Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Geometry ↓


Circles


Geometry is a field of mathematics that deals with shapes, sizes, and the properties of space. One of the simplest yet fascinating shapes in geometry is the circle. The study of circles is an important part of mathematics, especially at the grade 10 level. In this exploration of circles, we will take a deep look at various aspects of circles, including their properties, equations, and applications. This lesson is meant to provide a thorough understanding of circles in very simple language.

What is a circle?

A circle is the set of all points in a plane that are equidistant from a given point. This given point is called the center of the circle, and the constant distance from the center to any point on the circle is called the radius. Imagine a circle as the outline drawn by a perfectly round loop of rope.

radius Center

Key terms:

  • Centre: The fixed point from which every point on a circle is equidistant.
  • Radius: The distance from the center of a circle to any point on the circle.
  • Diameter: A line segment that passes through the center of a circle and whose endpoints are on the circle. It is twice the radius.
  • Circumference: The total distance around the circle.
  • Arc: A portion of the circumference of a circle.
  • Chord: A line segment with both endpoints on a circle.
  • Sector (Sec): The area bounded by two radii and an arc.
  • Tangent: A straight line that touches a circle at only one point.

Equation of a circle

The equation of a circle in the coordinate plane can be easily represented using the center and radius. If the center of the circle is at the point (h, k) and the radius is r, then the equation of the circle is given by:

(x - h)^2 + (y - k)^2 = r^2

This equation represents all points (x, y) that are at a distance r from the center (h, k).

Example:

Consider a circle with centre (3, 4) and radius 5 The equation of this circle is:

(x - 3)^2 + (y - 4)^2 = 25

Properties of circle

1. Perimeter

The circumference of a circle is the total distance around the circle. It is calculated using the formula:

c = 2πr

Where C is the circumference and r is the radius of the circle. For example, if the radius of a circle is 7, then the circumference is:

c = 2 * π * 7 = 14π

2. Area

The area of a circle is the space inside its circumference. The formula to calculate the area is:

a = πr^2

Where A is the area and r is the radius. So, if the radius of a circle is 3, the area is:

a = π * (3)^2 = 9π

Parts of a circle

Diameter

The diameter is a special chord that passes through the center of the circle. It is equal to twice the radius. If the radius is r, then the diameter D is:

d = 2r
D

Wire

A chord is a line segment whose endpoints lie on a circle. All diameters are chords, but not all chords are diameters.

Wire

Arch

An arc is a portion of the circumference of a circle. The measurement of an arc is given in degrees or radians.

Area

A sector is a part of a circle bounded by two radii and an arc. It looks like a "slice of pizza".

Area

Tangent line

The tangent to a circle is a straight line that touches the circle at only one point. The tangent is perpendicular to the radius at the point of contact.

tangent line

Special circles

Concentric circles

Concentric circles are two or more circles with the same center but different radii. They look like tree rings.

Thick

Incircle and circumcircle

The incircle of a triangle is the circle that lies inside the triangle and touches all its sides. The circumcircle is the circle that passes through all the vertices of the triangle.

Examples and applications of circles

Circles appear everywhere in real life and have many applications. Here are some examples and applications:

  • Wheels: Wheels are one of the most common and practical applications of circles, enabling vehicles to move efficiently.
  • Clocks: Many clocks use circular dials to show the time.
  • Architecture: Circular shapes are used for aesthetic and structural reasons in domes and circular buildings.
  • Technology: Compact discs (CDs), DVDs and other discs are made in circular shapes.

Practice problems

Problem 1:

Find the circumference and area of a circle of radius 8 cm.

Perimeter, C = 2πr = 2 * π * 8 = 16π cm
Area, A = πr^2 = π * (8)^2 = 64π cm²

Problem 2:

A circle is represented by the equation (x - 2)^2 + (y + 3)^2 = 36 What is the center and radius of the circle?

The equation is as follows: (x - h)^2 + (y - k)^2 = r^2
center (h, k) = (2, -3)
Radius, r = √36 = 6

Hopefully this detailed exploration of circles has enhanced your understanding of the topic. With these concepts, you can solve various geometric problems and appreciate the beauty of circles in different contexts.


Grade 10 → 5.3


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Applications of Similarity in Real-Life
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Properties of a Circle

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Circles

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