Playing with Constructions | CBSE Class 6 Maths Notes

This note covers compass constructions, circles and arcs, geometric artwork, properties and naming of squares and rectangles, construction from given measurements, distances across rectangles, equal squares inside rectangles, diagonals, and locating points at equal distances from two given points.

How does a compass help us construct a circle?

A construction is a drawing made using instruments to satisfy given geometric conditions. A ruler helps draw straight lines and measure lengths. A compass has a pointed tip and a pencil; it helps draw circles and parts of circles.

A curve is a shape that can be drawn on paper with a pencil. Curves include straight lines, circles and other shapes. A curve need not look like a bent line.

Property: Every point on a circle has the same distance from its centre

Definition: A circle consists of points at the same distance from a fixed point, its centre. The distance between the centre and any point on the circle is its radius.

Let P name a fixed point. The letters used in a drawing label points. The abbreviation cm means centimetre, a unit of length. Points that are all 4 cm from P form a circle with centre P and radius 4 cm.

Worked example 1. Construct the curve containing all points 4 cm from a point P.

Answer: Set the compass opening to 4 cm against a ruler. Fix its pointed tip at P and turn the pencil around P without changing the opening. The resulting circle has radius 4 cm.

How can you check the drawing?

1. Mark P clearly so that the compass tip can remain at the same point.
2. Measure the distance between the tip and pencil against the ruler.
3. Keep the pointed tip fixed while moving the pencil around it.
4. Choose points on the completed curve and check that each is 4 cm from P.

The compass opening controls the radius. The location of its fixed tip controls the centre. These are separate choices: deciding how wide to open the compass does not by itself decide where the circle will be drawn.

One drawing shows the compass against a ruler. The other shows a circle with centre P and a segment from the centre to the circle labelled radius.

See Fig. 8.2 in your NCERT textbook

How can circles and arcs produce geometric artwork?

An arc is part of a circle. It can be drawn by turning the compass through part of a turn. A half circle is one of two equal parts of a circle separated by a straight line through its centre.

A line segment is the straight part between two endpoints. If A and B name its endpoints, AB names the segment or its length, depending on the sentence. The statement AB = 8 cm means its length is 8 cm; the sign = means “is equal to”.

How is the Wavy Wave planned?

The wave uses two matching half circles on opposite sides of a central segment. Its central length can be chosen when no measurement is imposed. For the 8 cm version, point X separates the two equal parts of AB.

Worked example 2. Draw a Wavy Wave on AB of length 8 cm, using two identical half circles, the first from A to X and the second from X to B.

Answer: AX is 4 cm. Each half circle has radius 2 cm. Place the compass tip halfway along AX to draw the first half circle above AB. Use the same opening, with the tip halfway along XB, to draw the second below AB.

The midpoint of a segment is the point that divides it into two equal lengths. Locating the two midpoints gives the centres for the wave. Arcs smaller than half circles can also be tried. Getting both waves identical may be tricky.

How are A Person and Eyes approached?

The artwork called A Person combines a circular head and a short straight neck with a lower part bounded by straight lines and a curved top. Estimate a centre and choose a radius for the curved top, then try the compass in different positions.

For Eyes, upper and lower arcs should form a symmetrical figure, meaning matching parts on opposite sides of a dividing line. Supporting lines can help position the curves. Try to make the eyes as symmetrical and identical as possible; this might need many trials.

One often constructs supporting curves or figures that are not part of the final artwork. Their purpose is to help locate the points and arcs that are required.

What properties identify squares and rectangles?

The sides of a square or rectangle are the straight segments forming its boundary. The corners are the points at which consecutive sides meet. An angle describes the opening between two lines meeting at a point.

The symbol ∠ means angle. Thus ∠A means the angle at corner A. The symbol ° means degrees, the unit used to measure angles. A right angle measures 90°.

Property: A rectangle has equal opposite sides and four right angles

Consider a rectangle ABCD, with its corners encountered in that order around the boundary. Its sides are AB, BC, CD and DA. Opposite sides lie across the figure from one another: AB and CD form one pair, while AD and BC form the other.

The two rectangle conditions are equal lengths within each pair of opposite sides and a right angle at every corner. These conditions guide both recognition and construction. Checking only the lengths leaves the angles unchecked.

Property: A square has four equal sides and four right angles

A square has all four sides equal in length, and each of its four angles is 90°. Both conditions matter. A four-sided figure with equal sides does not qualify as a square if its angles fail the right-angle condition.

A square is the special case of a rectangle in which all four sides are equal. The word adjacent describes sides that share a corner. Making two adjacent sides of a rectangle equal gives this special case.

A and B label the upper corners, and D and C label the lower corners. The upper side AB and lower side CD are blue, identifying one opposite pair.

See Fig. 8.4 in your NCERT textbook

How should figures be named and checked after rotation?

A valid name follows the corners in order around the boundary. You can start at any corner and travel in either direction. You must keep travelling around the sides, rather than jumping across the figure to another corner.

Which names follow the boundary?

For a rectangle with consecutive corners A, B, C and D, ABCD, BCDA, CDAB and DABC follow one direction. The names ADCB, DCBA, CBAD and BADC follow the reverse direction. Each describes the same rectangle.

The combinations ABDC and ACBD are not valid names for that rectangle. They do not follow its boundary in order. Labelling a figure correctly also makes instructions such as “join these points” or “measure this side” easier to follow.

For a square whose corners occur as P, Q, R and S around the boundary, SPQR, RSPQ and QRSP are valid names. PQSR is not: after Q, it jumps to S rather than continuing around the boundary.

Does turning a figure change what it is?

Rotation means turning a figure. Rotating a square does not change its side lengths or angles. Its four sides remain equal and its angles remain right angles, so the rotated figure is still a square.

A rotated rectangle is still a rectangle for the same reason. Its appearance on the page may change, but both rectangle properties remain satisfied. A sloping side is therefore not a reason to reject a shape as a square or rectangle.

A dot grid is an arrangement of regularly spaced dots used to guide drawings. Draw rotated squares and rectangles with their corners on dots, then check their defining properties. Positions of corners on the grid can also help you reason about lengths and angles.

Note: Judge a proposed square by both equal sides and right angles. Its orientation on the page, or resemblance to a familiar outline, cannot replace these checks.

How are squares and rectangles constructed from their sides?

Two lines are perpendicular when they meet at a right angle. A protractor is an instrument for measuring and marking angles. Perpendiculars establish the right angles needed when constructing a square or rectangle.

Start with a rough diagram, a planning sketch that shows the intended figure and known measurements. Decide which side can be drawn first and which point can be located next. The finished construction must then satisfy the required measurements.

How is a square of side 6 cm constructed?

Worked example 3. Construct square PQRS with each side 6 cm.

Answer: Use the following sequence, taking P, Q, R and S as consecutive corners.

1. Draw the base segment PQ of length 6 cm.
2. Draw a perpendicular to PQ through P.
3. Mark S on this perpendicular so that PS is 6 cm.
4. Draw a perpendicular to PQ through Q on the same side of PQ.
5. Mark R on it so that QR is 6 cm, using the ruler or the same compass opening.
6. Join R to S. Check all four sides and all four angles.

The compass can transfer the 6 cm length to both perpendiculars without resetting its opening. The completed top side RS is 6 cm, and the angles at R and S are 90°. Check these as well as the parts explicitly marked during construction.

What changes for a rectangle?

Worked example 4. Construct rectangle ABCD with AB of length 4 cm and BC of length 6 cm.

Answer: Draw AB as 4 cm. At A and B, draw perpendiculars on the same side of AB. Mark D and C respectively 6 cm from A and B, then join CD. Check AB and CD are 4 cm, AD and BC are 6 cm, and every angle is 90°.

The same method constructs a rectangle with sides 2 cm and 10 cm. One opposite pair has length 2 cm and the other has length 10 cm. Unlike the square construction, the two chosen side lengths differ.

It is not possible to construct a four-sided figure with all four angles 90° but unequal opposite sides. The right-angle construction must be checked together with the resulting opposite side lengths.

What can moving points reveal about distances in a rectangle?

Construct rectangle ABCD with AB of length 7 cm and BC of length 4 cm. Let X be a movable point on AD, and Y a movable point on BC. The endpoints are allowed positions too.

The distance XY is the length of the straight segment joining X and Y. Move the points and measure this segment. The abbreviation mm means millimetre, another unit of length. Record the positions as distances from A and B respectively.

How can the observations be recorded?

A table saves repeating the same sentences for every pair of positions. Each row must keep three observations together: X's distance from A, Y's distance from B, and the measured length XY. Changing either position can change the distance between the points.

These positions place X and Y the same distance down their respective sides. In each listed case, XY has the same length as AB, and the four-sided figure ABYX is a rectangle. Follow A, B, Y and X in that order to trace its boundary.

Worked example 5. In rectangle ABCD, AB is 7 cm and BC is 4 cm. X lies on AD, 1 cm from A; Y lies on BC, 1 cm from B. Find XY and identify ABYX.

Answer: XY is 7 cm, the same as AB. The figure ABYX is a rectangle because X and Y occupy corresponding positions on the opposite sides.

Which positions give the greatest separation?

A diagonal joins two opposite corners of a square or rectangle. The greatest distance between X and Y equals the length of diagonal AC or diagonal BD. Compare the measured diagonal with XY when X and Y occupy opposite corners.

Explore several other positions before deciding where the points are nearest. Compare each measurement with AB and keep a record of equal-position cases. This separates a guess based on appearance from an observation supported by measurements.

How can a rectangle be divided into identical squares?

Identical squares have the same shape and side length. To construct a rectangle made from them, sketch the final arrangement first. The rough sketch shows which lengths must be equal and how many square sides fit along the longer side.

How does the two-square arrangement work?

Place A, B and C in order along the upper side, and F, E and D beneath them along the lower side. The two squares are ABEF and BCDE; the outer rectangle is ACDF. The segment BE separates the squares.

In square ABEF, AF, AB, BE and FE have equal lengths. In square BCDE, BE, BC, CD and ED have equal lengths. The shared side BE connects these equalities, so all the shorter segments have the same length.

A small equal-length mark, drawn as a short stroke across a segment, records that it has the same length as other segments bearing that mark. This is useful on the rough sketch before deciding the construction sequence.

Worked example 6. Rectangle ACDF consists of identical squares ABEF and BCDE placed side by side. If AF is 4 cm, find AC and explain how to locate B and C.

Answer: AB and BC are each 4 cm, so AC is 8 cm. Draw a perpendicular to AF at A. Open the compass to AF and transfer that length from A to mark B, then from B to mark C along the same line.

How can the idea be extended?

For three identical squares in a row, make the rectangle's longer side three times its shorter side. When no size is imposed, the first side can be drawn without assigning it a numerical length. Transfer that chosen length repeatedly with a compass.

A centred square inside a rectangle of sides 8 cm and 4 cm gives another planning task. The drawn square spans the rectangle's 4 cm width. Its side is therefore 4 cm, leaving equal gaps of 2 cm at the two ends along the longer side.

Falling Squares uses a sequence of squares meeting at corners. One arrangement uses side lengths of 4 cm throughout; another uses 7 cm, 5 cm and 3 cm. Preserve their alignment as well as their individual side lengths.

Further artwork combines a large square with smaller squares and shading, or places circular holes inside squares. For a centred circular hole, locate the meeting point of the square's diagonals as the circle's centre.

The Square with Curves activity uses a square of side 8 cm. Try compass positions that make the four arcs bulge uniformly inwards from the sides. Plan the centres and radii before attempting to match all four curves.

How do diagonals help explore rectangles and squares?

In rectangle PQRS, the segments PR and QS are its diagonals. Each joins opposite corners. Draw and measure both, then compare their lengths. Prediction followed by construction and measurement is the basis of this exploration.

Opposite angles are angles at opposite corners. The angle at P is opposite the angle at R; the angle at Q is opposite the angle at S. Each is a right angle before a diagonal divides it into smaller parts.

Does a diagonal divide a right angle equally?

A diagonal divides each of the two angles at its endpoints into two smaller angles. Equal division is a condition to investigate, rather than something to assume for every rectangle. Measure the smaller angles and compare different rectangles.

Record the side lengths and the eight smaller angles formed at the corners by the two diagonals. Include a square in the investigation. When adjacent sides of the rectangle are equal, it is a square, and its diagonals divide the opposite right angles equally.

How can a prescribed angle split guide construction?

Worked example 7. Construct rectangle ABCD so diagonal AC divides the right angle at A into 60° and 30°. No side length is specified.

Answer: Draw AB with any chosen length. At B draw a perpendicular to AB. From A draw a line making 60° with AB towards that perpendicular; name their meeting point C. At A draw a perpendicular to AB. Draw a perpendicular to BC through C to meet it at D.

At A, the angle between AB and AC is 60°. The remaining part between AC and AD is 30°, since the complete angle is 90°. Check the corresponding split at the opposite corner C in the completed rectangle.

There are at least two ways of locating D once A, B and C are fixed. Instead of drawing the perpendicular through C, transfer length BC onto the perpendicular at A to mark D, then join CD.

The same construction idea can be explored with angle pairs 50° and 40°, or 45° and 45°. In the equal-split case, observe that the resulting sides are equal. Do not assign numerical side lengths when the construction leaves the initial length free.

How is a rectangle constructed from a side and a diagonal?

A given diagonal provides a distance from one corner to the opposite corner. A perpendicular through a known endpoint provides the line on which another corner must lie. The required point is located by satisfying both conditions together.

Why is a circle useful here?

For a rectangle with side DC of length 5 cm and diagonal DB of length 7 cm, B must lie on the perpendicular to DC at C. It must also lie 7 cm from D. Name this perpendicular line l, where l is simply its label.

All points 7 cm from D lie on a circle with centre D and radius 7 cm. An intersection is a point where lines or curves meet. The required corner B is an intersection of that circle with line l, on the chosen side of DC.

Worked example 8. Construct rectangle ABCD with DC of length 5 cm and DB of length 7 cm.

Answer: Locate B using the intersection of a perpendicular and a circle, then complete the rectangle.

1. Draw DC of length 5 cm.
2. Draw line l perpendicular to DC through C.
3. Set the compass to 7 cm and draw a circle, or the required arc, with centre D.
4. Mark B where the circle or arc meets l on the chosen side of DC.
5. Draw perpendiculars to DC through D and to BC through B. Name their intersection A.
6. Check the opposite side lengths, the four right angles, and diagonal DB of length 7 cm.

Must the whole circle be drawn?

Only the arc near the intended intersection is needed. A full circle shows the distance condition clearly, but a suitable arc locates the same point with less drawing. The centre and compass opening remain unchanged.

Trying different ruler positions to find B involves trial and error. The circle method collects all points at the required distance before selecting the one that also lies on the perpendicular. This explains why the intersection satisfies both requirements.

Practise the same method with a side of 4 cm and diagonal of 8 cm, or a side of 3 cm and diagonal of 7 cm. In each case, the compass opening comes from the diagonal, while the first segment comes from the given side.

How do intersecting circles locate the roof of a house?

A point is equidistant from two given points when its distances from them are equal. The house construction uses this idea to locate a roof corner that is 5 cm from each of the two upper wall corners.

Label the lower corners D and E, the upper wall corners B and C, and the roof tip A. Draw DE, DB and EC each 5 cm long, with the walls perpendicular to DE on the same side. The doorway shown is 1 cm wide and 2 cm high.

How is the roof tip found?

Worked example 9. The house walls DB and EC are perpendicular to base DE on the same side. Each is 5 cm long, and DE is 5 cm. Locate A above B and C so that AB and AC are both 5 cm, then draw the roof arc.

Answer: With B and C as centres, draw arcs of radius 5 cm above the walls. Name their upper intersection A. Join AB and AC. With centre A and radius 5 cm, draw the lower arc from B to C.

A lies on the circle centred at B, so AB is 5 cm. It also lies on the circle centred at C, so AC is 5 cm. Its membership of both circles establishes the two distances without repeatedly trying ruler positions.

How is the curved part completed?

The final arc uses A as its centre. Because B and C are each 5 cm from A, the same compass opening reaches both points. Draw the part between them that curves down towards the body of the house.

Draw the base DE and upright walls DB and EC, all 5 cm. Locate A above the walls using intersecting arcs of radius 5 cm. Join AB and AC, then draw the arc from B to C with centre A.

Two full circles are not necessary to locate A: the intersecting portions suffice. The same idea can be used to explore a bigger house with border sides of 7 cm and to revisit the earlier artwork.

The equal-distance idea also helps explore a four-sided figure whose sides are all equal but which is not a square. Equal lengths alone do not enforce right angles. Check the angles separately before deciding whether the constructed figure is a square.

Glossary

• Construction — A drawing made with instruments to satisfy specified lengths, angles or other geometric conditions.
• Compass — An instrument with a pointed tip and pencil, used to draw circles and their parts.
• Curve — A shape drawable on paper with a pencil, including straight lines, circles and other figures.
• Circle — A curve whose points are all at the same distance from a fixed centre.
• Radius — The distance between the centre of a circle and any point on that circle.
• Arc — A part of a circle drawn without completing the whole circular curve.
• Rectangle — A four-sided figure with equal opposite sides and four angles measuring 90° each.
• Square — A four-sided figure with all sides equal and all four angles measuring 90°.
• Perpendicular lines — Two lines that meet to form a right angle, measuring 90°.
• Diagonal — A line segment joining two opposite corners of a rectangle or a square.
• Midpoint — The point on a segment that divides its length into two equal parts.
• Equidistant — Describes a point whose distances from two given points are equal.
• Intersection — A point common to lines or curves where they meet or cross.
• Rough diagram — A planning sketch used to identify measurements, relationships and a possible order of construction.

Common errors and misconceptions

• Misconception: A curve must be bent. Correct: Curves include straight lines as well as circles and other shapes drawable with a pencil.
• Misconception: Four equal sides are enough to identify a square. Correct: All four angles must also be 90°; check both requirements.
• Misconception: A square stops being a square when turned. Correct: Rotation preserves its lengths and angles, so its defining properties remain satisfied.
• Misconception: The corners of a rectangle can be named in any order. Correct: A valid name follows the boundary in either direction from a chosen starting corner.
• Misconception: A diagonal divides every rectangle's corner into equal angles. Correct: A rectangle can have a 60° and 30° split. Equal splitting occurs in the square case.
• Misconception: A full circle must be drawn whenever a compass locates a point. Correct: Suitable intersecting arcs can be enough to locate the required point.
• Misconception: The 7 cm measurement in the side-and-diagonal construction is another side. Correct: With DC of length 5 cm and DB of length 7 cm, DB joins opposite corners and is the diagonal.
• Misconception: A compass opening alone determines an arc's position. Correct: Its radius and the position of its centre must both be selected to obtain the intended arc.

Exam-style questions with model answers

Q1. State the two properties used to identify a square. [2 marks]

1. All four sides of the figure must be equal in length.
2. Each of its four angles must be a right angle, measuring 90°.

Q2. A Wavy Wave on segment AB of length 8 cm consists of two identical half circles, one on AX and one on XB, on opposite sides of AB. Find AX and the compass radius. [2 marks]

1. AX is half the central segment, so its length is 4 cm.
2. The centre of each half circle is halfway along its 4 cm straight boundary, giving a compass radius of 2 cm.

Q3. A square has consecutive corners P, Q, R and S. Explain why PQSR is not a valid name, give one valid name starting at S, and explain why rotating the square does not change its identity. [3 marks]

1. PQSR is invalid because after travelling from P to Q it jumps across to S instead of following the next side around the boundary.
2. SPQR is valid: starting at S, the name follows the corners consecutively around the square and returns to the starting point.
3. Rotation changes neither the side lengths nor the angles. The sides remain equal and all angles remain 90°, so the figure remains a square.

Q4. Rectangle ABCD has AB of length 7 cm and BC of length 4 cm. X lies on AD, 1 cm from A, and Y lies on BC, 1 cm from B. Find XY, identify ABYX, and state the greatest possible XY when X and Y may move along AD and BC, including their endpoints. [3 marks]

1. XY is 7 cm. X and Y are at equal distances from A and B on the opposite sides, so XY has the same length as AB.
2. ABYX is a rectangle. Its boundary is followed in the order A, B, Y and X, with X and Y in corresponding positions on AD and BC.
3. The greatest possible distance equals diagonal AC or BD, reached when the movable points occupy opposite corners of the original rectangle.

Q5. Describe the construction of square PQRS of side 6 cm, using a ruler, protractor and compass as needed. Include the final checks. [6 marks]

1. Draw the base segment PQ of length 6 cm with a ruler, and label its two endpoints clearly.
2. At P, mark a right angle with the protractor and draw a line perpendicular to PQ through P.
3. On this perpendicular mark S so that PS is 6 cm. A ruler or a compass set to 6 cm can locate S.
4. At Q, draw another perpendicular to PQ. Choose the side of PQ that already contains S for the next corner.
5. Mark R on the second perpendicular so that QR is 6 cm, then join R to S to close the figure.
6. Check that PQ, QR, RS and SP are each 6 cm and that every corner angle is 90°. Both square properties must hold.

Q6. Construct rectangle ABCD with side DC of length 5 cm and diagonal DB of length 7 cm. Describe five stages, including the reason for using a compass. [5 marks]

1. Draw DC of length 5 cm. This gives a known side and fixes the two lower corners for the intended rectangle.
2. Through C draw line l perpendicular to DC. Corner B must lie on this line because the angle at C is a right angle.
3. With centre D and radius 7 cm, draw an arc meeting l on the chosen side of DC. Label that intersection B; it is 7 cm from D. The compass locates B at the required distance without the trial and error of moving a ruler.
4. Draw a perpendicular to DC through D and a perpendicular to BC through B. Name their meeting point A to complete the rectangle.
5. Check equal opposite sides and all four right angles. Also check DC is 5 cm and DB is 7 cm, as required.

Q7. Construct rectangle ABCD in which diagonal AC divides the right angle at A into 60° between AB and AC and 30° between AC and AD. No side length is imposed. Describe four stages. [4 marks]

1. Choose a length for AB and draw it. Through B draw a perpendicular to AB on the side intended for the rectangle.
2. From A draw a line making 60° with AB towards that perpendicular. Name their intersection C.
3. Through A draw a perpendicular to AB. Through C draw a perpendicular to BC, meeting the line through A at D.
4. The angle at A is 90°. Since the part between AB and AC is 60°, the remaining part between AC and AD is 30°. Check the completed rectangle.

Q8. A house has base DE of length 5 cm and walls DB and EC, each 5 cm, perpendicular to DE on the same side. Locate roof tip A above the walls with AB and AC each 5 cm, join the roof sides, and draw the lower arc from B to C with centre A. Explain five stages. [5 marks]

1. Set the compass to 5 cm. With B as centre, draw an arc above the walls; every point on it is 5 cm from B.
2. Without changing the opening, draw another arc with C as centre so that it crosses the first arc above the walls.
3. Name the upper intersection A. Because it belongs to both arcs, it is 5 cm from B and also 5 cm from C.
4. Join A to B and A to C with straight segments. These give the two roof sides with the required lengths.
5. Place the compass tip at A with radius 5 cm. Draw the lower arc between B and C, curving towards the house body.

Key takeaways

• Every point on a circle has the same distance from its centre; this distance is called the radius.
• A compass can draw circles and arcs, transfer lengths, and locate points satisfying given distance conditions.
• A square requires four equal sides and four right angles; a rectangle requires equal opposite sides and four right angles.
• Valid names follow the boundary of a figure, and rotation leaves the defining lengths and angles unchanged.
• A rough diagram helps reveal equal lengths and decide the order in which points should be constructed.
• A rectangle can be constructed from its side lengths or from one side and a diagonal.
• Intersecting circles or arcs locate a point that satisfies two specified distances at the same time.
• Check the finished side lengths and angles instead of judging the success of a construction from appearance alone.

Test yourself

What curve is formed by all points 4 cm from P?

A circle with centre P and radius 4 cm.

Why must the compass opening remain unchanged while drawing one circle?

The opening fixes the radius, so every point drawn must remain the same distance from the centre.

Can a rotated rectangle still be a rectangle?

Yes. Rotation changes neither its side lengths nor its angles, so both rectangle properties remain satisfied.

Why is checking equal side lengths insufficient to identify a square?

A square must also have four right angles. An equal-sided figure can fail this angle requirement.

Two identical squares of side 4 cm lie side by side. What is the length of the outer rectangle?

Its length is 8 cm, made from two consecutive sides of 4 cm each.

What do PR and QS represent in rectangle PQRS?

They are its diagonals, joining the two pairs of opposite corners.

For rectangle ABCD with DC of length 5 cm and diagonal DB of length 7 cm, where is the compass tip placed to locate B?

Place it at D and use radius 7 cm. The arc meets the perpendicular through C at B.

Why can arcs replace full circles in the house construction?

Only the portions containing the required intersection are needed to locate the point at both specified distances.