Area of a Rhombus

Learn how to calculate the area of a rhombus using its diagonals.

Intermediate20 minLesson
Printable worksheet

Definition

A rhombus is a quadrilateral with all four sides equal in length. To find its area, we use the diagonals - the two line segments that connect opposite corners.
Area Formula:
A=d1×d22A = \frac{d_1 \times d_2}{2}
Where:
  • d1d_1 = length of the first diagonal
  • d2d_2 = length of the second diagonal
Why does this formula work? The diagonals of a rhombus bisect each other at right angles (90°). This divides the rhombus into 4 congruent right triangles. When rearranged, these triangles form a rectangle with dimensions d12\frac{d_1}{2} and d2d_2, giving us area = d1×d22\frac{d_1 \times d_2}{2}.

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What is the formula for the area of a rhombus?

Worked Examples

A rhombus has diagonals of length 8 cm and 6 cm. Find its area.
1

Identify the diagonals

d1=8d_1 = 8 cm, d2=6d_2 = 6 cm → Both diagonals identified
2

Write the area formula

A=d1×d22A = \frac{d_1 \times d_2}{2} → Formula ready
3

Substitute the values

A=8×62A = \frac{8 \times 6}{2} → A=482A = \frac{48}{2}
4

Calculate the area

A=24A = 24 → Area found

Common Mistakes

Forgetting to divide by 2

Why it's wrong: Students sometimes calculate d1×d2d_1 \times d_2 but forget the final step of dividing by 2.

Correct: Always remember: Area = d1×d22\frac{d_1 \times d_2}{2}. The division by 2 is essential because the diagonals create triangles, not a full rectangle.

Using the side length instead of diagonals

Why it's wrong: The rhombus has 4 equal sides, but the area formula uses diagonals, not sides.

Correct: Look for the diagonals (lines connecting opposite corners), not the sides. If only sides are given, you need additional information.

Confusing rhombus with rectangle

Why it's wrong: Both are quadrilaterals, but they have different area formulas.

Correct: Rectangle: A=l×wA = l \times w. Rhombus: A=d1×d22A = \frac{d_1 \times d_2}{2}. For rhombus, always use diagonals!

Interactive Visual

Area and perimeter

d₁ = 8d₂ = 6

Area24

A=d1⋅d22=8⋅62=24A = \dfrac{d_1 \cdot d_2}{2} = \dfrac{8 \cdot 6}{2} = 24

Perimeter20

P=4s=4(d12)2+(d22)2=20P = 4s = 4\sqrt{(\tfrac{d_1}{2})^2 + (\tfrac{d_2}{2})^2} = 20

Side5

s=5s = 5

The diagonals cross at right angles and cut the rhombus into 4 equal right triangles.

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Practice Problems

16 problems
Problem 1 of 16
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What is the formula for the area of a rhombus?

Why It Matters

The rhombus appears everywhere in design and architecture:
  • Kite Design: Kites are often rhombus-shaped, and knowing the area helps determine how much material is needed
  • Tile Patterns: Many floor and wall tiles use rhombus shapes for decorative patterns
  • Road Signs: Diamond-shaped warning signs are rhombuses
  • Jewelry: Diamond cuts often follow rhombus geometry
  • Sports: Baseball diamonds are actually squares (a special rhombus!)
Understanding the rhombus area formula also builds foundation for more complex polygon calculations.

Real World Applications

Kite Making

Kites are often designed as rhombuses or similar diamond shapes.

Example:

If a kite has diagonals of 80 cm and 50 cm, the fabric needed for one side is 80×502=2000\frac{80 \times 50}{2} = 2000 cm² (plus extra for seams).

1Try It Yourself

You're making a decorative kite with diagonals of 60 cm and 40 cm.

How much fabric do you need for one side of the kite?

Step 1: Write the mathematical expression

Use the formula: A=d1×d22A = \frac{d_1 \times d_2}{2}

Floor Tile Design

Architects use rhombus-shaped tiles to create eye-catching patterns.

Example:

A tile with diagonals of 20 cm and 15 cm has area 20×152=150\frac{20 \times 15}{2} = 150 cm². To cover 3 m² (30,000 cm²), you need 30000150=200\frac{30000}{150} = 200 tiles.

2Try It Yourself

Each rhombus tile has diagonals of 16 cm and 10 cm. You need to tile a 1.6 m² area.

How many tiles do you need?

Step 1: Write the mathematical expression

First find area of one tile, then divide total area by tile area

Key Takeaways

  • 1A rhombus has 4 equal sides and diagonals that bisect each other at 90°
  • 2Area formula: A=d1×d22A = \frac{d_1 \times d_2}{2} where d1d_1 and d2d_2 are the diagonals
  • 3Always use the diagonals (not the sides) in the area formula
  • 4Remember to divide by 2 after multiplying the diagonals

Frequently Asked Questions

Yes! A square is a rhombus where both diagonals are equal in length. If a rhombus has diagonals d1=d2=dd_1 = d_2 = d, its area is d22\frac{d^2}{2}.
Yes! A square is a rhombus where both diagonals are equal in length. If a rhombus has diagonals d1=d2=dd_1 = d_2 = d, its area is d22\frac{d^2}{2}.
Yes, A=b×hA = b \times h works too, where bb is any side and hh is the perpendicular height. But the diagonal formula is usually easier when diagonals are given.
Because all sides are equal, the diagonals create 4 congruent triangles. These triangles have equal angles at the center, and since they sum to 360°, each must be 90°.

Glossary

Rhombus
A quadrilateral with all four sides equal in length
Diagonal
A line segment connecting two non-adjacent vertices (opposite corners)
Bisect
To divide into two equal parts
Perpendicular
At a right angle (90°)
Today's challengeOne problem, three tries

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