Class 6 Mathematics · Chapter 8 NotesPlaying with Constructions
Learn Class 6 Mathematics Chapter 8 Playing with Constructions. Understand circles, squares, rectangles, diagonals and step-by-step compass constructions.
Playing with Constructions is a hands-on geometry chapter where you move beyond freehand drawing and learn to draw accurate figures using a ruler and a compass. The chapter begins with artwork such as 'A Person', 'Wavy Wave' and 'Eyes', which helps you get familiar with the compass and with circles. You then study the properties of squares and rectangles, and learn step-by-step methods to construct them when sides, or a side and a diagonal, are given. Along the way you explore diagonals of rectangles and squares, discover how to locate a point that is the same distance from two given points, and use these ideas to construct figures like a house. By the end, you will be able to plan a construction with a rough diagram, choose the right instrument, and check whether your figure satisfies the required properties.
What you'll learn
1Identify the basic geometric instruments and describe how a compass is used
2Define centre, radius and circle, and draw a circle of a given radius
3State and apply the properties of squares and rectangles
4Construct squares and rectangles using a ruler and a compass
5Construct a rectangle when one side and one diagonal are given
6Locate a point that is equidistant from two given points using arcs
7Plan a construction by first drawing a rough diagram
8Recreate simple artwork and figures using circles, arcs and straight lines
Chapter at a glance
01Chapter Overview
02Basic Geometric Instruments and Their Uses
03Construction of Line Segments and Angles
04Construction of Triangles
05Practical Applications and Problem Solving
Detailed chapter notes
01
Artwork and Getting to Know the Compass
The chapter opens with figures such as 'A Person', 'Wavy Wave' and 'Eyes'. You first try to draw them freehand, then repeat the work with a ruler and a compass. A compass has a pointed tip and a pencil holder; by fixing the tip at one point and moving only the pencil, you can draw a smooth curve. If you mark all the points that are 4 cm away from a fixed point P, the curve you get is a circle. The fixed point P is the centre, and the distance from the centre to any point on the circle is the radius. To draw a circle of radius 4 cm, open the compass against a ruler so that the gap between the tip and the pencil is 4 cm.
Centrethe fixed point from which all points of a circle are at the same distance
Radiusthe distance between the centre and any point on the circle
A compass is used to draw circles and parts of circles (arcs)
02
Properties of Squares and Rectangles
A rectangle is a four-sided figure whose corners are its vertices and whose sides form four angles. In a rectangle, the opposite sides are equal in length and all four angles are 90°. A square is a special rectangle in which all four sides are equal and all four angles are 90°. A figure can be named by going around its corners in order, so rectangle ABCD may also be called BCDA or CDAB, but not ABDC. Rotating a square or a rectangle does not change its side lengths or its angles, so a rotated square is still a square and a rotated rectangle is still a rectangle.
Rectangleopposite sides equal, all angles 90°
Squareall sides equal, all angles 90°
Names of a figure must follow the corners in order around the figure
03
Constructing Squares and Rectangles
To construct a square PQRS of side 6 cm, first draw PQ = 6 cm. At P, draw a perpendicular to PQ and mark S on it so that PS = 6 cm. At Q, draw a perpendicular to PQ, and then mark R so that QR = 6 cm. Join R to S. The side RS comes out to 6 cm and angles R and S are 90°, so the figure satisfies the properties of a square. The same steps work for a rectangle if the two given lengths are different. A perpendicular can be drawn with a ruler and set square, or with a compass, and a compass can also be used to transfer a length from one line to another.
Step 1draw the base of the given length
Step 2draw perpendiculars at both ends of the base
Step 3mark the given lengths on the perpendiculars and join the last two points
04
Rectangles, Diagonals and Angles
A diagonal of a rectangle joins two opposite corners. In rectangle PQRS, PR and QS are the diagonals, and they are equal in length. A diagonal splits each of the two opposite angles it touches into two smaller angles. For example, diagonal PR divides angle P into angles c and d, and angle R into angles g and h. By measuring, you can compare these smaller angles and look for equal pairs. If a rectangle is constructed so that a diagonal divides the opposite angles into equal parts, then the adjacent sides become equal, which means the figure is a square. This is why a square is the special case where the diagonal divides each opposite angle into two equal angles of 45°.
Diagonals of a rectangle are equal in length
A diagonal divides the opposite angles into two smaller angles
In a square, each diagonal divides the opposite angles into 45° and 45°
05
Constructing a Rectangle from a Side and a Diagonal
Suppose you must construct a rectangle whose side is 5 cm and whose diagonal is 7 cm. Start by drawing the side DC = 5 cm. At C, draw a line l perpendicular to DC. The opposite corner B must lie on l and must be 7 cm away from D. To find all points that are 7 cm from D, draw a circle of radius 7 cm with centre D. The point where this circle meets the line l is B. Now draw a perpendicular to DC at D and a perpendicular to BC at B; the point where these two lines meet is A. Joining the points gives rectangle ABCD. Instead of a full circle, you may draw only the arc near the line l.
Draw the given side first
Draw a perpendicular at one end of the side
Use an arc of radius equal to the diagonal to locate the opposite corner
Complete the rectangle using perpendiculars
06
Points Equidistant from Two Given Points
The 'House' construction shows how to find a point that is the same distance from two given points. In the house, all border lines are 5 cm, and point A must be 5 cm from both B and C. Draw an arc of radius 5 cm with centre B, and another arc of radius 5 cm with centre C. The two arcs meet at A, which is 5 cm from B and also 5 cm from C. This method avoids trial and error. The same idea is used to draw the curved roof: with A as centre and radius 5 cm, draw an arc that touches B and C. The hint for 'Eyes' uses the same technique, with two points A and B as centres for the upper and lower curves.
Draw arcs of the same radius from both given points
The point where the arcs intersect is equidistant from both points
This method is used in the House, Eyes and A Person constructions
07
Planning a Construction
Before constructing any figure, it helps to draw a rough diagram and plan the order of steps. A rough diagram shows which lines are equal, which angles are right angles, and where the compass tip should be placed. For example, to construct a rectangle that can be divided into two identical squares, you can first decide the side of each square. If AF = 4 cm, then AB = 4 cm and BC = 4 cm, so the whole rectangle has length 8 cm and breadth 4 cm. A compass can transfer the length AF to mark B and C without using a ruler. This kind of planning makes the actual construction smooth and accurate.
Draw a rough diagram before constructing
Mark equal sides and right angles on the rough diagram
Use a compass to transfer lengths accurately
Want the complete chapter resources?Topic notes, quizzes and flashcards for Playing with Constructions.
Q1. Explain the concept of a circle and its components as introduced in the chapter. How does a compass help in its construction?
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Model answer
A circle is a curve where every point on it is at an equal distance from a fixed point called the centre. This fixed distance is known as the radius. A compass is the ideal tool for this construction because it allows the user to keep the tip fixed at the centre while the pencil moves at a constant distance (the radius) to complete the curve.
Sample question3 marks
Q2. Describe the process of using a compass and a ruler to draw a circle with a radius of 4 cm, as explained in the text.
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Model answer
To draw a circle of 4 cm radius, first open the compass against a ruler so that the distance between the metal tip and the pencil lead is exactly 4 cm. Mark a point P on the paper to serve as the centre. Place the metal tip of the compass firmly on point P. Keeping the tip fixed, move only the pencil to draw a full curve. This curve is a circle where every point is 4 cm away from the centre P.
Sample question3 marks
Q3. A student wants to draw a circle with a radius of 4 cm using a compass and a ruler. Describe the steps they should follow and explain the geometric property that ensures the shape is a circle.
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Model answer
To draw a 4 cm circle, first open the compass against a ruler so the distance between the metal tip and the pencil lead is exactly 4 cm. Mark a point P as the center. Keeping the metal tip fixed at P, rotate the pencil to draw a closed curve. The geometric property is that every point on the resulting curve is at an equal distance (the radius) of 4 cm from the fixed center P.
Sample question3 marks
Q4. In a rectangle PQRS, the diagonal PR divides the corner angle P into two smaller angles. Under what specific condition will these two smaller angles be equal, and what will be their measure? Explain based on your exploration of squares.
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Model answer
According to the properties of rectangles and squares, the diagonal divides the opposite angles into equal parts only when the rectangle is a square (where all four sides are equal). Since all angles in a square are 90 degrees, if the diagonal divides the angle into two equal parts, each angle will measure 45 degrees (90 divided by 2).
Sample question3 marks
Q5. Describe the steps to locate a point 'A' that is exactly 5 cm away from two given points 'B' and 'C' using a compass, as explained in the 'House' construction section.
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Model answer
To locate point A without trial and error, set the compass to a radius of 5 cm using a ruler. Place the tip of the compass at point B and draw an arc. Keeping the same radius, place the tip at point C and draw another arc. The point where these two arcs intersect is the required point A, as it is exactly 5 cm from both B and C.
Want more questions with answers?Get the full practice set for this chapter.
What is a compass used for in Class 6 constructions?
A compass is used to draw circles and parts of circles called arcs. By fixing the pointed tip at a centre and moving only the pencil, you can draw a curve where every point is at the same distance from the centre. This distance is the radius. A compass can also transfer a length from one line to another.
What is the difference between a square and a rectangle?
A rectangle has opposite sides equal and all four angles 90°. A square also has all four angles 90°, but in addition all four sides are equal. So every square is a rectangle, but not every rectangle is a square.
How do you construct a square of side 6 cm?
Draw a line segment PQ = 6 cm. At P, draw a perpendicular and mark S on it so that PS = 6 cm. At Q, draw a perpendicular and mark R so that QR = 6 cm. Join R to S. The figure PQRS is a square of side 6 cm.
How do you construct a rectangle when one side and one diagonal are given?
Draw the given side DC. At C, draw a perpendicular line l. With D as centre and radius equal to the diagonal, draw an arc that cuts l at B. Then draw perpendiculars at D and B; their meeting point is A. Join the points to complete rectangle ABCD.
How do you find a point that is the same distance from two given points?
Draw an arc of the given radius from the first point and another arc of the same radius from the second point. The point where the two arcs intersect is at the same distance from both points. This method is used in the House construction to locate point A.
Why is a rough diagram useful before constructing a figure?
A rough diagram helps you plan the order of steps. It shows which sides are equal, where the right angles are, and where the compass tip should be placed. This reduces trial and error and makes the final construction accurate.