What is the formula for the area of a parallelogram?

The most common formula for the area of a parallelogram is A = b × h, where b is the length of the base and h is the perpendicular height. If you know the lengths of two adjacent sides a and b and the angle θ between them, you can also use A = a × b × sin(θ). When vertices are given, the area equals the magnitude of the cross product of the two side vectors.

How do I find height if I only know area and base?

Rearrange the formula A = b × h to get h = A / b. As long as the area A and base b are expressed in consistent units, this gives the perpendicular height. The calculator includes an inverse mode to solve for base or height automatically.

Can I calculate parallelogram area from coordinates?

Yes. If A(x1, y1), B(x2, y2) and C(x3, y3) are three vertices that form two adjacent sides of a parallelogram, the area is |(x2 − x1)(y3 − y1) − (x3 − x1)(y2 − y1)|. This is the magnitude of the 2D cross product of vectors AB and AC.

What units does this parallelogram calculator use?

You can enter side lengths in millimeters, centimeters, meters, inches, or feet. The primary output is in the corresponding square unit, for example cm² or m². The tool also converts the result to square meters and square centimeters where appropriate to help with homework or engineering documentation.

Is the area of a rectangle a special case of the parallelogram formula?

Yes. A rectangle is a parallelogram with interior angle θ = 90°. Then sin(θ) = 1 and the general formula A = a × b × sin(θ) simplifies to A = a × b, the usual area formula for rectangles.

How accurate are the results of this parallelogram area calculator?

The calculator uses standard double-precision floating point arithmetic, which is more than sufficient for school work and most engineering and design tasks. Results are rounded sensibly for display. For highly sensitive or safety-critical applications, always confirm results using your organisation’s approved calculation tools.

Parallelogram Area Calculator

A = b × h = a × b × sin(θ)

Calculate the area of a parallelogram using base and height, two sides with the included angle, or three vertex coordinates. An inverse mode lets you solve for base or height from a known area – with clear formulas and step-by-step explanation.

Interactive parallelogram area calculator

Calculation mode

Choose the mode that matches the information you have. All formulas are mathematically equivalent and give the same area when used with consistent measurements.

Length unit

All lengths must use the same unit. The area result is reported in square of that unit (e.g. cm²), with automatic conversion to m² and cm² where helpful.

Area from base and height

Base b

Height h (perpendicular)

The height is the perpendicular distance from the base to the opposite side, not the length of the slanted side.

This tool uses exact formulas with IEEE double-precision arithmetic. For exams or coursework, you can match your teacher’s rounding conventions by rounding intermediate steps manually if needed.

Parallelogram area formulas

A parallelogram is a quadrilateral with two pairs of parallel opposite sides. There are several equivalent formulas for its area, depending on the data you know.

Base and height

The most common formula uses the base \(b\) and the perpendicular height \(h\):

\[ A = b \cdot h \]

Here, \(h\) is the length of the perpendicular dropped from one side to the opposite side (or its extension), not the slanted side length.

Two sides and included angle

If you know the lengths of two adjacent sides \(a\) and \(b\), and the interior angle \(\theta\) between them, you can use the trigonometric form:

\[ A = a \cdot b \cdot \sin(\theta). \]

Geometrically, \(b \sin(\theta)\) is exactly the perpendicular height when \(a\) is taken as the base.

Coordinates and vector cross product

When working in coordinate geometry, you can interpret two adjacent sides as vectors. Suppose:

A parallelogram has vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\);
Vectors \(\vec{AB}\) and \(\vec{AC}\) form the sides.

The area is given by the magnitude of the 2D cross product:

\[ A = \bigl| \vec{AB} \times \vec{AC} \bigr| = \left| (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) \right|. \]

This is closely related to the determinant of a 2×2 matrix built from the side vectors.

Relation to rectangles and triangles

A rectangle is a parallelogram with a right angle (\(\theta = 90^\circ\)), so \(\sin(\theta) = 1\) and:

\[ A = a \cdot b \cdot \sin(90^\circ) = a \cdot b. \]

Any triangle with base \(b\) and height \(h\) has area \(A_{\triangle} = \frac{1}{2}b h\). A parallelogram with the same base and height can be thought of as two congruent triangles, so its area is exactly double:

\[ A_{\text{parallelogram}} = 2 A_{\triangle} = 2 \cdot \frac{1}{2} b h = b h. \]

Units and common conversions

As long as all sides and heights use the same length unit, the area will be in the corresponding square unit:

mm → mm²
cm → cm² (1 cm² = 100 mm²)
m → m² (1 m² = 10,000 cm²)
in → in²
ft → ft² (1 ft² = 144 in²)

In school contexts, cm² and m² are most common. In engineering or construction, m² and ft² are typical. The calculator automatically converts between the internal square unit and m² / cm² where appropriate.

Example 1 – base and height

A parallelogram has base \(b = 9\ \text{cm}\) and height \(h = 4\ \text{cm}\).

Confirm base and height use the same unit (cm).
Compute \(A = b h = 9 \times 4 = 36\ \text{cm}^2\).

Example 2 – two sides and included angle

A parallelogram has side lengths \(a = 5\ \text{m}\), \(b = 3\ \text{m}\), and the angle between them is \(\theta = 60^\circ\).

Compute \(\sin(60^\circ) \approx 0.8660.\)
Compute \(A = a b \sin(\theta) = 5 \times 3 \times 0.8660 \approx 12.99\ \text{m}^2.\)

Example 3 – coordinates

Let \(A(0, 0)\), \(B(6, 2)\), \(C(3, 5)\). These three points define a parallelogram with sides along \(\vec{AB}\) and \(\vec{AC}\).

\(\vec{AB} = (6 - 0, 2 - 0) = (6, 2)\).
\(\vec{AC} = (3 - 0, 5 - 0) = (3, 5)\).
Compute the cross product magnitude: \[ A = |6 \cdot 5 - 3 \cdot 2| = |30 - 6| = 24. \]
The area is \(24\) square units (depending on the scale of the coordinate system).

Good practice when using a parallelogram area calculator

Check that all length measurements correspond to the same physical unit before multiplying.
Use the angle between the two sides you are using; a different angle will not give the correct area.
When using coordinates, make sure the three vertices indeed form two adjacent sides, not a diagonal.
In assessment or design contexts, match the requested level of precision and show intermediate steps clearly.

Related geometry and area calculators

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