Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 ↓


Circles


In this lesson, we will dive deep into the world of circles. Circles are one of the most fascinating and ubiquitous shapes in mathematics and the world around us. Understanding circles helps us understand things like wheels, coins, and even the concept of cycles. Let's begin our exploration.

What is a circle?

A circle is a set of points that are equidistant from one point. This central point is called the center of the circle. The distance from the center to any point on the circle is called the radius. The word "radius" comes from the Latin word meaning "ray" because it radiates out from the center point.

radius

In the figure above, we have a simple circle. The red line represents the radius, which is the distance from the center (point) to the point on the circle.

Basic terms and parts of a circle

Center

The midpoint of a circle. Every point on a circle is the same distance from the center.

Radius

The straight line from the center to any point on the circle. It is constant for any given circle.

Diameter

The diameter is a straight line that passes from one side of a circle to the other through the center. The diameter is twice the length of the radius.

diameter = 2 × radius
Diameter

The blue line in the diagram represents the diameter. You can see that it passes through the center, where the two radii meet to form the diameter.

Circumference

The circumference is the total distance around the circle. It is like the perimeter in polygons.

Circumference = 2 × π × radius

This formula involves the mathematical constant π (pi), which is approximately 3.14159.

Some properties of circles

  • In a circle all radii are equal.
  • In a circle all diameters are equal.
  • The diameter is the longest chord of a circle. A chord is any line segment whose both ends are on the circle.

Chord and arc

Wire

A chord is a line segment whose both ends are on the circle. A diameter is a special type of chord that passes through the center of a circle.

Wire

Arch

An arc is a part of the circumference. It can be any part of the edge of the circle.

We define the arc as follows:

  • A minor arc is smaller than a semicircle.
  • A major arc is larger than a semicircle.

Tangent line

A tangent is a line that touches a circle at exactly one point. This point is known as the point of tangency. The tangent is always perpendicular to the radius at the point of contact.

tangent line radius

Sectors and sections

Area

The sector of a circle is like a "piece of pie". It is the area bounded by two radii and the arc between them.

Area

Section

A segment is the area inside a circle that is bounded by a chord and the arc (or intercept) it creates. It's like the smaller area you would get if you "cut off" a piece of the circle using a chord.

Section

Equation of a circle

The general equation of a circle on the Cartesian plane with radius r and center (h, k) is:

(x – h)² + (y – k)² = r²

If the center is at the origin (0, 0), the equation simplifies to:

x² + y² = r²

Let us solve a simple equation problem for better understanding: Suppose a circle has center (3, 4) and radius 5. Write the equation for this circle.

Solution: Using the formula (x - h)² + (y - k)² = r², we substitute h = 3, k = 4, and r = 5:

(x - 3)² + (y - 4)² = 25

Area and perimeter with circles

Area of a circle

The area of a circle is the space within its circumference. It can be calculated using the formula:

Area = π × radius²

Let the radius of the circle be 7 cm. The area will be:

Area = π × 7² = π × 49 ≈ 153.94 cm²

Conversion between diameter and radius

As we know from the definitions:

diameter = 2 × radius

On the contrary,

Radius = diameter / 2

Solving problems involving circles

Let's consider some problem-solving scenarios using our knowledge of circles:

Example 1

The radius of a bicycle wheel is 14 inches. Find its circumference.

Solution:

Perimeter = 2 × π × 14 = 28π ≈ 87.96 inches

Example 2

The diameter of a circular park is 200 m. Find the area of the park.

Solution:

First, convert the diameter to a radius:

Radius = diameter / 2 = 200 / 2 = 100 meters

Then calculate the area:

Area = π × 100² = 10000π ≈ 31415.93 square meter

Circles in real life

Circles are not just mathematical concepts; they are everywhere in real life. For example, they are used in designing wheels, clocks, coins, plates, and even buildings. Recognizing the properties of circles is helpful in both the aesthetic and functional aspects of design.

Exercise: Observe your surroundings

The next time you go out, try to identify circular objects around you using their properties. Consider the radius, diameter, circumference, and area of a circle to gain a deeper understanding of their utility and beauty.

This comprehensive guide will give you a strong foundation in understanding circles, their properties, and applications. By building on these basics, you can model and solve many geometric problems involving circles.


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Applications of Area Theorems
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Circles

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