Calculate all properties of a rhombus from side and angle, or from diagonals. Includes area, perimeter, diagonals, incircle radius, height, and angles.
A rhombus is a quadrilateral with all four sides equal in length — essentially a "pushed-over" square. While every square is a rhombus, not every rhombus is a square; the key difference is the angles. In a general rhombus, opposite angles are equal, adjacent angles are supplementary (sum to 180°), and the diagonals bisect each other at right angles but are typically unequal in length.
The area of a rhombus can be calculated two ways: A = s² sin(α), where s is the side length and α is any interior angle, or A = (d₁ × d₂) / 2, where d₁ and d₂ are the diagonals. The diagonals split the rhombus into four congruent right triangles, making it easy to derive any property from just two known values.
Rhombuses appear in everyday life more often than you might think: diamond patterns on playing cards, argyle textile patterns, certain kite shapes, floor tiles, chain-link fences, and crystallographic lattices all feature rhombus geometry. In architecture, rhombus-shaped windows and decorative panels add visual interest while maintaining structural efficiency.
This calculator supports two input methods: (1) side length and one angle, or (2) both diagonals. Either pair fully determines the rhombus. The tool computes area, perimeter, both diagonals, the incircle radius, height, and both interior angles. Presets for common rhombus shapes let you explore the relationship between angles and dimensions instantly.
Rhombus problems are easy to state and easy to get wrong. A single side length does not determine the shape, because a narrow rhombus and a wide rhombus can share the same side while having very different diagonals, heights, and area. That makes this calculator useful whenever you need to move between the two most common descriptions of a rhombus: side-plus-angle and the pair of diagonals.
It is especially practical for pattern design, tile layouts, kites, and classroom geometry, where you may be given whichever measurements are easiest to observe rather than the ones that fit a neat textbook formula. Instead of deriving every property from scratch, you can enter the known dimensions once and inspect the full geometry immediately.
Area: A = s²sin(α) or A = (d₁ × d₂)/2 Perimeter: P = 4s Diagonals: d₁ = 2s sin(α/2), d₂ = 2s cos(α/2) Height: h = s sin(α) Incircle radius: r = (d₁ × d₂)/(4s) Side from diagonals: s = √((d₁/2)² + (d₂/2)²)
Result: Area ≈ 21.65 cm², Perimeter = 20 cm, Diagonals ≈ 5 cm & 8.66 cm
For a rhombus with s = 5 cm and α = 60°: Area = 25 × sin(60°) ≈ 21.65 cm². Diagonals: d₁ = 2×5×sin(30°) = 5 cm, d₂ = 2×5×cos(30°) ≈ 8.66 cm. Perimeter = 20 cm. Height = 5 × sin(60°) ≈ 4.33 cm. Incircle radius ≈ 2.165 cm.
A rhombus is fully determined either by a side and one interior angle or by its two diagonals. Those descriptions are equivalent, but they emphasize different features. Side-and-angle form is common in textbook geometry and trigonometry, while diagonals are often easier to measure from a drawing or real object. Switching between them is useful because each form makes some properties easier to compute than others.
The diagonals of a rhombus do more than split the shape visually. They intersect at right angles, bisect each other, and divide the rhombus into four congruent right triangles. That structure explains why the area formula $A = frac{d_1 d_2}{2}$ works and why the side length can be reconstructed from half-diagonals with the Pythagorean theorem. If you understand the diagonals, most other rhombus relationships become much easier to remember.
Rhombuses with the same side length can look dramatically different because the angle controls the height. Acute angles create flatter shapes with smaller area and a long diagonal, while angles near 90 degrees produce a more square-like figure with larger area. When you compare presets or adjust the angle manually, watch how the height, diagonals, and incircle radius respond together. That gives you a better geometric picture than memorizing formulas in isolation.
Both have four equal sides. A square additionally has four 90° angles and equal diagonals. A rhombus has two pairs of equal angles (acute and obtuse) and unequal diagonals.
Area = (d₁ × d₂) / 2. The diagonals split the rhombus into 4 right triangles, and this formula computes the total area. For d₁ = 10, d₂ = 8: A = 80/2 = 40 square units.
Yes. Each diagonal bisects the pair of opposite angles it connects. This means the diagonal through the acute vertices splits each acute angle in half.
The incircle is the largest circle that fits inside the rhombus, tangent to all four sides. Its radius is r = (d₁ × d₂)/(4s), or equivalently r = Area / (2s).
In geometry, "diamond" informally refers to a rhombus oriented with one vertex pointing up. A baseball diamond is actually a square (a special rhombus). Playing card diamond symbols are rhombuses.
If a rhombus has one right angle, all angles must be 90° (since opposite angles are equal and adjacent angles are supplementary). So a rhombus with a right angle is necessarily a square.