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In this video, we will learn how to find the area of a circular segment.
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We will begin by looking at the definition of a circular segment.
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We will then identify formulas that can be used to calculate the area of a circular segment and then use these to solve some problems.
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What is the definition of a circular segment?
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Is it (A) a region of a circle bounded by an arc and a chord passing through the end points of the arc?
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(B) A region of a circle bounded by two radii and an arc.
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(C) A region of a circle bounded by a chord and a central angle subtended by that arc.
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(D) A region of a circle bounded between two chords and two arcs.
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Or (E) an arc which is half of the circumference.
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The circle shown has been split into two segments.
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The smaller part of the circle is known as the minor segment.
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The larger part is known as the major segment.
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Both of these segments are bounded by an arc.
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They are also bounded by a common chord.
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This means that the correct definition is option (A).
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A circular segment is a region of a circle bounded by an arc and a chord passing through the end points of the arc.
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In our diagram, we have a minor segment in orange and a major segment in pink.
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We will now look at the formulas that can be used to calculate the area of a segment.
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Which formula can be used to find the area of a circular segment, given radius π and a central angle π?
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We recall that any circle can be split into a minor and major segment as shown.
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Weβre also told that the radius of the circle is π and the central angle is π.
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If we label the two points at the end of the chord π΄π΅ and the center of the circle π, then the area of the segment will be equal to the area of the sector π΄ππ΅ minus the area of triangle π΄ππ΅.
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It is important to note at this point that our angle π might be given in degrees or in radians.
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180 degrees is equal to π radians.
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We know that a circle has a total of 360 degrees, which means it will have a total of two π radians.
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As a result, the area of a circular segment can be calculated using two linked formulas, one for degrees and one when the angle is in radians.
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When our angle was measured in degrees, the area of a sector is equal to π out of 360 multiplied by ππ squared.
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As already mentioned, 360 degrees is equal to two π radians.
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This means that the area of a sector when π is in radians is π over two π multiplied by ππ squared.
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In this case, the πs cancel.
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We are left with π over two multiplied by π squared, which is often written as a half π squared π.
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As weβve worked out a formula for the area of a sector in degrees and radians, we will now look at the area of a triangle.
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The area of any triangle can be calculated using the formula a half of ππ multiplied by sin πΆ.
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In our diagram, we can see that the lengths π and π are both equal to the radius or π.
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The angle πΆ is equal to π.
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Therefore, the area of a triangle inside a circle can be calculated using the formula half π squared multiplied by sin π.
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We will now clear some space to work out the formula that can be used to find the area of a circular segment.
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Letβs consider when π is measured in radians first.
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The area of the sector is a half π squared π, and the area of the triangle is a half π squared sin π.
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We can factor out a half π squared as this is common in both terms.
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Inside the parentheses or bracket, weβre left with π minus sin π.
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When the central angle π is given in radians, then the area of the circular segment can be calculated by multiplying a half π squared by π minus sin π.
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If the central angle is given in degrees, then our formula is equal to π over 360 multiplied by ππ squared minus a half π squared sin π.
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Whilst the π squared is common in both terms, we tend not to factor it out here but instead calculate the area of the sector and area of triangle separately.
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We then subtract our two answers to calculate the area of the circular segment.
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Either one of these formulas can be used depending on the context of the question.
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We will now use these to find the area of a segment given different properties of a circle.
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The area of a circle is 227 square centimeters and the central angle of a segment is 120 degrees.
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Find the area of the segment, giving the answer to two decimal places.
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Weβre told in the question that the central angle of a segment is 120 degrees.
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And we need to calculate the area of this segment.
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When the angle of a segment is given in degrees, we can calculate the angle of this segment by subtracting the area of the triangle from the area of the sector.
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The area of the sector is equal to π over 360 multiplied by ππ squared.
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The area of the triangle is equal to a half π squared multiplied by sin π.
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We are told in the question that the area of the circle is equal to 227 square centimeters.
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This means that ππ squared equals 227.
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Dividing both sides of this equation by π gives us π squared is equal to 227 over π.
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We can now substitute these into both of our formulas.
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The area of the sector is equal to 120 over 360 multiplied by 227.
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This can be simplified to one-third multiplied by 227 or 227 over three.
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The area of the triangle can be calculated by multiplying a half by 227 over π by sin of 120 degrees.
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Sin of 120 degrees is equal to root three over two.
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The area of the segment can therefore be calculated by subtracting a half multiplied by 227 over π multiplied by root three over two from 227 over three.
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Typing this into the calculator gives us 44.37875 and so on.
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As we need to round our answer to two decimal places, the key or deciding number is the eight.
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This means that we round up to 44.38.
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The area of the segment is 44.38 square centimeters.
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As this is the area of the minor segment, we could calculate the area of the major segment by subtracting this answer from 227 square centimeters.
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We will now look at another question where we are given the radius and the chord.
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A circle has a radius of 10 centimeters.
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A chord of length 14 centimeters is drawn.
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Find the area of the major segment, giving the answer to the nearest square centimeter.
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We are told that the circle has a radius of 10 centimeters.
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A chord of length 14 centimeters is drawn on the circle.
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If we let the two ends of the chord be points π΄ and π΅ and the center point π, then the area of the minor segment is equal to the area of the sector minus the area of the triangle.
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In order to calculate both of these, we firstly need to work out the central angle π.
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This can be done in either radians or degrees.
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In this question, we will use radians.
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So, it is important that our calculator is in the correct mode.
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The area of a sector, when π is in radians, is equal to a half π squared π.
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And the area of a triangle is equal to a half π squared sin π.
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This can be simplified by factoring, giving us the area of the segment equal to a half π squared multiplied by π minus sin π.
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We can now calculate the angle π by using right-angle trigonometry or the cosine rule.
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In order to calculate the angle in any triangle using the cosine rule, we use the following formula.
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Cos of π΄ is equal to π squared plus π squared minus π squared divided by two ππ, where π, π, and π are the three lengths of the triangle and π΄ is the one opposite the angle weβre trying to work out.
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Substituting in our values gives us cos of π equals 10 squared plus 10 squared minus 14 squared over two multiplied by 10 multiplied by 10.
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This simplifies to cos of π equals one fiftieth.
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Ensuring that our calculator is in radian mode, π is equal to the inverse cos of one fiftieth.
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This is equal to 1.55079 and so on radians.
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We can now substitute this value into our formula for the area of a segment.
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The area of the minor segment is equal to 27.5497 and so on.
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We have been asked to calculate the area of the major segment.
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This is the area of the whole circle minus the area of the minor segment.
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The area of a circle is equal to ππ squared.
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As our radius is equal to 10 centimeters, the area is equal to 100π.
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We need to subtract 27.5497 and so on from this.
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This is equal to 286.6095 and so on.
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Weβre asked to round our answer to the nearest square centimeter.
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The deciding number is the six in the tenths column.
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So, we round up to 287 square centimeters.
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This is the area of the major segment in the circle.
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We will now summarize the key points from this video.
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A segment is a region bounded by an arc and a chord passing through the end points of the arc.
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Drawing a chord on any circle splits it into two segments, a minor segment and a major segment.
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As suggested by the name, the major segment is the larger of the two parts.
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The area of the minor segment, shown in orange on the diagram, is equal to the area of the pink sector minus the area of the blue triangle.
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In order to calculate these areas, we need the angle π in either radians or degrees and the length of the radius π.
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When dealing in degrees, we tend to calculate the two areas separately.
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The area of the sector is equal to π over 360 multiplied by ππ squared.
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And the area of the triangle is equal to a half π squared multiplied by sin π.
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When dealing with radians, we can simplify by factoring.
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The area of the segment when π is in radians is a half π squared multiplied by π minus sin π.
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In any question, we can convert from one angle measurement to the other using the fact that 180 degrees is equal to π radians and, therefore, 360 degrees is equal to two π radians.
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In some questions, we might need to use trigonometry to find out the angle π first.