Quadrilaterals • Topic 2 of 3

Rhombus & Square

A rhombus is a parallelogram with all four sides equal. Its diagonals bisect each other AT RIGHT ANGLES and also bisect the vertex angles — but they are unequal. Because the diagonals are perpendicular bisectors, each side equals ½√(d₁² + d₂²) (the half-diagonals form a right triangle), and the area is ½ × d₁ × d₂. A square is the perfect quadrilateral: all sides equal AND all angles 90°, so it is simultaneously a rectangle and a rhombus. It therefore has diagonals that are equal, bisect each other, AND meet at right angles — the only quadrilateral with all three. A square of side s has diagonal s√2 and area s² = ½ × diagonal². A classic CAT trap: a rhombus and a square both give area = ½ d₁ d₂, but in a square d₁ = d₂, so the formula collapses to ½ d².

✅ Solved examples

1. A rhombus has diagonals 16 cm and 12 cm. Find its area and side.
Area = ½ × 16 × 12 = 96 cm². Side = ½√(16² + 12²) = ½√400 = ½ × 20 = 10 cm.
2. A square has diagonal 10√2 cm. Find its area.
Area = ½ × diagonal² = ½ × (10√2)² = ½ × 200 = 100 cm². (Side = 10 cm.)
3. A rhombus has side 13 cm and one diagonal 24 cm. Find the other diagonal.
Half-diagonals: one is 12. Side² = 12² + (d₂/2)² ⇒ 169 = 144 + (d₂/2)² ⇒ (d₂/2)² = 25 ⇒ d₂/2 = 5 ⇒ d₂ = 10 cm.
4. A square and a rhombus have the same perimeter 40 cm. The rhombus has one diagonal 16 cm. Which has the larger area, and by how much?
Side = 10 each. Square area = 100. Rhombus: half-diagonal 8, other half = √(10² − 8²) = 6 ⇒ d₂ = 12; area = ½ × 16 × 12 = 96. Square is larger by 4 cm².

✏️ Practice — try these, take hints as needed

1. A rhombus has diagonals 30 cm and 16 cm. Find its area.
Area = ½ d₁ d₂.
½ × 30 × 16.
15 × 16.
240 cm²
2. A square has area 144 cm². Find its diagonal.
Side = √area = 12.
Diagonal = side × √2.
12√2.
12√2 cm
3. A rhombus has side 10 cm and one diagonal 12 cm. Find the other diagonal.
Half of 12 is 6.
10² = 6² + (d₂/2)².
(d₂/2)² = 64.
16 cm
4. A square has diagonal 14 cm. Find its area.
Area = ½ × diagonal².
½ × 14².
½ × 196.
98 cm²
5. A rhombus has area 168 cm² and one diagonal 24 cm. Find the other diagonal.
Area = ½ d₁ d₂.
168 = ½ × 24 × d₂.
168 = 12 d₂.
14 cm

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Parallelogram & Rectangle Trapezium & Kite →