Is every square a rhombus?

Yes! A square is a special rhombus where all four interior angles are right angles (90 degrees) and both diagonals are equal in length. However, not every rhombus is a square.

Why do diagonals in a rhombus always intersect at 90 degrees?

Because all four sides of a rhombus are equal, the diagonals divide the shape into four congruent right-angled triangles sharing a common perpendicular intersection point.

How do you find the altitude or height of a rhombus?

Once you know the area and the side length, the altitude is simply Height = Area ÷ Side Length. Alternatively, if you know an interior angle θ, Height = Side × sin(θ).

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Rhombus Calculator

Calculate the area, perimeter, side length, diagonals, altitude, and inscribed circle radius (inradius) of a rhombus using perpendicular diagonals, side and height, or side and angle.

Interactive Rhombus Calculator Workspace

Live Solver

Rhombus Calculation: Solve for area, perimeter, side length, diagonals, altitude, and inscribed circle radius from perpendicular diagonals, side and height, or side and angle.

Input Method

Diagonal 1 (d₁)

Diagonal 2 (d₂)

Rhombus Metrics

Area (A)40.0000

Perimeter (P)25.6125

Side Length (s)6.4031

Altitude (h)6.2470

Inradius (r)3.1235

Acute Angle77.32°

Orthogonal Diagonal Proof

1. Area: Evaluated as Area = 40.0000

2. Perimeter: All 4 edges are identical: P = 4 × 6.4031 = 25.6125

3. Diagonals: d₁ = 10.0000, d₂ = 8.0000 intersecting at 90° right angles.

4. Inradius (Inscribed Circle): Maximum circular tangent radius inside the rhombus = r = Area / (2 × side) = 3.1235

Mathematical Formula & Variables

Equation ModelArea = (d₁ × d₂) / 2 = s × h = s² × sin(θ), Perimeter = 4 × s

Calculates all geometric properties of an equilateral parallelogram with mutually perpendicular diagonals bisecting each other at 90 degrees.

Variable Definitions

Symbol	Description
d₁, d₂ (Diagonals)	Perpendicular segments connecting opposite corners.
s (Side Length)	Length of each of the 4 equal sides: s = √((d₁/2)² + (d₂/2)²).
h (Altitude / Height)	Perpendicular distance between opposing parallel sides.
r (Inradius)	Radius of the inscribed tangent circle: r = (d₁ × d₂) / (4s).

How to Use the Rhombus Calculator

Choose your preferred input method: Diagonals (d₁, d₂), Side & Height, or Side & Angle.
Enter the known values into the corresponding input fields.
View the real-time calculations for Area, Perimeter, Side Length, Altitude, and Inradius.
Examine the step-by-step mathematical proof based on orthogonal diagonal triangles.

Practical Example Calculation

Scenario Context: A civil engineer is designing a rhombus-shaped paving tile with diagonals measuring 12 cm and 16 cm.

Step 1: Identify given values: Diagonal 1 (d₁) = 12 cm, Diagonal 2 (d₂) = 16 cm.

Step 2: Calculate Area: A = (d₁ × d₂) / 2 = (12 × 16) / 2 = 192 / 2 = 96 square centimeters.

Step 3: Calculate Half-Diagonals: d₁/2 = 6 cm, d₂/2 = 8 cm.

Step 4: Calculate Side Length (s) using Pythagorean theorem: s = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.

Step 5: Calculate Perimeter (P): P = 4 × s = 4 × 10 = 40 cm.

Step 6: Calculate Inradius (r): r = Area / (2 × s) = 96 / 20 = 4.8 cm.

The rhombus tile has an area of 96 cm², a side length of 10 cm, a perimeter of 40 cm, and an inradius of 4.8 cm.

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