Exercise 3.2 — SSSSD Construction

Constructing a quadrilateral when four sides and a diagonal are given.

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Class 8 · Chapter 3 · Exercise 3.2

Construction of Quadrilaterals — When Four Sides and a Diagonal Are Given (S.S.S.S.D.)

Master the step-by-step compass-and-ruler method to draw any quadrilateral accurately, including general quadrilaterals, parallelograms, and rhombuses. Aligned with CBSE, Telangana, and Andhra Pradesh syllabuses.

Class 8 Maths Chapter 3 CBSE Telangana Board AP Board Geometry S.S.S.S.D. Method

What Is the S.S.S.S.D. Method?

A quadrilateral is a closed figure with four sides, four angles, and two diagonals. To draw a unique quadrilateral, you need exactly five independent measurements. Exercise 3.2 focuses on the case where you are given the lengths of all four sides and one diagonal — this is known as the S.S.S.S.D. method (Side-Side-Side-Side-Diagonal).

Key Idea: A diagonal divides a quadrilateral into two triangles. Since a triangle is uniquely determined by three sides (SSS congruence), knowing four sides and one diagonal gives you both triangles — and therefore the complete quadrilateral.

Quadrilateral ABCD with diagonal AC
  → Triangle ABC (sides AB, BC, AC) is fixed by SSS
  → Triangle ACD (sides AC, CD, AD) is fixed by SSS
  → Joining both triangles gives the complete quadrilateral

Given Data

Four sides + one diagonal

Method

S.S.S.S.D.

Two SSS triangles combined

Tools Needed

Ruler + Compass

No protractor required

General Procedure for S.S.S.S.D. Construction

Regardless of which specific quadrilateral you are constructing, the same 7-step framework always applies. Study this general approach before attempting individual problems.

Step	Action	What It Does
1	Draw one side as the base	Establishes the starting edge
2	Draw arc from one endpoint using the diagonal length	Locates the third vertex
3	Draw arc from other endpoint using an adjacent side	Intersects previous arc to fix the diagonal endpoint
4	Join vertices to complete Triangle 1	First triangle (SSS) is complete
5	Draw arc from a vertex using remaining side	Begins locating the fourth vertex
6	Draw arc from diagonal endpoint using last side	Intersects to fix the fourth vertex
7	Join all vertices to complete the quadrilateral	Full quadrilateral is formed

Problem (a)

Quadrilateral ABCD

Given: AB = 4.5 cm | BC = 5.5 cm | CD = 4 cm | AD = 6 cm | Diagonal AC = 7 cm

The diagonal AC splits quadrilateral ABCD into △ABC (sides AB, BC, AC) and △ACD (sides AC, CD, AD). Construct each triangle using the SSS method.

Triangle	Sides Used	Vertices Fixed
△ABC	AB = 4.5 cm, BC = 5.5 cm, AC = 7 cm	A, B, C
△ACD	AC = 7 cm, CD = 4 cm, AD = 6 cm	A, C, D

Figure: Rough sketch of Quadrilateral ABCD with diagonal AC shown as a dashed line

Steps of Construction

Draw a line segment AB = 4.5 cm. This is your base side.
With B as centre, draw an arc of radius 5.5 cm (= BC).
With A as centre, draw an arc of radius 7 cm (= AC diagonal). The two arcs intersect at point C.
Join AC and BC to complete △ABC.
With A as centre, draw an arc of radius 6 cm (= AD).
With C as centre, draw an arc of radius 4 cm (= CD). These arcs intersect at point D.
Join AD and CD. The required quadrilateral ABCD is complete.

Problem (b)

Quadrilateral PQRS

Given: PQ = 3.5 cm | QR = 4 cm | RS = 5 cm | PS = 4.5 cm | Diagonal QS = 6.5 cm

Here the diagonal QS splits the quadrilateral into △PQS (sides PQ, QS, PS) and △QRS (sides QR, RS, QS). Notice that in this problem the base triangle uses PQ and the diagonal from Q — so the construction starts along the bottom of the figure.

Triangle	Sides Used	Vertices Fixed
△PQS	PQ = 3.5 cm, QS = 6.5 cm, PS = 4.5 cm	P, Q, S
△QRS	QR = 4 cm, RS = 5 cm, QS = 6.5 cm	Q, R, S

Figure: Rough sketch of Quadrilateral PQRS with diagonal QS

Steps of Construction

Draw a line segment PQ = 3.5 cm as the base.
With Q as centre, draw an arc of radius 6.5 cm (= QS diagonal).
With P as centre, draw an arc of radius 4.5 cm (= PS). The arcs intersect at S.
Join PS and QS to complete △PQS.
With Q as centre, draw an arc of radius 4 cm (= QR).
With S as centre, draw an arc of radius 5 cm (= RS). These arcs intersect at R.
Join QR and SR to complete the required quadrilateral PQRS.

Problem (c)

Parallelogram ABCD

Given: AB = 6 cm | AD = 4.5 cm | Diagonal BD = 7.5 cm

Important property used: In a parallelogram, opposite sides are equal. So AB = CD = 6 cm and AD = BC = 4.5 cm. This lets us derive the fourth measurement we need.

Parallelogram ABCD:
  AB = CD = 6 cm   (opposite sides equal)
  AD = BC = 4.5 cm (opposite sides equal)
  Diagonal BD = 7.5 cm (given)
  → Five measurements now available → S.S.S.S.D. applies

The diagonal BD splits the parallelogram into △ABD (sides AB = 6 cm, AD = 4.5 cm, BD = 7.5 cm) and △BCD (sides BC = 4.5 cm, CD = 6 cm, BD = 7.5 cm).

Figure: Parallelogram ABCD — opposite sides marked equal with tick marks

Steps of Construction

Draw a line segment AB = 6 cm as the base.
With B as centre, draw an arc of radius 7.5 cm (= BD diagonal).
With A as centre, draw an arc of radius 4.5 cm (= AD). These intersect at D.
Join AD and BD to complete △ABD.
With B as centre, draw an arc of radius 4.5 cm (= BC).
With D as centre, draw an arc of radius 6 cm (= DC). These intersect at C.
Join BC and DC to complete the required parallelogram ABCD.

Problem (d)

Rhombus NICE

Given: NI = 4 cm | Diagonal IE = 5.6 cm

Important property used: In a rhombus, all four sides are equal. Since NI = 4 cm, we immediately know NE = IC = CE = 4 cm as well. Only the diagonal IE = 5.6 cm is the extra piece of information needed.

Rhombus NICE:
  NI = IC = CE = NE = 4 cm  (all sides equal)
  Diagonal IE = 5.6 cm      (given)
  → Five measurements available → S.S.S.S.D. applies

The diagonal IE divides rhombus NICE into △NIE (sides NI = 4 cm, IE = 5.6 cm, NE = 4 cm) and △ICE (sides IC = 4 cm, CE = 4 cm, IE = 5.6 cm). Because all sides of the rhombus are equal, both triangles are actually isosceles.

Figure: Rhombus NICE — all sides equal (shown with tick marks), diagonal IE in red

Steps of Construction

Draw a line segment NI = 4 cm.
With I as centre, draw an arc of radius 5.6 cm (= IE diagonal).
With N as centre, draw an arc of radius 4 cm (= NE). These arcs intersect at E.
Join NE and IE to complete △NIE.
With I as centre, draw an arc of radius 4 cm (= IC).
With E as centre, draw an arc of radius 4 cm (= EC). These arcs intersect at C.
Join IC and EC to complete the required rhombus NICE.

Special Shapes — Properties Used in This Exercise

Two problems in Exercise 3.2 involve special quadrilaterals. Understanding their properties is essential because they reduce the number of measurements you need to measure yourself.

Shape	Property	Implication in Construction
Parallelogram	Opposite sides are equal (AB = CD, AD = BC)	Knowing 2 sides + 1 diagonal is enough for S.S.S.S.D.
Rhombus	All four sides are equal	Knowing 1 side + 1 diagonal is enough for S.S.S.S.D.
General Quadrilateral	No special equal-side relationship	All four sides must be given explicitly

Common Mistakes to Avoid

Wrong diagonal selected — Always identify which diagonal is given and which two triangles it creates. Draw the rough sketch first to be sure.
Forgetting opposite-side equality — In a parallelogram, students often treat it as a general quadrilateral and wait for measurements that aren't given. Derive the missing sides from the property before you start.
Arcs not intersecting — This happens when measurement values are wrong or arcs are too small. Double-check compass settings before drawing each arc.
Incorrect base side — The base (first line segment) must be one of the given sides, not the diagonal. The diagonal is used via arcs, not drawn directly as the starting line.
Skipping the rough sketch — Always draw a rough labelled sketch before the actual construction. CBSE, Telangana, and Andhra Pradesh examiners award marks for rough sketches.
Not labelling vertices — In board exams, all four vertices (A, B, C, D or P, Q, R, S) must be clearly labelled on the final figure.

Board Exam Tip: In CBSE, Telangana, and AP board construction questions, you must show all arcs clearly and leave them visible in your final answer. Do not erase construction arcs — they carry marks.

What This Exercise Prepares You For

Mastering S.S.S.S.D. construction builds the spatial thinking and compass-ruler precision needed for the rest of Chapter 3 and beyond. Once you are comfortable with this exercise, you are ready to move to more advanced constructions.

Exercise 3.1 — Construction when three sides and two diagonals are given (S.S.S.D.D.). A useful prerequisite if you found Exercise 3.2 challenging.
Exercise 3.3 — Construction when two diagonals are known and both split the figure differently — the next step after mastering 3.2.
Algebraic Expressions (Chapter 8) — Understanding variables and expressions supports the formula-based reasoning behind geometric proofs.
Class 9 Triangles — The SSS congruence rule you used in every construction here is formally proved in the Class 9 chapter on triangles.

Syllabus alignment: This exercise is part of Chapter 3 — Construction of Quadrilaterals as prescribed for Class 8 by the CBSE, Telangana State Board (SCERT), and Andhra Pradesh State Board (APSCERT) Mathematics curricula.