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Playing with Constructions

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces students to the world of geometric constructions using a ruler and a compass. It begins by exploring how to draw circles and curves, then moves to defining and constructing squares and rectangles based on their properties. The one big idea is that precise geometric shapes can be created using simple tools by understanding and applying their defining properties, such as equal sides, right angles, and equal distances.
Key Concepts (12)
Compass
A geometric tool used to draw circles and arcs, and to measure and transfer distances.
e.g. Opening a compass to 4 cm against a ruler to draw a circle of radius 4 cm.
Circle
A curve where every point on it is at the same distance from a fixed point called the centre.
e.g. All points that are 4 cm away from point P form a circle with centre P.
Centre of a circle
The fixed point from which all points on the circle are at an equal distance.
e.g. Point P in the figure is the centre of the circle.
Radius of a circle
The distance between the centre of a circle and any point on the circle.
e.g. If the distance from centre P to a point on the circle is 4 cm, then the radius is 4 cm.
Rectangle
A four-sided figure with opposite sides equal in length and all angles equal to 90°.
e.g. A book cover, a classroom door, or a 4 cm by 6 cm shape drawn on paper.
Square
A four-sided figure with all sides equal in length and all angles equal to 90°.
e.g. A chessboard square, a floor tile, or a 6 cm by 6 cm shape.
Opposite sides
In a rectangle or square, the sides that lie opposite to each other, such as AB and CD.
e.g. In rectangle ABCD, AB and CD are opposite sides, and AD and BC are the other pair.
Diagonals of a rectangle
The lines joining opposite corners of a rectangle, such as PR and QS in rectangle PQRS.
e.g. In a rectangle, the two diagonals are equal in length.
Perpendicular
A line that meets another line at a right angle (90°).
e.g. Drawing a line through point P that is perpendicular to line PQ to construct a square.
Arc
A part of the circumference of a circle.
e.g. Drawing only a part of a circle (an arc) from point B to locate point A in the house construction.
Rough diagram
A freehand sketch used to plan the construction of a figure before using tools.
e.g. Drawing a rough rectangle with labels A, B, C, D to decide which side to draw first.
Valid name for a rectangle/square
A name where the corners occur in an order of travel around the figure, starting from any corner.
e.g. For square PQRS, names like SPQR, RSPQ, QRSP are valid, but PQSR is not valid.
Formulas & Equations (5)
Distance from centre to any point on circle = Radius
Reads as: The distance from the centre to any point on the circle is equal to the radius.
Variables: Radius (r) — measured in cm or mm.
Use when: When drawing a circle with a compass or checking if a point lies on a circle.
⚠ Students often think the radius is the diameter or the full width of the circle.
Opposite sides of a rectangle are equal (R1)
Reads as: In a rectangle, the lengths of opposite sides are equal.
Variables: AB = CD and AD = BC, where AB, BC, CD, DA are sides.
Use when: Verifying if a four-sided figure is a rectangle or constructing a rectangle given two side lengths.
⚠ Students forget that all four sides are not equal in a rectangle, only opposite sides are equal.
All angles of a rectangle are 90° (R2)
Reads as: Every angle inside a rectangle measures 90 degrees.
Variables: ∠A = ∠B = ∠C = ∠D = 90°.
Use when: Checking if a figure is a rectangle or constructing perpendicular sides.
⚠ Students assume any four-sided figure with right angles is a square, forgetting side length conditions.
All sides of a square are equal (S1)
Reads as: In a square, every side has the same length.
Variables: PQ = QR = RS = SP, where PQ, QR, RS, SP are sides.
Use when: Identifying a square or constructing one with a given side length.
⚠ Students think a rotated square is no longer a square because it looks different.
All angles of a square are 90° (S2)
Reads as: Every angle inside a square measures 90 degrees.
Variables: ∠P = ∠Q = ∠R = ∠S = 90°.
Use when: Verifying a square after construction.
⚠ Students confuse the angles of a rhombus (which are not necessarily 90°) with a square.
How It Works
Drawing a circle using a compass
  1. Step 1: Open the compass against a ruler to the desired radius (e.g., 4 cm).
  2. Step 2: Place the pointed tip of the compass at the fixed point P (the centre).
  3. Step 3: Keep the tip fixed and rotate the pencil end around to draw the curve.
  4. Step 4: The resulting curve is a circle with centre P and the given radius.
💡 The compass tip must remain fixed at the centre; only the pencil moves.
Constructing a square with a given side length (e.g., 6 cm)
  1. Step 1: Draw a line segment PQ of length 6 cm using a ruler.
  2. Step 2: At point P, draw a line perpendicular to PQ.
  3. Step 3: On this perpendicular, mark point S such that PS = 6 cm (using ruler or compass).
  4. Step 4: At point Q, draw a line perpendicular to PQ.
  5. Step 5: On this perpendicular, mark point R such that QR = 6 cm.
  6. Step 6: Join points R and S to complete the square PQRS.
💡 All four sides must be exactly equal, and all angles must be 90°. Use a compass to transfer lengths accurately.
Constructing a rectangle given one side and a diagonal (e.g., side 5 cm, diagonal 7 cm)
  1. Step 1: Draw the base CD of length 5 cm.
  2. Step 2: Draw a line l perpendicular to DC at point C.
  3. Step 3: With D as centre, draw a circle (or arc) of radius 7 cm (the diagonal length).
  4. Step 4: Mark the intersection point of the circle/arc with line l as point B.
  5. Step 5: Draw a line perpendicular to DC at D, and another perpendicular to BC at B.
  6. Step 6: The intersection of these two perpendiculars gives point A.
  7. Step 7: Join A to D and A to B to complete rectangle ABCD.
💡 The circle method avoids trial and error — all points 7 cm from D lie on the circle.
Finding a point equidistant from two given points (e.g., point A from B and C, both 5 cm away)
  1. Step 1: With B as centre, draw a circle (or arc) of radius 5 cm.
  2. Step 2: With C as centre, draw another circle (or arc) of radius 5 cm.
  3. Step 3: The point(s) where the two circles/arcs intersect is the required point A.
  4. Step 4: Verify that AB = 5 cm and AC = 5 cm.
💡 Drawing full circles is not necessary; only arcs near the expected intersection point are sufficient.
Constructing a rectangle that can be divided into two identical squares
  1. Step 1: Draw a rough diagram of the rectangle ACDF divided into two squares ABEF and BCDE.
  2. Step 2: Identify that AB = BC and all shorter lines (AF, AB, BE, FE, BC, CD, ED) are equal.
  3. Step 3: Assign a length to AF (e.g., 4 cm) using a ruler or just by drawing.
  4. Step 4: Use a compass to transfer the length of AF to mark point B (so AB = AF).
  5. Step 5: Similarly, mark point C such that BC = AB.
  6. Step 6: Complete the rectangle by drawing perpendiculars and joining points.
💡 The length of the rectangle (AC) must be twice the side of one square (AF).
Exam Traps — Don't Make These Mistakes
⚠ Naming a square or rectangle incorrectly
✗ Students write names like PQSR or ACBD for a square or rectangle.
→ The corners must be listed in order of travel around the figure. PQSR jumps from P to Q to S, which is not a continuous path around the shape.
✓ For square PQRS, valid names are PQRS, QRSP, RSPQ, SPQR (or their reverses). Always go around the shape.
⚠ Thinking a rotated square is not a square
✗ A rotated square is a rhombus or diamond, not a square.
→ Rotating a shape does not change its side lengths or angles. The rotated figure still has all sides equal and all angles 90°, so it remains a square.
✓ A rotated square is still a square because its properties (equal sides and right angles) are unchanged by rotation.
⚠ Confusing radius with diameter
✗ When asked for the radius to draw a half-circle on an 8 cm line, students write 8 cm.
→ The half-circle's diameter is the full 8 cm line AB. The radius is half of that, which is 4 cm.
✓ Radius = half of the length of the central line. For AB = 8 cm, radius = 4 cm.
⚠ Assuming all four-sided figures with equal sides are squares
✗ A rhombus is a square because all sides are equal.
→ A square requires both conditions: all sides equal AND all angles 90°. A rhombus has equal sides but angles are not necessarily 90°.
✓ A four-sided figure with all sides equal is a rhombus. It is a square only if all angles are also 90°.
⚠ Forgetting to use a compass for accurate length transfer
✗ Students try to measure and mark points using only a ruler, leading to inaccuracies.
→ Using a ruler alone for marking equal lengths (like transferring AF to AB) can introduce measurement errors. A compass ensures exact transfer.
✓ Use a compass to 'pick up' the length of AF and then mark point B by drawing an arc from A with that same compass opening.
Compare & Contrast
Properties of sides
Square
Rectangle
A square has all four sides equal, while a rectangle has only opposite sides equal.
Completeness of the curve
Circle
Arc
A circle is the complete set of all points at a fixed distance from a centre, while an arc is only a part of that circle.
Exam Focus
Marks pattern: 2-3 MCQs (1 mark each) on properties and naming, 1 Short Answer (2-3 marks) on construction steps, 1 Long Answer (4-5 marks) on constructing a square/rectangle or a composite figure like 'House'.
Most asked: Properties of squares and rectangles (equal sides, right angles, opposite sides equal) · Construction of squares and rectangles given side lengths or a side and a diagonal · Identifying valid names for quadrilaterals · Concept of radius and centre of a circle · Using a compass to draw circles and transfer lengths
Don't skip: Construct a rectangle given one side and a diagonal (step-by-step construction with justification); Construct a square with a given side and verify its properties; Explain why a rotated square is still a square
One-Line Revision (read aloud before the exam)
  1. A circle is the set of all points at a fixed distance (radius) from a centre.
  2. The compass is used to draw circles and arcs, and to transfer lengths accurately.
  3. A rectangle has opposite sides equal and all angles 90°.
  4. A square has all sides equal and all angles 90°.
  5. Rotating a square or rectangle does not change its properties; it remains the same shape.
  6. A valid name for a quadrilateral lists corners in order around the shape.
  7. To construct a square, draw one side, then perpendiculars at both ends of equal length.
  8. To construct a rectangle with a side and diagonal, use a circle to locate the opposite corner.
  9. A point equidistant from two given points is found at the intersection of circles centred at those points.
  10. A rough diagram helps plan the sequence of construction steps.
  11. The two diagonals of a rectangle are equal in length.
  12. A four-sided figure with all sides equal but angles not 90° is a rhombus, not a square.
  13. When drawing a half-circle wave, the radius is half the length of the central line.
  14. Use a compass to transfer lengths instead of a ruler for greater accuracy.

MCQs — 10 Questions with Answers

Q1. What is the distance from the centre of a circle to any point on the circle called?

  • A. Radius ✓
  • B. Diameter
  • C. Circumference
  • D. Chord

Answer: A — Radius is defined as the distance between the centre and any point on the circle.

Q2. Which property is true for both squares and rectangles?

  • A. All sides are equal
  • B. All angles are 90° ✓
  • C. Opposite sides are equal and parallel
  • D. Only one pair of sides is equal

Answer: B — Both squares and rectangles have all four angles equal to 90°; however, only squares have all sides equal.

Q3. If you rotate a square 45°, what happens to it?

  • A. It becomes a rectangle
  • B. It becomes a diamond shape but stays a square ✓
  • C. It is no longer a square
  • D. Its sides become unequal

Answer: B — Rotation does not change the lengths of sides or angles, so a rotated square is still a square, even if it looks diamond-shaped.

Q4. To draw a circle using a compass, what must you keep fixed?

  • A. The pencil tip
  • B. The sharp point (centre) ✓
  • C. Both the sharp point and pencil equally
  • D. Only the ruler

Answer: B — You must keep the sharp point fixed at the centre P while rotating the pencil to draw the circle.

Q5. In rectangle ABCD, which sides are called opposite sides?

  • A. AB and BC
  • B. AB and CD, and also AD and BC ✓
  • C. All four sides are opposite
  • D. Only AB and AD are opposite

Answer: B — Opposite sides are sides that do not share a corner; AB is opposite to CD, and AD is opposite to BC.

Q6. Which of the following is NOT a valid name for square PQRS?

  • A. QRSP
  • B. RSPQ
  • C. PQSR ✓
  • D. SPQR

Answer: C — PQSR jumps from P to Q to S, skipping R in order; valid names must follow the corners in travelling order like PQRS, QRSP, RSPQ, or SPQR.

Q7. What are all the angles of a square equal to?

  • A. 60°
  • B. 45°
  • C. 90° ✓
  • D. 180°

Answer: C — A square has all four angles equal to 90° (right angles), which is one of its defining properties.

Q8. You need to construct a rectangle with sides 3 cm and 5 cm. After drawing it, what should you verify?

  • A. Only that all sides are equal
  • B. Only that all angles are 90°
  • C. That opposite sides are equal and all angles are 90° ✓
  • D. That all four sides are exactly equal

Answer: C — To verify a rectangle, you must check both rectangle properties: opposite sides equal (3 cm opposite 3 cm, 5 cm opposite 5 cm) and all angles 90°.

Q9. In the diagram, point P is the centre and the distance PQ is 4 cm. If R is another point on the circle, what is the distance PR?

  • A. 2 cm
  • B. 4 cm ✓
  • C. 8 cm
  • D. Cannot be determined

Answer: B — All points on a circle are at the same distance (radius) from the centre; since PQ = 4 cm and Q is on the circle, PR must also equal 4 cm.

Q10. A square has one side measuring 7 cm. What is the length of the opposite side?

  • A. 3.5 cm
  • B. 7 cm ✓
  • C. 14 cm
  • D. Cannot be determined

Answer: B — In a square, all four sides are equal; if one side is 7 cm, the opposite side (and all other sides) must also be 7 cm.

Flashcards

What is the centre of a circle?

The fixed point P from which all points on the circle are equally distant.

What is the radius of a circle?

The distance between the centre and any point on the circle.

Name two properties of a rectangle.

Opposite sides are equal in length, and all angles are 90 degrees.

Name two properties of a square.

All four sides are equal in length, and all angles are 90 degrees.

How do you use a compass to draw a circle?

Open the compass to the radius length, fix the sharp point at the centre, and rotate the pencil around while keeping the point fixed.

Is a rotated square still a square?

Yes, because rotation does not change the lengths of sides or the angles, so it still satisfies both square properties.

How should you name a rectangle ABCD in other valid ways?

You can use BCDA, CDAB, or DABC — corners must be listed in order as you travel around the rectangle.

What angle do all corners of a rectangle have?

All corners of a rectangle have a right angle of 90 degrees.

What is the difference between a square and a rectangle?

A square has all four sides equal, while a rectangle has only opposite sides equal; both have all angles of 90 degrees.

What tools do you need to construct a perfect square or rectangle?

A ruler to draw straight lines, a compass to create right angles and circles, and a set square to check angles.

Important Board Questions

What is a radius of a circle? [1 mark]

It is the distance from the centre to any point on the circle. All radii of the same circle are equal.

List any two properties of a square. [2 marks]

Think about the sides and the angles. A square must have all sides equal and all angles equal to 90°.

How would you use a compass to draw a circle with radius 5 cm? Explain the steps clearly with an example from your daily life where you might need to draw such a circle. [3 marks]

First, open the compass to 5 cm using a ruler. Then fix the sharp point at centre P and rotate the pencil around it. Example: drawing a circular plate design or a pizza cutter template.

Draw a rectangle ABCD with sides 4 cm and 6 cm. After drawing, verify that it satisfies both the properties of a rectangle by taking measurements. Write down what you measured and what you found. [5 marks]

Use ruler to draw sides: AB = 4 cm, BC = 6 cm, CD = 4 cm, DA = 6 cm. Check: (1) Are opposite sides equal? (2) Are all angles 90° using a set square? State your findings clearly.

True or False: A rotated square is no longer a square because it looks like a diamond. Give a reason for your answer. [2 marks]

Check if rotation changes the side lengths or angles. A square remains a square after rotation because its properties (equal sides and 90° angles) do not change.

Assertion-Reason Questions for Playing with Constructions

Q1. Assertion (A):

A circle is drawn with a compass where the distance between the tip and pencil is 5 cm. The radius of this circle is 5 cm.

Reason (R):

The distance set on the compass when drawing a circle equals the radius of that circle.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains why the assertion is true. The compass opening distance is exactly the radius.

Q2. Assertion (A):

A square rotated 45 degrees is no longer a square.

Reason (R):

Rotation changes the shape of any figure.

Answer: D — A is false but R is true — Both the assertion and reason are false. Rotation does not change side lengths or angles, so a rotated square is still a square with all sides equal and all angles 90°.

Q3. Assertion (A):

A rectangle must have all four angles equal to 90 degrees.

Reason (R):

Rectangles are special types of quadrilaterals where all four angles are right angles.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true, and the reason correctly explains the assertion. This is the defining property of rectangles.

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Case-Based Questions (CBQ) for Playing with Constructions

Case Study:

Ravi's mother is designing a new kitchen counter. She wants to create a rectangular cooking surface that is 120 cm long and 80 cm wide. The corners must be exactly at right angles so that pots and pans sit properly. Ravi offers to help by using a ruler and measuring the sides and angles to make sure the counter is constructed correctly according to the rectangle properties.

Q1. For the counter to be a perfect rectangle, what must be true about its opposite sides?

  • A. They must be equal in length ✓
  • B. They must be different in length
  • C. They must be perpendicular
  • D. They must be curved

Answer: A

Q2. Ravi measures the four corners of the counter. What angle should each corner measure to satisfy the rectangle property?

  • A. 45 degrees
  • B. 60 degrees
  • C. 90 degrees ✓
  • D. 180 degrees

Answer: C

Q3. If Ravi measures the length of the counter as 120 cm on one side, what should the opposite side measure?

  • A. 80 cm
  • B. 120 cm ✓
  • C. 60 cm
  • D. 240 cm

Answer: B

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