius r and center at O, let ∠POQ = θ (in degrees) be the angle of the sector. Then, the area of the circle is calculated using the...

Sometimes you might need to determine the area of a sector, say for math questions or for a project you are working on. A sector is a part of a circle that is shaped like a piece of pizza or pie. To find the area of this piece, you need to know the radius, arc length and the degree of the central angle. With this information, finding the area of a sector is a simple matter of plugging the numbers into given formulas. Set up the formula A = ( θ 360 ) π r 2 . When finding the area of a sector, you are really just calculating the area of the whole circle, and then multiplying by the fraction of the circle the sector represents. • A circle is 360 degrees, so when you place the measurement of the sector's central angle over 360, it gives you the fraction of the whole circle. X Research source Plug the sector's central angle measurement into the formula. Divide the central angle by 360. Doing this will give you what fraction or percent of the entire circle the sector represents. X Research source • For example, if the central angle is 100 degrees, you will divide 100 by 360, to get 0.28. (The area of the sector is about 28 percent of the area of the whole circle.) • If you don't know the measurement of the central angle, but you know what fraction of the circle the sector is, determine the measurement of the angle by multiplying that fraction by 360. For example, if you know the sector is one-fourth of the circle, multiply 360 by one-fourth (.25) to get 90 degrees. Plug the rad...

Explanation: Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle radians. If the radius of the circle is , what is the area of the semi-circular design? Recall the following formual for area of a sector: So, we plug in our knowns and solve for area! Therefore, our answer is... Explanation: Lucy is making a solar panel to cover a portion of a satellite dish. If the central angle of the sector the solar panel will cover is , and the satellite dish has a radius of , what area will the solar panel cover? To calculate area of a sector, use the following formula: Where the numerator of the fraction is the measure of the desired angle in radians, and r is the radius of the circle. Now, we know both our variables, so we simply need to plug them in and simplify. Now, this looks messy, but we can simplify it to get: Next, use your calculator to find a decimal answer, and then round to get our final answer. Making the area of the sector If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by ...

central angle 36 0 ∘ = arc length circumference = sector area circle area \dfrac 3 6 0 ∘ central angle ​ = circumference arc length ​ = circle area sector area ​ start fraction, start text, c, e, n, t, r, a, l, space, a, n, g, l, e, end text, divided by, 360, degrees, end fraction, equals, start fraction, start text, a, r, c, space, l, e, n, g, t, h, end text, divided by, start text, c, i, r, c, u, m, f, e, r, e, n, c, e, end text, end fraction, equals, start fraction, start text, s, e, c, t, o, r, space, a, r, e, a, end text, divided by, start text, c, i, r, c, l, e, space, a, r, e, a, end text, end fraction • Your answer should be • an integer, like 6 6 6 6 • a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5 • a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4 • a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4 • an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75 • a multiple of pi, like 12 pi 12\ \text 2 / 3 pi 2, slash, 3, space, start text, p, i, end text Check Explain For example, in the figure above, O A ‾ \overline O C start overline, O, C, end overline are radii of the circle, so O A = O C OA=OC O A = O C O, A, equals, O, C . Triangle A O C AOC A O C A, O, C is an isosceles triangle, and the measures of ∠ O A C \angle OAC ∠ O A C angle, O, A, C and ∠ O C A \angle OCA ∠ O C A angle, O, C, A are both 3 0 ∘ 30^\circ 3 0 ∘ 30, degrees . • Your answer should be • an integer, like 6 6 6 6 • a simplified proper fraction, like...

A sector of a circle is a closed figure bounded by an arc of a circle and two of its radii. Each sector has a unique central (sector) angle that it subtends at the centre of the circle. Minor sectors subtend angles less than 180° while major sectors subtend angles more than 180°. The semicircular sector subtends an angle of 180°. The following diagram shows a minor sector of a circle of radius r units whose central angle is θ. (Image will be uploaded soon) Firstly, it is important to realise that a circle is a perfectly symmetric planar figure. It has infinite lines of symmetry passing through its centre. This means that all sectors of the same circle or of congruent circles, which have the same central angles are congruent sectors. We will use this fact to derive the formulas for the perimeter and the area of a sector. Consider the following diagram which shows two adjacent congruent sectors of a circle. (Image will be uploaded soon) Sectors OAB and OCB have the same central angle (θ), hence their arc lengths AB and BC are equal (l), as well as their areas. Together they form a bigger sector OAC. Clearly the central angle of sector OAC is 2θ, while its arc length AC is 2l . Therefore, we realise that the arc length of a sector is directly proportional to its central angle. The same applies for the area of the sector as well. So let’s consider a sector of a circle of radius r and central angle θ. We know that for a central angle of 360°, the sector is actually the complete...

The Lesson The area of a sector of a In this formula, r is the θ is the How to Find the Area of a Sector of a Circle (Radians) Finding the area of a sector of a circle, when the angle is in radians, is easy. Question What is the area of the sector with an angle of 2 radians and a radius of 5 cm, as shown below? Step-by-Step: Lesson Slides The slider below shows another real example of how to find the area of a sector of a circle when the angle is in radians. What Is a Sector? A sector is a region of a circle bounded by two radii and the arc lying between the radii. What Are Radians? Radians are a way of measuring angles. 1 radian is the angle found when the radius is wrapped around the circle. Why Does the Formula Work? The area of a sector is just a πr 2, where r is the radius. For example, a sector that is half of a circle is half of the area of a circle. A sector that is quarter of a circle has a quarter of the area of a circle. In each case, the fraction is the angle of the sector divided by the When measured in radians, the full angle is 2π. Hence for a general angle θ, the formula is the fraction of the angle θ over the full angle 2π multiplied by the area of the circle:

\( \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...