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 → Mensuration ↓


Areas of Plane Figures


Measurement is a branch of mathematics that deals with measuring the dimensions of plane and solid figures. Understanding the area of plane figures is an essential part of measurement, especially for class 10 students who are beginning to explore more advanced mathematical concepts. In this detailed guide, we will review the area of various plane figures such as triangle, rectangle, square, parallelogram, trapezoid, circle, etc. using a simple and comprehensive approach.

Introduction to plane figures

Plane figures are flat, two-dimensional shapes that lie on a plane. They have length and width but no depth. Examples of plane figures include triangles, rectangles, circles, and more. The area of a plane figure is the measure of the area enclosed within its boundary lines. It is usually measured in square units such as square centimeters, square meters, etc.

Calculating the area

Every plane figure has a specific formula to find its area. These formulas help determine how much space is occupied by the figure. Let us now look at the area formulas for some common plane figures.

1. Area of a rectangle

A rectangle is a four-sided shape whose opposite sides are equal and parallel, and each angle is a right angle.

Formula: Area of a rectangle = length × width
L × B

For example, if the length of a rectangle is 10 units and the width is 5 units, then its area is:

Area = 10 × 5 = 50 square units

2. Area of the square

A square is a four-sided shape in which all sides are equal and each angle is a right angle.

Formula: Area of a square = side × side = side²
S²

For example, if the side length of a square is 6 units, then its area will be:

Area = 6 × 6 = 36 square units

3. Area of a triangle

A triangle is a polygon with three sides. The area can be calculated using various formulas based on the given data. A general formula for right-angled triangles is:

Formula: Area = ½ × base × height
½ × B × H

For example, if a right triangle has a base of 8 units and a height of 5 units, its area is:

Area = ½ × 8 × 5 = 20 square units

4. Area of parallelogram

A parallelogram is a four-sided figure whose opposite sides are equal and parallel.

Formula: Area = Base × Height
B × H

For example, if the base of a parallelogram is 9 units and the height is 4 units, then its area is:

Area = 9 × 4 = 36 square units

5. Area of trapezium

A trapezoid (or trapezium) is a four-sided shape with a pair of parallel sides.

Formula: Area = ½ × (base1 + base2) × height
½ × (b₁ + b₂) × h

For example, if the bases of a trapezoid are 12 units and 8 units long, and the height is 5 units, then its area is:

Area = ½ × (12 + 8) × 5 = ½ × 20 × 5 = 50 square units

6. Area of a circle

A circle is a circular figure, every point on whose boundary is at the same distance from its center.

Formula: Area = π × radius²
πr²

For example, if the radius of a circle is 7 units, then its area will be:

Area = π × 7² = 154 square units (using π ≈ 22/7 or 3.14)

Combined figures

Compound shapes are those shapes that contain more than one basic shape. Calculating their area involves adding or subtracting the areas of the known shapes. Let us look at an example to understand this concept better.

Example of a combined shape

Consider a figure that has a rectangle on one side and a semicircle on the other.

If the length of the rectangle is 200 units and width is 100 units, and the radius of the semicircle is 100 units, then we have to find the area of the composite figure.

First, find the area of the rectangle:

Area of the rectangle = length × breadth = 200 × 100 = 20,000 square units

Next, find the area of the semicircle:

Area of a circle = π × radius² = π × 100² = 31,400 square units (approx.)
Area of the semicircle = ½ × 31,400 = 15,700 square units (approx.)

Finally, add the areas of the rectangle and the semicircle to get the total area:

Total area = 20,000 + 15,700 = 35,700 sq. units (approx.)

Conclusion

The study of plane figure areas is fundamental to understanding geometry and its applications in real-life scenarios. By mastering these basic concepts and formulas, students can solve a wide variety of problems not only in mathematics but also in fields such as architecture, engineering, and design, where such knowledge is applied practically. With practice, the calculations become intuitive, allowing further exploration into more complex geometric shapes and their properties.


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