Rhombus Area Calculator

Compute the area of a rhombus from diagonals, base and height, side and angle, or vertex coordinates. See all formulas and step-by-step workings.

1. Choose what you know

2. Enter dimensions

3. Result

Area of the rhombus:

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Step-by-step solution

Enter your values and click “Calculate area” to see the working.

Rhombus area formulas

A rhombus is a quadrilateral with all four sides equal in length. Its area can be computed in several equivalent ways depending on what you know.

1. From diagonals

Formula

\[ A = \frac{d_1 \cdot d_2}{2} \]

where \(d_1\) and \(d_2\) are the lengths of the diagonals.

This is often the easiest formula when the diagonals are known, because in a rhombus they are perpendicular and bisect each other.

2. From base and height

Formula

\[ A = b \cdot h \]

where \(b\) is any side (used as the base) and \(h\) is the perpendicular distance between two opposite sides (the height).

3. From side and interior angle

Formula

\[ A = a^2 \sin(\theta) \]

where \(a\) is the side length and \(\theta\) is any interior angle of the rhombus.

This follows from the fact that a rhombus is a special parallelogram and the area of a parallelogram is \(a \cdot a \sin(\theta)\).

4. From coordinates (shoelace formula)

If the vertices of the rhombus are \(A(x_1,y_1), B(x_2,y_2), C(x_3,y_3), D(x_4,y_4)\) in order around the shape, then the area is

\[ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - y_1x_2 - y_2x_3 - y_3x_4 - y_4x_1 \right| \]

This is a special case of the shoelace formula for polygons.

Worked examples

Example 1 – Area from diagonals

Problem. A rhombus has diagonals \(d_1 = 10\text{ cm}\) and \(d_2 = 8\text{ cm}\). Find its area.

Solution.

  1. Use the diagonal formula: \(A = \dfrac{d_1 d_2}{2}\).
  2. Substitute the values: \(A = \dfrac{10 \cdot 8}{2}\).
  3. Compute: \(10 \cdot 8 = 80\), then \(80 / 2 = 40\).

Answer: \(A = 40\text{ cm}^2\).

Example 2 – Area from side and angle

Problem. A rhombus has side length \(a = 6\text{ m}\) and one interior angle \(\theta = 60^\circ\). Find its area.

Solution.

  1. Use \(A = a^2 \sin(\theta)\).
  2. Compute \(a^2 = 6^2 = 36\).
  3. \(\sin(60^\circ) = \dfrac{\sqrt{3}}{2} \approx 0.866\).
  4. So \(A \approx 36 \cdot 0.866 \approx 31.18\).

Answer: \(A \approx 31.2\text{ m}^2\) (to one decimal place).

Properties of a rhombus

  • All four sides are equal in length.
  • Opposite sides are parallel (so it is a parallelogram).
  • Diagonals are perpendicular and bisect each other.
  • Diagonals bisect the interior angles.
  • A square is a special rhombus with all angles \(90^\circ\).

Common mistakes to avoid

  • Using the wrong angle. In \(A = a^2 \sin(\theta)\), \(\theta\) must be an interior angle of the rhombus, not an angle between a diagonal and a side.
  • Mixing units. Make sure all lengths (sides, diagonals, height) are in the same unit before calculating area.
  • Forgetting to divide by 2 in the diagonal formula \(A = d_1 d_2 / 2\).

FAQ about rhombus area

How do I know if a quadrilateral is a rhombus?

A quadrilateral is a rhombus if any of the following holds:

  • All four sides are equal in length.
  • It is a parallelogram and two adjacent sides are equal.
  • Its diagonals are perpendicular and bisect each other.

Can the area of a rhombus be negative?

No. Area is always non‑negative. Some coordinate formulas produce a signed area; we take the absolute value to get the geometric area.

Which formula is best to use?

Use whichever matches the information you have:

  • Know both diagonals → \(A = d_1 d_2 / 2\).
  • Know base and height → \(A = b h\).
  • Know side and interior angle → \(A = a^2 \sin(\theta)\).
  • Know coordinates → shoelace formula (the calculator handles this for you).