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 ↓


Area of a Rhombus


The rhombus is a fascinating geometric shape, characterized by its equal-length sides and opposite equal angles. In this comprehensive guide, we will discuss the concept of the area of a rhombus in depth, exploring its formula, how it is derived, and the various ways it can be applied. By the end of this article, you will understand everything you need to know about calculating the area of a rhombus.

What is a rhombus?

A rhombus is a type of quadrilateral in which all four sides are of equal length. It is easiest to think of a rhombus as a flattened square - it has the property of equal sides but does not have the right angles of a square. The opposite angles in a rhombus are equal, making it an equilateral quadrilateral.

Properties of rhombus

  • All sides are of equal length.
  • Opposite angles are equal.
  • The diagonals bisect each other at right angles (90 degrees).
  • The diagonals also bisect the vertex angle.
A Rhombus Diagram:
+--------+
/|        |
/ |        | 
+--+--------+--+
 |        | /
|        |/
+--------+
Note: All sides are equal in length.

Formula for area of rhombus

The most common formula for calculating the area of a rhombus depends on the lengths of its diagonals:

Area = (d1 * d2) / 2

In this formula:

  • d1 and d2 are the lengths of the diagonals.
  • To find the area, the product of the diagonals is divided by 2.

Why does this formula work?

The reason this formula works so well with rhombuses is because of its geometric properties. The diagonals of a rhombus bisect each other at right angles, effectively dividing the rhombus into four right triangles. Each of these triangles can be calculated separately, and their combined areas give the total area of the rhombus.

Understanding with illustration:
A
/ 
/   
B +-----+ C
   /
 /
D

Here, AC and BD are diagonals. The rhombus can be split into triangles ABC and ACD. Using property of right angles at intersection,

Total Area = Triangle ABC + Triangle ACD = (1/2 * AC * BD)

Step-by-step calculation

Let's consider the practical application of this formula step by step:

Example:

Find the area of a rhombus whose diagonals are 10 cm and 12 cm.

  1. First, identify which measurements serve as the diagonal. In this case, we have d1 = 10 cm and d2 = 12 cm.

  2. Apply the formula: 

    Area = (d1 * d2) / 2 = (10 * 12) / 2

  3. Multiply the lengths of the diagonals: 

    10 * 12 = 120

  4. Divide the product by 2 to find the area: 

    120 / 2 = 60
    Therefore, the area of the rhombus is 60 square centimeters.

Alternative calculation method: sides and angles

While the diagonal method is the most common, the area of a rhombus can also be found using the side lengths and interior angles. The formula for this method is:

Area = a^2 * sin(θ)

Where:

  • a is the length of one side.
  • θ is any interior angle.
  • sin(θ) is the sine of the angle.

Example: Using sides and angles

Suppose the length of the side of a rhombus is 6 cm and one of its angles is 60 degrees. Find its area.

Given:
a = 6 cm
θ = 60 degrees
Area = a^2 * sin(θ) = 6^2 * sin(60) = 36 * (√3/2) = 36 * 0.866 = 31.176 cm²

Visualizing the area

To better understand how the formula measures area, imagine four triangles formed by the diagonals:

Diagonal Intersection Division:
+---+
/|    |
/ |    | 
+--+---+--+
 |    | /
|    |/
+---+
Each section is a right-angled triangle, demonstrating how the total sum gives the rhombus area.

Applications of rhombus area

The formula for the area of a rhombus is useful in architecture, agriculture, art, and other diverse fields. Design professionals often consider such geometric properties to optimize layout designs and develop sophisticated algorithms for computational uses.

Conclusion

Understanding how to find the area of a rhombus is not just an academic exercise; it is a practical skill that has applications in real-world problem-solving. Whether using the diagonal length or the side/angle approach, the principles remain consistent, reflecting the underlying order and harmony of geometry.


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Area of a Trapezium
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Area of a Rhombus

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