Key takeaway: A rhombus is a quadrilateral with four congruent sides whose diagonals cross at right angles, cut each other in half and bisect the corner angles they meet.
Grade 10 – GeometryG-CO.C.11Geometry and Measurement
Definition of Rhombus
Think of a playing-card diamond: a rhombus has four sides of one common length. It still belongs to the parallelogram family, so opposite sides are parallel and opposite corners match.
Equal sides add extra rules. The diagonals cross at \(90^{\circ}\) and cut the corners they join in half. They also chop the figure into four identical right triangles, so the side length and the area can be found from the two diagonals alone. Give it four square corners and you get a square.
Formula and notation
Call the two diagonals \(p\) and \(q\). The area is half their product:
\[ A = \frac{p\,q}{2} \]
A side \(s\) is the long edge of a right triangle whose legs are \(\frac{p}{2}\) and \(\frac{q}{2}\), so \(s^2 = (\frac{p}{2})^2 + (\frac{q}{2})^2\).
Key facts
- All four sides are equal, and the perimeter is four times one side.
- The diagonals are perpendicular and bisect each other.
- Each diagonal cuts the vertex angles at its ends into two equal parts.
- Every rhombus is a parallelogram, but a parallelogram needs equal sides to be one.
Rhombus example
A kite-shaped glass tile is a rhombus with diagonals of 18 cm and 80 cm. Find its side length, its perimeter and its area.
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The half-diagonals are 9 cm and 40 cm, and the diagonals cross at right angles.
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Side: \(s = \sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41\) cm.
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Perimeter: \(4(41) = 164\) cm. Area: \(\frac{1}{2}(18)(80) = 720\) cm².
Side 41 cm, perimeter 164 cm, area 720 cm².
Common mistake
Watch out
Lena uses the whole diagonals as legs: \(s = \sqrt{18^2 + 80^2} \approx 82\) cm for the side.
Why it is wrong
The diagonals cross at their midpoints, so each right triangle has legs equal to half of each diagonal, 9 cm and 40 cm. The side is 41 cm. Her 82 cm is the length of a hypotenuse for a triangle twice as large.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement