MathsICSE Class 8 · Construction of Quadrilaterals

Constructing Quadrilaterals: Four Cases with Steps and Checks (ICSE Class 8 Maths)

Updated 17 October 2026 · Fact-checked

A quadrilateral needs five independent measurements. The CISCE cases are: four sides and one diagonal; three sides and two diagonals; three sides and two included angles; two adjacent sides and three angles. Draw a rough sketch first, build a triangle with the given diagonal, then locate the fourth vertex with arcs or angles.

Build triangle ABC first, then use arcs from A and C to find D on the other side of diagonal AC.

Illustration drawn for QuestionBuddy.

Understand Constructing Quadrilaterals from Five Measurements

The curriculum asks for construction of quadrilaterals given:
1. four sides and one diagonal,
2. three sides and two diagonals,
3. three sides and two included angles,
4. two adjacent sides and three angles.

Why five measurements? A quadrilateral has 4 vertices, and fixing its shape needs five numbers. Four sides alone are not enough because the shape can be "squeezed"; adding a diagonal makes it rigid.

Rough sketch first. Draw a quick sketch and label the given measurements. It shows which triangle to start with.

Case 1: four sides and a diagonal (AB, BC, CD, DA, AC). Triangle $A B C$ from $A B$ , $B C$ , $A C$ by arcs. Then triangle $A C D$ on the other side of $A C$ from $A D$ and $C D$ . Join to finish.

Case 2: three sides and two diagonals (AB, BC, CD, AC, BD). Triangle $A B C$ from $A B$ , $B C$ , $A C$ . Then $D$ is at distance $B D$ from $B$ and $C D$ from $C$ (arcs).

**Case 3: three sides and two included angles (AB, BC, CD, $∠ B$ , $∠ C$ ).** Draw $B C$ ; at $B$ make $∠ B$ and mark $A$ on the ray at distance $A B$ ; at $C$ make $∠ C$ and mark $D$ at distance $C D$ ; join $A D$ .

**Case 4: two adjacent sides and three angles (AB, BC, $∠ A$ , $∠ B$ , $∠ C$ ).** Draw $A B$ , make $∠ B$ and mark $C$ at distance $B C$ ; make $∠ A$ at $A$ and $∠ C$ at $C$ ; the two rays meet at $D$ . The fourth angle is $36 0^{\circ} - (A + B + C)$ .

Feasibility. Every triangle used must satisfy the triangle inequality: the sum of any two sides is greater than the third. If not, the figure cannot be drawn.

Checking. Measure a quantity you did not use (for example the other diagonal) and compare it with a calculated value; and the angle sum must be $36 0^{\circ}$ .

Constructing Quadrilaterals from Five Measurements: formulas to remember

Triangle inequality: $a + b > c for every pair of sides$; Needed before drawing each triangle.
Fourth angle: $∠ D = 36 0^{\circ} - (∠ A + ∠ B + ∠ C)$; Case 4 check.
Parallelogram diagonals (check): $A C^{2} + B D^{2} = 2 (A B^{2} + B C^{2})$; Calculate the other diagonal.

How to solve Constructing Quadrilaterals from Five Measurements questions

Sketch, build the first triangle, then fix the last vertex with arcs or angles.

1Draw a rough sketch and label the given measurements.
2Check the triangle inequality for each triangle you will draw.
3Draw the base and the first triangle with a compass and ruler.
4Locate the fourth vertex with arcs (cases 1, 2) or angles (cases 3, 4).
5Join the vertices and measure a spare quantity to check.

Constructing Quadrilaterals from Five Measurements 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.

Fixing a diagonal brace

Suppose a four-bar frame has sides 5 cm, 4.5 cm, 5.5 cm and 4 cm and is held rigid by a 6.5 cm brace from corner A to corner C.

What does the other diagonal BD measure?

Try it first, then show the solutionSolution

Triangle $A B C$ with $A B = 5$ , $B C = 4.5$ , $A C = 6.5$ is valid ( $5 + 4.5 > 6.5$ ).
Triangle $A C D$ with $A D = 4$ , $C D = 5.5$ is on the other side of $A C$ ; measuring $B D$ gives about $7.0$ cm (calculated $6.98$ cm).

Answer: About 7.0 cm.

Memory hook: One brace makes the shape rigid.

Tricks to remember Constructing Quadrilaterals from Five Measurements

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

Triangle then triangle
Case 1 and 2 are two triangles sharing the diagonal.
Use it when: Planning arcs.
Angles at both ends
Cases 3 and 4 use rays: draw the angles with a protractor at the base ends.
Use it when: Planning rays.

Constructing Quadrilaterals from Five Measurements: solved examples

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

Example 1: How many measurements? · 1 mark · MCQBoard level

The minimum number of independent measurements needed to construct a unique quadrilateral is:
(a) 3 (b) 4 (c) 5 (d) 6

Try it first, then show the solutionSolution

Four sides can be squeezed; one more fixes it: five.

Answer: (c)

Example 2: Case 1: four sides and a diagonal · 3 marks · Long answerBoard level

Construct quadrilateral $A B C D$ with $A B = 5$ cm, $B C = 4.5$ cm, $C D = 5.5$ cm, $D A = 4$ cm and $A C = 6.5$ cm. Write the steps and state the length of $B D$ .

Try it first, then show the solutionSolution

Check: $5 + 4.5 > 6.5$ and $4 + 5.5 > 6.5$ , so both triangles exist.
Draw $A B = 5$ . With centre $A$ radius $6.5$ and centre $B$ radius $4.5$ , draw arcs meeting at $C$ . Join $A C$ , $B C$ .
With centre $A$ radius $4$ and centre $C$ radius $5.5$ draw arcs meeting at $D$ on the side of $A C$ opposite to $B$ . Join $A D$ , $C D$ .
Measure $B D$ : about $7.0$ cm (calculated $6.98$ cm).

Answer: Quadrilateral drawn; $B D \approx 7.0$ cm

Example 3: Case 2: three sides and two diagonals · 3 marks · Long answerHarder board level

Construct $A B C D$ with $A B = 4.5$ cm, $B C = 5$ cm, $C D = 4$ cm, $A C = 6$ cm and $B D = 5.5$ cm. State the length of $A D$ .

Try it first, then show the solutionSolution

Check: $4.5 + 5 > 6$ for triangle $A B C$ .
Draw $A B = 4.5$ ; arcs from $A$ (radius $6$ ) and $B$ (radius $5$ ) meet at $C$ .
$D$ : arc of radius $5.5$ from $B$ and radius $4$ from $C$ , on the side of $B C$ away from $A$ . Join $A D$ , $C D$ .
Measure $A D$ : about $8.8$ cm (calculated $8.78$ cm).

Answer: Quadrilateral drawn; $A D \approx 8.8$ cm

Example 4: Case 3: two included angles · 4 marks · Long answerHarder board level

Construct $A B C D$ with $A B = 5$ cm, $B C = 4.5$ cm, $C D = 4.8$ cm, $∠ B = 7 5^{\circ}$ and $∠ C = 11 0^{\circ}$ . State $D A$ and $∠ A$ .

Try it first, then show the solutionSolution

Draw $B C = 4.5$ . At $B$ make $7 5^{\circ}$ and mark $A$ with $B A = 5$ .
At $C$ make $11 0^{\circ}$ (on the same side as $A$ ) and mark $D$ with $C D = 4.8$ . Join $A D$ .
Measure $D A$ : about $4.9$ cm. Also $∠ A \approx 10 1^{\circ}$ , and $∠ D = 360 - (75 + 110 + 101) \approx 7 4^{\circ}$ .

Answer: DA about 4.9 cm; angle A about 101 degrees

Example 5: Case 4: two sides and three angles · 4 marks · Long answerHOTS / competency

Construct $A B C D$ with $A B = 4.5$ cm, $B C = 5$ cm, $∠ A = 7 5^{\circ}$ , $∠ B = 10 5^{\circ}$ and $∠ C = 10 0^{\circ}$ . Find $∠ D$ , name the type of quadrilateral and measure $A D$ and $C D$ .

Try it first, then show the solutionSolution

$∠ D = 360 - (75 + 105 + 100) = 8 0^{\circ}$ .
Draw $A B = 4.5$ ; at $B$ make $10 5^{\circ}$ and mark $C$ with $B C = 5$ . At $A$ draw a ray at $7 5^{\circ}$ and at $C$ a ray at $10 0^{\circ}$ to $C B$ ; they meet at $D$ .
$∠ A + ∠ B = 18 0^{\circ}$ , so $A D ∥ B C$ : the figure is a trapezium.
Measure: $A D \approx 6.9$ cm, $C D \approx 4.4$ cm.

Answer: Angle D = 80 degrees; trapezium; AD about 6.9 cm, CD about 4.4 cm

Where students lose marks in Constructing Quadrilaterals from Five Measurements

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

Mistake: Not checking the triangle inequality first
If the sides cannot form a triangle the arcs never meet.
Wrong working
Trying $A B = 5$ , $B C = 4$ , $A C = 10$ .
Right working
$5 + 4 < 10$ : stop and report that the construction is impossible.
Do this: Write the inequality in your steps.
Usually costs: Whole construction.
Mistake: Placing D on the same side of the diagonal as B
This folds the quadrilateral over itself.
Wrong working
Arcs for $D$ drawn above $A C$ with $B$ also above.
Right working
$D$ is on the opposite side of $A C$ from $B$ .
Do this: Mark which side each vertex goes on.
Usually costs: Figure and accuracy marks.

Exam tips

Keep construction arcs visible; do not erase them.
Label every vertex and write the given measurements next to the figure.
Use a sharp pencil and measure the spare quantity to check accuracy.

Practice questions on Constructing Quadrilaterals from Five Measurements

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
A unique quadrilateral cannot be constructed from:
(a) four sides and a diagonal (b) three sides and two diagonals (c) four sides only (d) two adjacent sides and three angles
Hint
Which option has fewer than five independent measurements?
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Can a quadrilateral $A B C D$ be constructed with $A B = 5$ cm, $B C = 4$ cm and diagonal $A C = 10$ cm? Give a reason.
Hint
Triangle inequality for ABC.
Answer
No. In triangle $A B C$ , $A B + B C = 9 < 10 = A C$ , so the triangle cannot be drawn.
Q3 · 3 marks · Long answerHarder board level
Construct parallelogram $A B C D$ with $A B = 5$ cm, $B C = 4$ cm and diagonal $A C = 6$ cm. Write the steps and find $B D$ by the relation $A C^{2} + B D^{2} = 2 (A B^{2} + B C^{2})$ .
Hint
D is found with arcs radius BC from A and AB from C.
Answer
Draw triangle $A B C$ ; arc radius $4$ from $A$ and radius $5$ from $C$ meet at $D$ . $B D^{2} = 2 (25 + 16) - 36 = 46$ , so $B D \approx 6.8$ cm (measure to check).
Q4 · 3 marks · Long answerHarder board level
Construct $A B C D$ with $A B = 6$ cm, $B C = 5$ cm, $C D = 4$ cm, $D A = 4.5$ cm and $A C = 7$ cm. Verify that it can be drawn and state $B D$ .
Hint
Two triangle inequalities.
Answer
$6 + 5 > 7$ and $4 + 4.5 > 7$ , so possible. Triangles $A B C$ and $A C D$ on opposite sides of $A C$ give $B D \approx 6.6$ cm.
Q5 · 4 marks · Long answerHOTS / competency
Construct rhombus $A B C D$ with side 5 cm and diagonal $A C = 6$ cm (four sides and a diagonal). Calculate the other diagonal and verify by measurement.
Hint
Rhombus diagonals bisect at right angles.
Answer
Triangle $A B C$ with $A B = B C = 5$ , $A C = 6$ ; $D$ from arcs of radius $5$ at $A$ and $C$ . Half of $B D = 25 - 9 = 4$ , so $B D = 8$ cm. Measuring gives $8$ cm.

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.How many measurements fix a quadrilateral?Show answer
Five.
2.List the four cases in the curriculum.Show answer
Four sides and a diagonal; three sides and two diagonals; three sides and two included angles; two adjacent sides and three angles.
3.Triangle inequality?Show answer
The sum of any two sides is greater than the third.
4.Fourth angle in case 4?Show answer
360 degrees minus the sum of the other three.

Constructing Quadrilaterals: Four Cases with Steps and Checks (ICSE Class 8 Maths)

Understand Constructing Quadrilaterals from Five Measurements

Constructing Quadrilaterals from Five Measurements: formulas to remember

How to solve Constructing Quadrilaterals from Five Measurements questions

Constructing Quadrilaterals from Five Measurements in real life

Tricks to remember Constructing Quadrilaterals from Five Measurements

Constructing Quadrilaterals from Five Measurements: solved examples

Where students lose marks in Constructing Quadrilaterals from Five Measurements

Exam tips

Practice questions on Constructing Quadrilaterals from Five Measurements

Test your recall

Constructing Quadrilaterals from Five Measurements: frequently asked questions

Why are four sides not enough?

Do I need to show a rough sketch?