Rhombus
In short: A rhombus is a four-sided figure with all four sides the same length, and its diagonals cross at right angles while halving each other and the corners they pass through.
GeometryGeometryG-CO.C.11
What is Rhombus?
Think of a diamond on a playing card: that is a rhombus. It is a parallelogram whose four sides all have one length, so everything true of a parallelogram stays true, and extra facts appear.
The diagonals meet at \(90^{\circ}\), each cuts the other in half, and each bisects the two corners it connects. These perpendicular diagonals chop the figure into four congruent right triangles, which gives two handy formulas: side \(=\sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}\) and area \(=\frac{d_1 d_2}{2}\).
How to spot it
- A figure shows a slanted four-sided shape with the same tick mark on all four sides.
- The problem lists the two diagonals of a kite-like or diamond-shaped object, such as a tile or a flag.
- The wording says all sides are congruent, or that the diagonals are perpendicular.
- A perimeter is requested after only one side is known, since the other three match it.
The method
- Halve each diagonal to get the two short legs of one inner right triangle.
- Apply the Pythagorean theorem to those legs; the hypotenuse is the side of the figure.
- Multiply the side by 4 for the perimeter.
- For the area, multiply the diagonals and take half: \(\frac{d_1 d_2}{2}\).
Rhombus: solved example
Problem
A rhombus-shaped floor tile has diagonals of 16 inches and 30 inches. Find its side length, perimeter and area.
Solution
Half-diagonals: 8 and 15. The diagonals meet at a right angle, so the side is \(\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17\) inches.
Perimeter: \(4 \times 17 = 68\) inches. Area: \(\frac{16 \times 30}{2} = 240\) square inches. The tile has side 17 in, perimeter 68 in and area 240 square in.
Try it
Question 1
Each side of a rhombus-shaped sign is 13 cm. What is the perimeter?
Show the answer
All four sides match, so \(4 \times 13 = 52\) cm.
Question 2
A rhombus has side 13 and one diagonal of 10. Find the other diagonal.
Show the answer
The half-diagonal is \(\sqrt{13^2 – 5^2} = \sqrt{144} = 12\), so the other diagonal is \(24\).
Rhombus vs. Parallelogram
Every rhombus is a parallelogram, but not the other way around. A parallelogram only promises that opposite sides are equal, and its diagonals usually meet at a slant. The rhombus adds equal adjacent sides and perpendicular diagonals. A square is both a rhombus and a rectangle.
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