Area of Rhombus Using Diagonal Lengths with Examples

Area of Rhombus is the region enclosed by a rhombus. It can be calculated from its diagonals, a side and perpendicular height, or a side and an included angle. The diagonal formula is A = ½d₁d₂, while the base-height formula is A = bh. These forms help solve direct and missing-value questions.

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What Is the Area of a Rhombus?

The area of a rhombus is the amount of plane enclosed by its four equal sides. If its diagonals are d₁ and d₂, its area is A = ½d₁d₂.

A rhombus has two diagonals that bisect each other at right angles. These diagonals divide the rhombus into four right triangles, so the total area is half the product of the diagonals. The area can also be found as base × perpendicular height. For example, diagonals of 12 cm and 8 cm give an area of ½ × 12 × 8 = 48 cm².

Area of Rhombus Video Lessons

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Rhombus Area: Formula and Calculation

Learn how to calculate the area of a rhombus using its diagonals, with a clear explanation of the formula and its application in mensuration problems.

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Area of Rhombus Formula & Tricks

Important Formulas

Using the diagonals
A = ½ × d₁ × d₂

Here, d₁ and d₂ are the lengths of the two diagonals of the rhombus.

Using base and height
A = b × h

Here, b is any side taken as the base and h is the perpendicular distance between the parallel sides.

Using side and included angle
A = a² sin θ

Here, a is the side length and θ is the angle between two adjacent sides.

Finding a missing diagonal
d₂ = 2A ÷ d₁

Rearrange A = ½d₁d₂ when the area and one diagonal are known.

Quick Tricks

Use half the product of diagonals

When both diagonals are given, multiply them and divide by 2. No side length or angle is required.

Example: For d₁ = 15 cm and d₂ = 10 cm, A = 15 × 10 ÷ 2 = 75 cm².
Check whether height is perpendicular

In the base-height formula, use the perpendicular height, not the slanting side. A slanting side can be used directly only when the included angle is also known.

Example: If b = 9 cm and h = 7 cm, A = 9 × 7 = 63 cm².
Keep units consistent

Convert both diagonals, or the base and height, to the same unit before multiplying. The final area has square units.

Example: If diagonals are 0.5 m and 40 cm, convert 0.5 m to 50 cm. Then area = ½ × 50 × 40 = 1000 cm².

Area of Rhombus Concepts

Area Using the Two Diagonals

The area of a rhombus is half the product of its two diagonals: A = ½d₁d₂.

The diagonals intersect at right angles and bisect each other. Their four triangular parts together form the complete rhombus. Therefore, multiply the diagonal lengths and divide the result by 2.

Example: For diagonals 18 m and 7 m, area = ½ × 18 × 7 = 63 m².

Area Using Base and Perpendicular Height

When a side is used as the base, area equals the base multiplied by its perpendicular height: A = bh.

The height is the shortest distance between the pair of parallel sides. It is not generally equal to the other side of the rhombus unless the rhombus is a square.

Example: If the base is 14 cm and the perpendicular height is 5 cm, area = 14 × 5 = 70 cm².

Area Using Side and Included Angle

If the side length is a and the included angle is θ, area is A = a² sin θ.

The rhombus can be treated as a parallelogram with base a and perpendicular height a sin θ. The angle must be between two adjacent sides.

Example: For side 6 cm and included angle 30°, area = 6² × sin 30° = 36 × ½ = 18 cm².

Finding a Missing Measurement

A missing diagonal can be found by rearranging A = ½d₁d₂, giving d₂ = 2A ÷ d₁.

Similarly, a missing base is b = A ÷ h, and a missing perpendicular height is h = A ÷ b. The unknown length must use the same unit as the known length.

Example: If area = 96 cm² and one diagonal is 12 cm, the other diagonal is 2 × 96 ÷ 12 = 16 cm.

Area of Rhombus Visual Guide

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Rhombus with Two Perpendicular Diagonals

Visual guide: Rhombus with Two Perpendicular Diagonals
d₁: 10 cm d₂: 6 cm
The diagonals divide the rhombus into four right triangles. For d₁ = 10 cm and d₂ = 6 cm, area = ½ × 10 × 6 = 30 cm².

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Practice Area of Rhombus Questions

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1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Area of Rhombus: Revision Points

Use these formulas and properties to revise calculations involving a rhombus.

  • Area from diagonals: A = ½d₁d₂.
  • Area from base and perpendicular height: A = bh.
  • Area from side and included angle: A = a² sin θ.
  • The diagonals of a rhombus bisect each other at right angles.
  • Use square units for area, such as cm², m² or km².
  • For a missing diagonal, use d₂ = 2A ÷ d₁.
  • Convert all given lengths to the same unit before applying a formula.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Area of Rhombus FAQs

What is the area of a rhombus with diagonals 16 cm and 9 cm?

Area = ½ × 16 × 9 = 72 cm².

What is the area of a rhombus with side 10 cm and perpendicular height 6 cm?

Using A = bh, area = 10 × 6 = 60 cm².

One diagonal of a rhombus is 14 cm and its area is 84 cm². Find the other diagonal.

d₂ = 2A ÷ d₁ = 2 × 84 ÷ 14 = 12 cm.

Can the side length alone determine the area of a rhombus?

No. The angle between adjacent sides or the perpendicular height is also needed. The side length alone determines the area only for a specified shape such as a square.

What is the area of a rhombus with side 8 cm and included angle 60°?

A = a² sin θ = 8² × sin 60° = 64 × √3/2 = 32√3 cm², approximately 55.43 cm².

Are the diagonals of a rhombus always equal?

No. They are always perpendicular bisectors of each other, but they are equal only when the rhombus is a square.

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