MathsICSE Class 8 · Understanding Quadrilaterals

Diagonal Properties of Parallelogram, Rectangle, Rhombus and Square (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

The diagonals of a parallelogram bisect each other; of a rectangle are equal and bisect each other; of a rhombus bisect each other at right angles; of a square are equal and bisect each other at right angles. In a rhombus, side $= \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}$.

Understand Diagonal Properties of Special Quadrilaterals

The curriculum lists these properties, to be arrived at by verification (measurement and paper folding).

Quadrilateral	Diagonal property
Parallelogram	Bisect each other
Rectangle	Equal; bisect each other
Rhombus	Bisect each other at right angles
Square	Equal; bisect each other at right angles

Parallelogram. The point $O$ where the diagonals meet is the midpoint of both: $A O = O C$ and $B O = O D$ . The diagonals are not equal in general.

Rectangle. Every angle is $9 0^{\circ}$ , which makes the two diagonals equal: $A C = B D$ . Each diagonal is a "hypotenuse", so $A C^{2} = A B^{2} + B C^{2}$ .

Rhombus. All four sides are equal. The diagonals cut at right angles, and each diagonal bisects the angles at its ends. The four small triangles are right-angled and equal, each with legs $\frac{d _{1}}{2}$ and $\frac{d _{2}}{2}$ , so the side is the hypotenuse.

Square. A square is both a rectangle and a rhombus, so its diagonals are equal and perpendicular.

Using the properties. Set equal halves to find unknowns (for a parallelogram: $A O = O C$ ); set equal diagonals for a rectangle; use right-angled triangles with Pythagoras for rhombus and rectangle sums. A rhombus has area $\frac{1}{2} d_{1} d_{2}$ .

Angle chasing. In a rhombus, a diagonal bisects the angle at a vertex; together with the $9 0^{\circ}$ at $O$ , this gives the angles of the small triangles.

Diagonal Properties of Special Quadrilaterals: formulas to remember

Rhombus side: $s = (\frac{d _{1}}{2})^{2} + (\frac{d _{2}}{2})^{2}$; From a right-angled triangle at O.
Rectangle diagonal: $d = l^{2} + b^{2}$; Both diagonals are equal.
Rhombus area: $\frac{1}{2} d_{1} d_{2}$; Also true for a square.

How to solve Diagonal Properties of Special Quadrilaterals questions

Match the quadrilateral to its diagonal property, then use halves, right angles and Pythagoras.

1Draw the figure and mark the point where the diagonals meet.
2Write the property that applies (bisect, equal, perpendicular).
3Equate the equal parts to form an equation.
4Use a right-angled triangle with half-diagonals for sides.
5Verify with the angle or length sum.

Diagonal Properties of Special Quadrilaterals 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.

Diamond (rhombus) frame check

Suppose a carpenter makes a diamond-shaped (rhombus) frame and checks that its two cross-bars are 18 cm and 24 cm, crossing at their midpoints at right angles.

Find the length of each side of the frame.

Try it first, then show the solutionSolution

Half-bars: $9$ and $12$ .
Side $= 81 + 144 = 225 = 15$ cm.

Answer: 15 cm (perimeter 60 cm).

Memory hook: Half-diagonals are the legs, the side is the hypotenuse.

Tricks to remember Diagonal Properties of Special Quadrilaterals

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

R for equal, R for right
Rectangle diagonals are Equal; Rhombus diagonals meet at Right angles; the square has both.
Use it when: Remembering the table.
Halve first
Always work with half the diagonals when a rhombus is involved.
Use it when: Side and angle problems.

Diagonal Properties of Special Quadrilaterals: solved examples

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

Example 1: Identify the property · 1 mark · MCQBoard level

In which quadrilateral do the diagonals bisect each other at right angles?
(a) rectangle (b) parallelogram (c) rhombus (d) trapezium

Try it first, then show the solutionSolution

A rhombus has perpendicular, bisecting diagonals. A rectangle's are equal but not perpendicular.

Answer: (c)

Example 2: Parallelogram diagonals · 3 marks · Short answerBoard level

In parallelogram $A B C D$ the diagonals meet at $O$ , with $A O = (3 x - 1)$ cm and $O C = (x + 5)$ cm. Find $x$ and the length of $A C$ .

Try it first, then show the solutionSolution

Diagonals bisect each other: $A O = O C$ .
$3 x - 1 = x + 5 \Rightarrow 2 x = 6 \Rightarrow x = 3$ .
$A O = 8$ cm, so $A C = 16$ cm.

Answer: $x = 3$ ; $A C = 16$ cm

Example 3: Rectangle diagonals · 3 marks · Short answerHarder board level

In rectangle $P QR S$ the diagonals are $P R = (2 y + 3)$ cm and $QS = (3 y - 5)$ cm. Find $y$ , the diagonals and the distance from $P$ to the centre.

Try it first, then show the solutionSolution

Diagonals are equal: $2 y + 3 = 3 y - 5 \Rightarrow y = 8$ .
Each diagonal $= 19$ cm.
Distance from $P$ to the centre $= \frac{19}{2} = 9.5$ cm.

Answer: $y = 8$ ; diagonals $19$ cm; $9.5$ cm

Example 4: Rhombus from diagonals · 4 marks · Short answerHarder board level

The diagonals of a rhombus are 18 cm and 24 cm. Find its side, perimeter and area.

Try it first, then show the solutionSolution

Half-diagonals $9$ and $12$ cm, meeting at right angles.
Side $= 9^{2} + 1 2^{2} = 225 = 15$ cm; perimeter $= 4 \times 15 = 60$ cm.
Area $= \frac{1}{2} \times 18 \times 24 = 216$ cm $^{2}$ .

Answer: Side 15 cm; perimeter 60 cm; area 216 cm $^{2}$

Example 5: Angles in a rhombus · 4 marks · Long answerHOTS / competency

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

Try it first, then show the solutionSolution

A diagonal bisects the vertex angle: $∠ B A C = \frac{70}{2} = 3 5^{\circ}$ .
Adjacent angles are supplementary: $∠ A B C = 11 0^{\circ}$ , so $∠ A B D = 5 5^{\circ}$ .
Diagonals are perpendicular: $∠ A O B = 9 0^{\circ}$ . Check in $△ A O B$ : $35 + 55 + 90 = 180$ .
Opposite angles are equal: $∠ B C D = 7 0^{\circ}$ .

Answer: $3 5^{\circ}$ ; $5 5^{\circ}$ ; $9 0^{\circ}$ ; $7 0^{\circ}$

Where students lose marks in Diagonal Properties of Special Quadrilaterals

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

Mistake: Saying the diagonals of a rhombus are equal
Among parallelograms, only rectangles and squares have equal diagonals.
Wrong working
Rhombus with diagonals $18$ and $24$ : "diagonals are equal".
Right working
Diagonals of a rhombus are perpendicular bisectors, not equal.
Do this: Learn the table: equal = rectangle/square.
Usually costs: Property mark.
Mistake: Using full diagonals in the Pythagoras step for a rhombus
The right angle is formed by the half-diagonals.
Wrong working
Side $= 1 8^{2} + 2 4^{2} = 30$ .
Right working
Side $= 9^{2} + 1 2^{2} = 15$ .
Do this: Mark the halves on the diagram.
Usually costs: Answer mark.

Exam tips

Name the property used in each line (for example "diagonals of a parallelogram bisect each other").
Check with Pythagoras when numbers form a triple like 5-12-13.
Draw the diagram with O marked.

Practice questions on Diagonal Properties of Special Quadrilaterals

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 diagonals that are equal and bisect each other at right angles?
(a) rectangle (b) rhombus (c) parallelogram (d) square
Hint
Both properties together.
Answer
(d)
Q2 · 2 marks · Short answerBoard level
In parallelogram $A B C D$ , $A C = 14$ cm and $B D = 10$ cm, meeting at $O$ . Find $O A$ and $O B$ .
Hint
Diagonals bisect each other.
Answer
$O A = 7$ cm and $O B = 5$ cm.
Q3 · 3 marks · Short answerHarder board level
One diagonal of a rectangle is 26 cm and one side is 24 cm. Find the other side and the perimeter.
Hint
Right triangle with hypotenuse the diagonal.
Answer
Other side $= 2 6^{2} - 2 4^{2} = 676 - 576 = 100 = 10$ cm. Perimeter $= 2 (24 + 10) = 68$ cm.
Q4 · 3 marks · Short answerHarder board level
The diagonal of a square is 12 cm. Find its area.
Hint
Square is a rhombus with equal diagonals.
Answer
Area $= \frac{1}{2} \times 12 \times 12 = 72$ cm $^{2}$ .
Q5 · 4 marks · Long answerHOTS / competency
A rhombus has side 13 cm and one diagonal 24 cm. Find the other diagonal, its area and the angle between the diagonals. Explain why it is not a square.
Hint
Half the given diagonal is 12.
Answer
Half other diagonal $= 1 3^{2} - 1 2^{2} = 5$ , so the other diagonal is $10$ cm. Area $= \frac{1}{2} (24) (10) = 120$ cm $^{2}$ . The diagonals meet at $9 0^{\circ}$ . Diagonals $24 \neq = 10$ are unequal, so it is not a square.

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.Do rectangle diagonals bisect at right angles?Show answer
Not in general; only if it is a square.
2.Rhombus area formula?Show answer
Half the product of the diagonals.
3.Diagonals of a square?Show answer
Equal, bisect each other at right angles.
4.Parallelogram diagonals are equal?Show answer
No, they just bisect each other.

Diagonal Properties of Parallelogram, Rectangle, Rhombus and Square (ICSE Class 8)

Understand Diagonal Properties of Special Quadrilaterals

Diagonal Properties of Special Quadrilaterals: formulas to remember

How to solve Diagonal Properties of Special Quadrilaterals questions

Diagonal Properties of Special Quadrilaterals in real life

Tricks to remember Diagonal Properties of Special Quadrilaterals

Diagonal Properties of Special Quadrilaterals: solved examples

Where students lose marks in Diagonal Properties of Special Quadrilaterals

Exam tips

Practice questions on Diagonal Properties of Special Quadrilaterals

Test your recall

Diagonal Properties of Special Quadrilaterals: frequently asked questions

Is every square a rhombus?

Is every rectangle a square?