Rhombus Shape Solver

A rhombus is a quadrilateral with all four sides equal in length. That single condition forces every other property:

• Opposite sides are parallel, so every rhombus is a parallelogram (a "diamond" tilted parallelogram, not necessarily upright).
• Opposite angles are equal; consecutive angles are supplementary, summing to $180^\circ$.
• The diagonals are perpendicular bisectors of each other, meeting at $90^\circ$ and cutting each other exactly in half.
• Each diagonal bisects the two angles it passes through.
• It has 2 lines of symmetry — the diagonals — and rotational symmetry of order 2.

Because the perpendicular diagonals split the rhombus into four congruent right triangles with legs $d_1/2$ and $d_2/2$, the side satisfies

$s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}$

and the perimeter is simply $P = 4s$.

Three formulas, each matched to what you are given.

From the diagonals — the one to reach for first:

$A = \frac{d_1 \cdot d_2}{2}$

From base and height, since a rhombus is a parallelogram:

$A = b \cdot h$

Here $h$ is the perpendicular distance between two parallel sides, not a slanted side.

From a side and an included angle:

$A = s^2 \sin\theta$

This works for either angle, because $\sin\theta = \sin(180^\circ - \theta)$ and the two angles of a rhombus are supplementary.

Choosing and rearranging

Given	Use
Both diagonals	$A = d_1 d_2 / 2$
Side + angle	$A = s^2 \sin\theta$
Side + height	$A = s h$
Area + one diagonal	$d_2 = 2A/d_1$

All three assume the figure really is a rhombus — four equal sides. If only opposite sides match, use the parallelogram formulas instead.

The classification questions all follow from one hierarchy: square $\subset$ rhombus $\subset$ parallelogram $\subset$ quadrilateral.

• Is a square a rhombus? Yes, always. A square has four equal sides, which is the definition of a rhombus.
• Is a rhombus a square? Only when its angles are $90^\circ$ (equivalently, when its diagonals are equal). A tilted rhombus is not a square.
• Is a rectangle a rhombus? Only if the rectangle is a square. A $3 \times 5$ rectangle has unequal adjacent sides, so it fails the definition.
• Is a rhombus a parallelogram? Yes — equal opposite sides force both pairs to be parallel.
• Is a rhombus a polygon? Yes: a closed four-sided plane figure, so a quadrilateral.

Common mistakes

• Using $A = s^2$. That is the square's formula; a rhombus with the same side has a smaller area unless $\theta = 90^\circ$.
• Multiplying the diagonals and forgetting the $\tfrac{1}{2}$.
• Treating a slanted side as the height in $A = bh$.
• Claiming the diagonals are equal — they are perpendicular, not congruent, unless the rhombus is a square.