The diagonals of a rhombus cut each other in half at right angles. Enter both diagonals to get the area, the side length and the perimeter.
Enter both diagonals.
Read the area, side and perimeter.
Area: A = ½ × d₁ × d₂
Side: s = √((d₁/2)² + (d₂/2)²)
A rhombus is a parallelogram with all four sides equal, the shape of a diamond on a playing card. This calculator needs only the two diagonals. From them it finds the area, the length of each side and the perimeter.
Diagonals are often the easiest thing to measure on a real rhombus. On a floor tile, a kite frame or a rangoli pattern you can stretch a tape from corner to corner in both directions far more easily than you can find a perpendicular height. Class 8 mensuration and Class 9 quadrilateral chapters lean heavily on this, and many competitive exam questions give the diagonals and ask for the side or perimeter.
1. Measure the longer diagonal d₁ and the shorter diagonal d₂ from corner to opposite corner.
2. Multiply them and halve the product to get the area: A = ½ × d₁ × d₂.
3. Halve each diagonal. These halves are the two legs of a right triangle, because the diagonals of a rhombus bisect each other at 90°.
4. Use Pythagoras for the side: s = √((d₁/2)² + (d₂/2)²).
5. Multiply the side by 4 for the perimeter: P = 4s.
6. If you need an angle of the rhombus, use tan(θ/2) = (d₂/2) ÷ (d₁/2), where θ is the angle at the ends of the longer diagonal.
Draw a rectangle around the rhombus so that each corner of the rhombus touches the middle of one side of the rectangle. The rectangle measures d₁ by d₂. The rhombus splits it into the rhombus itself plus four corner triangles, and those four triangles are exactly the same size as the four triangles inside the rhombus. So the rhombus fills precisely half of the rectangle: A = ½ × d₁ × d₂. The same argument works for any quadrilateral whose diagonals are perpendicular, which is why a kite uses the same area formula.
Because a rhombus is a parallelogram, its diagonals bisect each other. Because all its sides are equal, the two triangles on either side of a diagonal are congruent, which forces the diagonals to cross at 90°. The four small triangles are therefore right triangles with legs d₁/2 and d₂/2 and hypotenuse equal to the side. That is where s = √((d₁/2)² + (d₂/2)²) comes from. Diagonals of 24 and 10 give legs of 12 and 5, a 5-12-13 Pythagorean triple, which is why textbook problems favour such numbers.
A rhombus is also a parallelogram, so its area equals side × perpendicular height, or s² × sin θ where θ is any interior angle. All three formulas must agree, which makes a good self-check. When the diagonals are equal the rhombus becomes a square, and the formula reduces to d²/2, the familiar result that a square's area is half the square of its diagonal. When one diagonal shrinks towards zero the rhombus flattens and its area heads to zero, even though the sides stay long.
Kavya is cutting rhombus-shaped mirror pieces for a craft panel. Each piece has diagonals of 24 cm and 10 cm, and she needs the area of glass and the length of edging tape for one piece.
Area = ½ × d₁ × d₂: = 0.5 × 24 × 10 = 120
Side (diagonals bisect at right angles): √((24/2)² + (10/2)²) = 13
Perimeter = 4 × side: = 52
Answer: Area 120; Side 13; Perimeter 52
Forgetting the half and reporting d₁ × d₂, which is the area of the surrounding rectangle, twice the rhombus.
Using the full diagonals in Pythagoras instead of their halves, which doubles the side length.
Applying the side formula to a kite. A kite has perpendicular diagonals but they do not bisect each other, so its sides differ.
Measuring a diagonal along the surface of a slightly bent or curved object, which gives a longer value than the straight-line distance.
Estimating material for rhombus-shaped tiles, jaali panels and mirror work.
Finding the side and perimeter of a kite or diamond frame from its cross-sticks.
Class 8 and Class 9 mensuration and quadrilateral property questions.
Laying out diamond patterns in rangoli, embroidery and flooring designs.
Crystallography and design work, where rhombic lattices and faces appear.
Is a square a rhombus?
Yes, a rhombus with equal diagonals and right angles.
Can I use this for a kite?
The area formula ½ × d₁ × d₂ also works for a kite, but the side formula does not.