A rhombus is a quadrilateral with four equal sides. This calculator solves it three ways — from both diagonals, from a side and height, or from a side and an interior angle — returning area, perimeter, side, height, the diagonals, and an angle.
Properties of a rhombus
A rhombus shares its equal-side property with a square but differs in one key way: a square forces all four angles to be 90°, while a rhombus allows the shape to lean. The four angles of a rhombus come in two equal-opposite pairs (for example, 60°, 120°, 60°, 120°), and opposite angles are always equal. Two additional geometric facts make computation easier:
- The diagonals always bisect each other at right angles — they cross at their midpoints and form four right triangles in the interior.
- Each diagonal bisects the vertex angles it connects, making the half-angles available directly from the diagonal lengths.
How it works
The tool applies the standard rhombus relationships to whichever pair of values you provide:
- From diagonals p and q: the half-diagonals (p/2 and q/2) are the legs of a right triangle whose hypotenuse is the side — so
side = √((p/2)² + (q/2)²). Area is simply(p × q) ÷ 2. - From side and height h:
area = side × h. The height is the perpendicular distance between two parallel sides. - From side and interior angle θ:
area = side² × sin θ(same as the cross-product of two edge vectors). The height equalsside × sin θ, and the diagonals follow from the law of cosines applied to the triangles formed.
Perimeter is always 4 × side because all four sides are equal.
Worked example
A rhombus with diagonals p = 6 and q = 8:
- Area =
(6 × 8) ÷ 2 = 24 - Side =
√(3² + 4²) = √25 = 5(this is a 3-4-5 right triangle inside the rhombus) - Perimeter =
4 × 5 = 20 - Acute angle:
2 × arctan(q/2 ÷ p/2) = 2 × arctan(4/3) ≈ 106.3°(the obtuse angle); the acute angle is about73.7°
| Known inputs | Area formula | Notes |
|---|---|---|
| Diagonals p, q | (p × q) ÷ 2 | Most direct; no trig needed |
| Side a, height h | a × h | Height is perpendicular between parallel sides |
| Side a, angle θ | a² × sin θ | Use interior (not exterior) angle |
When rhombus calculations come up
Rhombuses appear in practical contexts more often than people expect: diamond-shaped floor tiles, lattice patterns in engineering drawings, gem facet geometry, and rhombus-truss roof structures all require computing the same area or diagonal relationships. They also appear in physics as the parallelogram law for adding vectors, where the diagonal of the rhombus (a special parallelogram where both sides are equal) represents the resultant vector when two equal forces are applied at an angle.
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