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Mensuration - Area of Rhombus

Grade 7ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A rhombus is a special type of parallelogram where all four sides are of equal length. Its opposite sides are parallel, and opposite angles are equal.

A rhombus showing all four sides equal to s.
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The diagonals of a rhombus, denoted as d1d_1 and d2d_2, bisect each other at right angles (90∘90^\circ). This property allows the area to be calculated as half the product of the diagonals.

Rhombus with diagonals d1 and d2 intersecting at 90 degrees.
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The area of a rhombus can also be calculated using its base (bb) and its vertical height or altitude (hh), just like any other parallelogram: Area=b×hArea = b \times h.

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Because the diagonals bisect each other at right angles, each side of the rhombus forms the hypotenuse of a right-angled triangle with legs d12\frac{d_1}{2} and d22\frac{d_2}{2}.

📐Formulae

Area=frac12timesd1timesd2Area = \\frac{1}{2} \\times d_1 \\times d_2

Area=textbasetimestextaltitudeArea = \\text{base} \\times \\text{altitude}

Perimeter=4timessPerimeter = 4 \\times s

s=sqrt(fracd12)2+(fracd22)2s = \\sqrt{(\\frac{d_1}{2})^2 + (\\frac{d_2}{2})^2}

💡Examples

Problem 1:

Calculate the area of a rhombus if the lengths of its diagonals are 16textcm16\\text{ cm} and 12textcm12\\text{ cm}.

Solution:

  1. Identify the given diagonal lengths: d1=16textcmd_1 = 16\\text{ cm} and d2=12textcmd_2 = 12\\text{ cm}.
  2. Use the area formula for a rhombus: Area=frac12timesd1timesd2Area = \\frac{1}{2} \\times d_1 \\times d_2.
  3. Substitute the values into the formula: Area=frac12times16times12Area = \\frac{1}{2} \\times 16 \\times 12.
  4. Calculate the product: Area=8times12=96textcm2Area = 8 \\times 12 = 96\\text{ cm}^2.

Explanation:

To find the area when both diagonals are given, we multiply the diagonals together and divide the result by 2.

Problem 2:

The area of a rhombus is 135textcm2135\\text{ cm}^2 and its altitude is 9textcm9\\text{ cm}. Find the length of each side.

Solution:

  1. Given: Area=135textcm2Area = 135\\text{ cm}^2 and Altitude=9textcmAltitude = 9\\text{ cm}.
  2. Use the parallelogram formula: Area=textbasetimestextaltitudeArea = \\text{base} \\times \\text{altitude}.
  3. Substitute the known values: 135=textbasetimes9135 = \\text{base} \\times 9.
  4. Solve for the base: textbase=frac1359=15textcm\\text{base} = \\frac{135}{9} = 15\\text{ cm}.
  5. Conclusion: Since all sides of a rhombus are equal, each side is 15textcm15\\text{ cm}.

Explanation:

Since a rhombus is also a parallelogram, dividing its total area by its perpendicular height (altitude) gives the length of the base, which is equal to its side length.

Problem 3:

Find the area of a rhombus where one side measures 10 cm10\text{ cm} and the altitude is 7.5 cm7.5\text{ cm}.

Rhombus with base 10 cm and altitude 7.5 cm.

Solution:

Given: Base (bb) = 10 cm10\text{ cm} Altitude (hh) = 7.5 cm7.5\text{ cm}

Using the formula: Area=Base×AltitudeArea = \text{Base} \times \text{Altitude} Area=10×7.5Area = 10 \times 7.5 Area=75 cm2Area = 75\text{ cm}^2

The area of the rhombus is 75 cm275\text{ cm}^2.

Explanation:

When the base and altitude are known, we treat the rhombus as a parallelogram to find the area.

Problem 4:

One diagonal of a rhombus is 24 cm24\text{ cm} and its area is 120 cm2120\text{ cm}^2. Find the length of the other diagonal.

Rhombus with one diagonal 24 cm and unknown diagonal marked with a question mark.

Solution:

Given: Area = 120 cm2120\text{ cm}^2 d1=24 cmd_1 = 24\text{ cm}

Using the formula: Area=12×d1×d2Area = \frac{1}{2} \times d_1 \times d_2 120=12×24×d2120 = \frac{1}{2} \times 24 \times d_2 120=12×d2120 = 12 \times d_2 d2=12012d_2 = \frac{120}{12} d2=10 cmd_2 = 10\text{ cm}

The length of the other diagonal is 10 cm10\text{ cm}.

Explanation:

We use the diagonal-based area formula and solve for the unknown variable d2d_2.