Areas Related to Circles Revision Notes

A circle is a simple geometrical figure surrounded by a curved line, which contains points separated by radius from a static point known as a centre. When a line and a circle are lying on the same plane and farther from each other, they do not intersect. But when the line comes in contact with the circle, it is tangent to the circle. That line intersects the two points of the curved line of the circle.

Keyterms: Circle, Tangent, Chords, Diameters, Area, Diameter, Radius, Circumference

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What is a Circle?

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A circle is a simple geometrical figure surrounded by a curved line, which contains points separated by radius from a static point known as a centre.

When a line and a circle are lying on the same plane and farther from each other, they do not intersect. But when the line comes in contact with the circle, it is tangent to the circle. That line intersects the two points of the curved line of the circle.

The circle is a figure enclosed within a curved line which divides a plane into two regions : exterior and interior.

A circle can also be understood as an ellipse where the two foci are concurrent and eccentricity is zero.

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Tangent

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A line is understood to be tangent to a circle when it touches the circle at a certain point. The point where the tangent and the circle meet refers to be the point of tangency. A tangent has its own mathematical equation.

Basic equation is: x²+ y² = a² at (x₁, y₁) is xx₁+yy₁= a²

The line is always perpendicular to the radius of the circle.

There are three criteria which determine the condition of tangency, they are:

• The point of intersection is in the circle
• The point of intersection is exterior to the circle.
• The point of intersection is on top of the circle.

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Circumference of a Circle

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Circumference of the circle also refers to the perimeter of the circle that estimates the periphery of the circle. When the circle opens to transform it into a straight line, the circumference determines its length.

The units of measurement of the circumference of the circle are in meters or centimeters.

Formula for the circumference of the circle:

Perimeter of a circle = 2πR;

R stands for the radius of the circle; π stands for Pi and it has a value of 3.14, pi is constant mathematically in every equation.

Circumference of a Semicircle

When a circle splits into two equal halves, the circumference or the perimeter of the circle also gets half.

Equation = πr + 2r

Here, π stands for Pi with the value 3.14 and r stands for radius of the circle.

Area of a Circle

Area of the circle refers to be the enclosed section by the circle or space engaged by the circle itself.

Equation for the area of the circle = A = πr²

Area of a Semicircle

Area of a semicircle is the space engaged by the half circle in the two dimensional plane.

Equation = πr²/2

Sectors of a Circle

Sector of a circle refers to the part of a circle which lies among the two radii and the connecting arch. The commonest example which best explains the sector of the circle is Semi- circle.

There are two types of sector of a circle:

• Major sector
• Minor sector

Major sector refers to the part estimated greater than the semi circle.

Minor sector refers to the part estimated lesser than the semi circle.

The only thing calculated in the sector of a circle is the area.

Formula for the area of the sector of a circle = (θ/360º) × πr²

Area of the minor sector of the circle = (θ/360º) × πr² - r² sin θ /2

Area of the major sector of the circle is the difference between the area of the circle and area of the minor sector of the circle.

Here, the θ stands for theta and the angle formed in the center, r stands for radius and π (pi) has the value 3.14.

Segment of a Circle

Segment of a circle refers to the area formed by the chord and the corresponding arch. The corresponding arch shall lie between the two end points of the chord.

Formula to calculate area of the segment of the circle:

A = (½) × r² (θ – Sin θ)

Here, A is the considered area, r is radian, θ is the angle and sin θ has the value according to the value of θ.

Important Formulas: Areas Related to a Circle

1. Area of a ring – π (R+r) (R-r)
2. Length of the arch - πr²/ 180^o 
3. When the chord secant 90^o, area of the corresponding segment 
4. When the chord secant 60^o, area of the corresponding segment
5. When the chord secant 120^o, area of the corresponding segment
6. Number of revolutions in a minute Distance covered in 60 seconds/ circumference
7. Circumference or perimeter of the sector

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Revision Notes Benefits

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The benefits of the revision notes on Areas Related to Circle are:

• The notes provide good ideas to the students about the probable questions examiners will ask.
•  The question paper divides the solutions into various sections, so that the students understand better.
• With these revision notes, the students get a better understanding about the chapter in a simplified language.
• These revision notes are free from errors and well prepared for the students.
• These revision notes help students to answer all categories of questions long or short, asked in the exams with clear conception and in proper steps.
• These revision notes are highly beneficial as it covers the chapter to itscore and is convenient to go through right before the exams.

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Preparation Guidelines

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1. The students should read the questions asked carefully. Tough questions can complicate the thoughts and may lead to writing incorrect answers.
2. The students should memorize the basic formulas of the chapter like the circumference or the area of the circle as the chapter revolves round these two formulas.
3. The revision notes also include some practice questions and solving those sums will help the students become well prepared for the exams. Their concepts will be clear with the solving of the sums.
4. The practice sums are set by math experts who help to clear the doubts of the students. The doubts regarding the chapter areas related to a circle are clear under one roof. The students thereafter, solve the questions with understanding.