MathsICSE Class 9 · Rectilinear Figures

Rhombus, Rectangle and Square: Properties, Diagonals and Solved Problems (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

A rhombus has four equal sides and diagonals that bisect each other at right angles and bisect the angles. A rectangle has four right angles and equal diagonals. A square has all of these: four equal sides, four right angles, and equal diagonals that bisect each other at right angles.

Rhombus diagonals cross at 90°, rectangle diagonals are equal, and the square has every property of both.

Illustration drawn for QuestionBuddy.

Understand Rhombus, Rectangle and Square: Properties and Diagonals

Rhombus (a parallelogram with four equal sides). Properties: diagonals bisect each other at $9 0^{\circ}$ ; each diagonal bisects the angles at its ends; area $= \frac{1}{2} d_{1} d_{2}$ ; side $= (\frac{d _{1}}{2})^{2} + (\frac{d _{2}}{2})^{2}$ .

*Proof that diagonals are perpendicular.* In $△ A O B$ and $△ A O D$ : $A B = A D$ , $O B = O D$ (diagonals of a parallelogram bisect), $A O$ common, so they are congruent (SSS); $∠ A O B = ∠ A O D$ and they form a linear pair, so each is $9 0^{\circ}$ .

Rectangle (a parallelogram with one right angle, hence four). Properties: diagonals are equal and bisect each other; diagonal $= l^{2} + b^{2}$ ; area $= l b$ .

Square. A rectangle with equal adjacent sides, equivalently a rhombus with a right angle. Diagonals are equal, bisect each other at $9 0^{\circ}$ and bisect the angles into $4 5^{\circ}$ . Diagonal $= 2 s$ , area $= s^{2} = \frac{d ^{2}}{2}$ .

Which test proves which figure?

Parallelogram with equal adjacent sides $\Rightarrow$ rhombus.
Parallelogram with one right angle (or equal diagonals) $\Rightarrow$ rectangle.
Rhombus with equal diagonals, or rectangle with equal adjacent sides $\Rightarrow$ square.

**Rhombus with a $6 0^{\circ}$ angle.** If $∠ A = 6 0^{\circ}$ and $A B = A D$ , then $△ A B D$ is equilateral, so the short diagonal $B D =$ side, and the long diagonal $A C = 2 s^{2} - \frac{s ^{2}}{4} = s 3$ . For side $8$ : $B D = 8$ , $A C = 83$ , area $= \frac{1}{2} \times 8 \times 83 = 323$ .

Angle chasing in a rhombus. If $∠ A = 7 0^{\circ}$ then $∠ C = 7 0^{\circ}$ , $∠ B = ∠ D = 11 0^{\circ}$ ; diagonals halve the angles, so $∠ O A B = 3 5^{\circ}$ , $∠ O B A = 5 5^{\circ}$ and $∠ A O B = 9 0^{\circ}$ (check $35 + 55 + 90 = 180$ ).

Level. Class IX is a school-exam class. Basic property questions are routine; the 60° rhombus and mixed angle chase are at the hardest school-exam level.

Rhombus, Rectangle and Square: Properties and Diagonals: formulas to remember

Rhombus area: $\frac{1}{2} d_{1} d_{2}$; Diagonals perpendicular.
Rhombus side: $s = (d_{1} /2)^{2} + (d_{2} /2)^{2}$; Half-diagonals are legs.
Rectangle diagonal: $d = l^{2} + b^{2}$; Diagonals equal.
Square: $d = s 2, A = \frac{d ^{2}}{2}$; Diagonal bisects the angles.
Rhombus with 60° angle: $d_{s h or t} = s, d_{l o n g} = s 3$; Equilateral triangles.

How to solve Rhombus, Rectangle and Square: Properties and Diagonals questions

Name the figure, recall its diagonal property, then use triangles formed by the diagonals.

1Mark equal sides and right angles.
2Use the diagonal property: bisect, perpendicular, equal.
3For lengths, use Pythagoras in a triangle with the half-diagonals.
4For angles, use that diagonals bisect the angles and that the diagonals meet at $9 0^{\circ}$ in a rhombus.
5Check with the angle sum of the triangle (180°).

Rhombus, Rectangle and Square: Properties and Diagonals in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A rhombus-shaped kite frame

Suppose a kite frame is a rhombus with diagonals 24 cm and 10 cm, with ribbon along the diagonals and the border.

Find the side, perimeter and the length of ribbon for the diagonals.

Try it first, then show the solutionSolution

Side $= 1 2^{2} + 5^{2} = 13$ cm; perimeter $= 52$ cm.
Diagonals: $24 + 10 = 34$ cm of ribbon.

Answer: Side 13 cm; perimeter 52 cm; 34 cm ribbon.

Memory hook: Half-diagonals are the legs.

A rectangular screen

Imagine a rectangular display of ratio 3 : 4 whose diagonal is 20 cm.

Find its sides and area.

Try it first, then show the solutionSolution

Sides $3 k, 4 k$ with $9 k^{2} + 16 k^{2} = 400$ , so $k = 4$ .
Sides $12$ cm and $16$ cm; area $192$ cm².

Answer: 12 cm × 16 cm; 192 cm².

Memory hook: 3-4-5 scaled.

Tricks to remember Rhombus, Rectangle and Square: Properties and Diagonals

Short memory aids. Say each one aloud once, then use it on a question the same day.

Rhombus: perpendicular
Diagonals of a rhombus cross at right angles and halve the angles.
Use it when: Angle problems.
Rectangle: equal
Diagonals of a rectangle are equal.
Use it when: Length problems.
Square: both
A square has every diagonal property.
Use it when: Square questions.

Rhombus, Rectangle and Square: Properties and Diagonals: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Side of a rhombus · 1 mark · MCQBoard level

The diagonals of a rhombus are $12$ cm and $16$ cm. Its side is:
(a) 8 cm
(b) 10 cm
(c) 14 cm
(d) 20 cm

Try it first, then show the solutionSolution

Side $= 6^{2} + 8^{2} = 10$ cm.

Answer: (b) 10 cm

Example 2: Rectangle · 3 marks · Short answerBoard level

A diagonal of a rectangle is $13$ cm and one side is $5$ cm. Find the other side and the perimeter.

Try it first, then show the solutionSolution

Other side $= 169 - 25 = 12$ cm.
Perimeter $= 2 (5 + 12) = 34$ cm.

Answer: 12 cm; 34 cm

Example 3: Angle chasing · 4 marks · Short answerHarder board level

In rhombus $A B C D$ , $∠ A = 7 0^{\circ}$ and the diagonals meet at $O$ . Find $∠ B$ , $∠ O A B$ , $∠ O B A$ and $∠ A O B$ and check the angle sum.

Try it first, then show the solutionSolution

$∠ B = 180 - 70 = 11 0^{\circ}$ (adjacent angles).
Diagonals bisect angles: $∠ O A B = 3 5^{\circ}$ , $∠ O B A = 5 5^{\circ}$ .
Diagonals meet at $9 0^{\circ}$ : $∠ A O B = 9 0^{\circ}$ .
Check: $35 + 55 + 90 = 180$ . ✓

Answer: $11 0^{\circ}, 3 5^{\circ}, 5 5^{\circ}, 9 0^{\circ}$

Example 4: Rhombus from side and a diagonal · 4 marks · Long answerHarder board level

The side of a rhombus is $13$ cm and one diagonal is $24$ cm. Prove that the diagonals are perpendicular, then find the other diagonal and the area.

Try it first, then show the solutionSolution

Diagonals of a parallelogram bisect each other; with $A B = A D$ and $O B = O D$ , $△ A O B ≅ △ A O D$ (SSS), so $∠ A O B = ∠ A O D = 9 0^{\circ}$ .
Half the other diagonal $= 1 3^{2} - 1 2^{2} = 5$ , so the diagonal is $10$ cm.
Area $= \frac{1}{2} \times 24 \times 10 = 120$ cm².

Answer: Proved; 10 cm; 120 cm²

Example 5: The 60° rhombus · 5 marks · Long answerHOTS / competency

A rhombus of side $8$ cm has $∠ A = 6 0^{\circ}$ . Prove that the short diagonal equals the side, and find the long diagonal and the area.

Try it first, then show the solutionSolution

In $△ A B D$ , $A B = A D = 8$ and $∠ A = 6 0^{\circ}$ , so the base angles are $6 0^{\circ}$ each: it is equilateral, so $B D = 8$ cm.
Diagonals bisect at right angles: half the long diagonal $= 8^{2} - 4^{2} = 48 = 43$ , so $A C = 83 \approx 13.86$ cm.
Area $= \frac{1}{2} \times 8 \times 83 = 323 \approx 55.43$ cm².

Answer: BD = 8 cm; AC = 8√3 cm; area 32√3 cm² ≈ 55.43 cm²

Where students lose marks in Rhombus, Rectangle and Square: Properties and Diagonals

The slips examiners see most often, and what to write instead.

Mistake: Full diagonals in Pythagoras
Half-diagonals are the legs.
Wrong working
$1 2^{2} + 1 6^{2} = 20$ as the side.
Right working
$6^{2} + 8^{2} = 10$ .
Do this: Halve first.
Usually costs: Whole answer.
Mistake: Saying diagonals of a rhombus are equal
They are equal only in a square.
Wrong working
Rhombus diagonals are equal.
Right working
They bisect at right angles; they are equal only if it is a square.
Do this: Know which figure has which property.
Usually costs: Reason mark.
Mistake: Wrong angle bisected
Diagonals bisect the vertex angles of a rhombus.
Wrong working
$∠ O A B = 7 0^{\circ}$ .
Right working
$∠ O A B = 3 5^{\circ}$ .
Do this: Halve the angle.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; the perpendicular-diagonals proof is a standard theorem.
Always mark the right angle at O in rhombus diagrams.
Check angle questions with the triangle sum.
Leave surds exact unless a decimal is requested.

Practice questions on Rhombus, Rectangle and Square: Properties and Diagonals

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Which quadrilateral has equal diagonals that bisect each other at right angles?
(a) Rectangle
(b) Rhombus
(c) Square
(d) Parallelogram
Hint
Needs both equal and perpendicular.
Answer
(c) Square
Q2 · 3 marks · Short answerHarder board level
The sides of a rectangle are in the ratio $3 : 4$ and its diagonal is $20$ cm. Find its sides and area.
Hint
$(3 k)^{2} + (4 k)^{2} = 400$ .
Answer
$k = 4$ : sides $12$ cm and $16$ cm; area $192$ cm².
Q3 · 3 marks · Short answerHarder board level
In rhombus $A B C D$ , $∠ A B C = 13 0^{\circ}$ . Find $∠ B A C$ and $∠ B D C$ .
Hint
Diagonals bisect the angles.
Answer
$∠ A = 5 0^{\circ}$ , so $∠ B A C = 2 5^{\circ}$ ; $B D$ bisects $∠ B$ , so $∠ C B D = 6 5^{\circ}$ , and in the isosceles $△ B C D$ (with $B C = C D$ ), $∠ B D C = ∠ C B D = 6 5^{\circ}$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Every rectangle is a parallelogram.
Reason (R): In a rectangle, opposite sides are equal and parallel.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
A parallelogram has opposite sides parallel.
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A kite frame is a rhombus with diagonals $24$ cm and $10$ cm. Ribbon is run along the border and both diagonals.
(a) Find the side of the rhombus.
(b) Find the length of ribbon for the border.
(c) Find the length of ribbon for the two diagonals.
(d) Find the area of cloth needed to cover it.
Hint
Half-diagonals are 12 and 5.
Answer
(a) 13 cm (b) 52 cm (c) 34 cm (d) 120 cm²

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Area of a rhombus?Show answer
½ × product of diagonals.
2.What are the diagonals of a rectangle like?Show answer
Equal and bisecting each other.
3.What is special about the diagonals of a square?Show answer
Equal, perpendicular, bisecting each other and the angles.
4.Short diagonal of a 60° rhombus?Show answer
Equal to its side.

Rhombus, Rectangle and Square: Properties and Diagonals: frequently asked questions

Is a square a rhombus?

Yes, a rhombus with a right angle.

Is every rectangle a square?

No; only when adjacent sides are equal.

Are constructions part of this chapter?

The chapter also includes constructing quadrilaterals; this page covers properties and calculations.