How to find the area of the Rhombus – 10 easy methods

A rhombus is a special type of quadrilateral where all four sides are equal in length. There are multiple ways to calculate the area of the rhombus, depending on the information available, such as diagonals, base & height, or angles. Understanding these methods helps in solving various geometry problems efficiently.

Estimated reading time: 6 minutes

Using Diagonals (Most Common Method)

Formula:

\( A = \frac{d_1 \times d_2}{2} \)

Steps:

1. Measure the two diagonals (d₁ and d₂).
2. Multiply them together.
3. Divide the result by 2.

Example:
If d₁ = 8 cm and d₂ = 6 cm,

\( A = \frac{8 \times 6}{2} = 24 \text{ cm}^2 \)

Using Base and Height to Find the Area of the Rhombus

Formula:

\( A = \text{Base} \times \text{Height} \)

Steps:

1. Measure one side (base).
2. Measure the perpendicular height.
3. Multiply them together.

Example:
If Base = 10 cm, Height = 5 cm,

\( A = 10 \times 5 = 50 \text{ cm}^2 \)

Using Side and Angle (Trigonometric Formula)

Formula:

\( A = a^2 \times \sin(\theta) \)

Steps:

1. Measure the side a.
2. Measure the included angle θ (in degrees).
3. Take the sine of the angle.
4. Multiply it with a².

Example:
If a = 7 cm, θ = 60°,

\( A = 7^2 \times \sin 60^\circ = 49 \times 0.866 = 42.43 \text{ cm}^2 \)

Using a Perimeter and One Diagonal

Formula:

\( A = \frac{p \times d}{4} \)

Steps:

1. Measure the perimeter (p).
2. Measure one diagonal (d).
3. Multiply them and divide by 4.

Example:
If p = 20 cm, d = 8 cm,

\( A = \frac{20 \times 8}{4} = 40 \text{ cm}^2 \)

Using Side and Circumradius to Find the Area of the Rhombus

Formula:

\( A = 4 \times R^2 \times \sin 45^\circ \)

Steps:

1. Measure the circumradius (R).
2. Square it.
3. Multiply by 4 and sin 45° (0.7071).

Example:
If R = 5 cm,

\( A = 4 \times 5^2 \times 0.7071 = 70.71 \text{ cm}^2 \)

Using Inscribed Circle Radius (Inradius)

Formula:

\( A = 4 \times r \times s \)

Steps:

1. Measure the inradius (r).
2. Measure the side (s).
3. Multiply them by 4.

Example:
If r = 3 cm, s = 6 cm,

\( A = 4 \times 3 \times 6 = 72 \text{ cm}^2 \)

Using a Product of Half-Diagonals

Formula:

\( A = \left(\frac{d_1}{2} \times \frac{d_2}{2}\right) \times 4 \)

Steps:

1. Measure both diagonals (d₁ and d₂).
2. Divide each by 2.
3. Multiply and then multiply by 4.

Example:
If d₁ = 10 cm, d₂ = 6 cm,

\( A = \left(\frac{10}{2} \times \frac{6}{2}\right) \times 4 = (5 \times 3) \times 4 = 60 \text{ cm}^2 \)

Using a Determinant Formula (If Coordinates are Given) to Find the Area of the Rhombus

Formula:

\( A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 – (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \)

Steps:

1. Assign the (x, y) coordinates to four vertices.
2. Use the determinant formula.
3. Solve and take the absolute value.

Example:
For (1,2), (4,5), (7,2), (4,-1)

\( A = \frac{1}{2} | 1(5) + 4(2) + 7(-1) + 4(2) – (2(4) + 5(7) + 2(4) + (-1)(1)) | \)
\( = \frac{1}{2} | 5 + 8 – 7 + 8 – (8 + 35 + 8 -1) | \)
\( = \frac{1}{2} \times 36 = 18 \text{ cm}^2 \)

Using Vectors (Cross Product) to Find the Area of the Rhombus

Formula:

\( A = \frac{1}{2} | \vec{AB} \times \vec{AD} | \)

Steps:

1. Represent two adjacent sides as vectors.
2. Compute the cross-product.
3. Take the absolute value and divide by 2.

Using Heron’s Formula (If Split into Triangles)

Formula:

\( A = \sqrt{s(s-a)(s-b)(s-c)} \)

Steps:

1. Divide the rhombus into two triangles using one diagonal.
2. Find s (semi-perimeter)

\( s = \frac{a + b + c}{2} \)

1. Use Heron’s formula for each triangle.
2. Multiply by 2.

Example:
For a triangle with sides 5, 5, 6.

\( s = \frac{5+5+6}{2} = 8 \)
\( A = \sqrt{8(8-5)(8-5)(8-6)} = \sqrt{8 \times 3 \times 3 \times 2} = \sqrt{144} = 12 \)
\( \text{Total Area} = 12 \times 2 = 24 \text{ cm}^2 \)

Conclusion

The area of a rhombus can be determined using various formulas, depending on the available measurements. Knowing multiple methods ensures a deeper understanding and flexibility in solving geometry problems. 🚀

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Frequently Asked Questions

What is the formula for the area of a rhombus?

The most common formula is:

\( A = \frac{1}{2} \times d_1 \times d_2 \)

where d₁ and d₂ are the diagonals.

How do you find the area of a rhombus with side length?

Use the formula:

\( A = a^2 \times \sin(\theta) \)

where a is the side length, and θ is the included angle.

How do you find the area when the base and height are given?

Multiply:

\( A = \text{Base} \times \text{Height} \)

Can a rhombus have right angles?

Yes, the rhombus becomes a square if all angles are 90°.

How is the rhombus different from a square?

A square is a rhombus where all angles are 90°, while a rhombus can have different angles.

Read more on the area of the rhombus here: Cuemath