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5 Understanding Elementary Shapes
5.1 Introduction
All the shapes we see around us are formed using curves or lines. We can see corners, edges, planes, open curves and closed curves in our surroundings. We organise them into line segments, angles, triangles, polygons and circles. We find that they have different sizes and measures. Let us now try to develop tools to compare their sizes.
5.2 Measuring Line Segments
We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. To compare any two line segments, we find a relation between their lengths. This can be done in several ways.
(i) Comparison by observation:
By just looking at them can you tell which one is longer?
You can see that is longer.
But you cannot always be sure about your usual judgment.
For example, look at the adjoining segments :
The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary.
In this adjacent figure, and have the same lengths. This is not quite obvious.
So, we need better methods of comparing line segments.
(ii) Comparison by Tracing
To compare and,we use a tracing paper, trace and place the traced segment on .
Can you decide now which one among and is longer?
The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them.
(iii) Comparison using Ruler and a Divider
Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider.
Divider
Ruler
Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm. Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm.
2 mm is 0.2 cm and so on .
2.3 cm will mean 2 cm and 3 mm.
How many millimeters make one centimetre? Since 1cm = 10 mm, how will we
write 2 cm? 3mm? What do we mean by 7.7 cm?
Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB. Suppose the length is 5.8 cm, we may write,
Length AB = 5.8 cm or more simply as AB = 5.8 cm.
There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it.
To get correct measure, the eye should be correctly positioned, just vertically above the mark. Otherwise errors can happen due to angular viewing.
Can we avoid this problem? Is there a better way?
Let us use the divider to measure length.
Open the divider. Place the end point of one of its arms at A and the end point of the second arm at B. Taking care that opening of the divider is not disturbed, lift the divider and place it on the ruler. Ensure that one end point is at the zero mark of the ruler. Now read the mark against the other end point.
5.3 Angles – ‘Right’ and ‘Straight’
You have heard of directions in Geography. We know that China is to the north of India, Sri Lanka is to the south. We also know that Sun rises in the east and sets in the west. There are four main directions. They are North (N), South (S), East (E) and West (W).
Do you know which direction is opposite to north?
Which direction is opposite to west?
Just recollect what you know already. We now use this knowledge to learn a few properties about angles.
Stand facing north.
 What is the angle name for half a revolution?
 What is the angle name for onefourth revolution?
 Draw five other situations of onefourth, half and threefourth revolution on a clock.
5.4 Angles – ‘Acute’, ‘Obtuse’ and ‘Reflex’
We saw what we mean by a right angle and a straight angle. However, not all the angles we come across are one of these two kinds. The angle made by a ladder with the wall (or with the floor) is neither a right angle nor a straight angle.
Step 1  Step 2  Step 3 
Take a piece of paper 
Fold it somewhere paper 
Fold again the straight edge. Your tester is ready 
Observe your improvised ‘rightangletester’. [Shall we call it RA tester?]
Does one edge end up straight on the other?
Suppose any shape with corners is given. You can use your RA tester to test the angle at the corners.
Do the edges match with the angles of a paper? If yes, it indicates a right angle.
 The hour hand of a clock moves from 12 to 5. Is the revolution of the hour hand more than
1 right angle?
 What does the angle made by the hour hand of the clock look like when it moves from 5 to 7. Is the angle moved more than 1 right angle?

Draw the following and check the angle with your RA tester.
(a) going from 12 to 2 (b) from 6 to 7
(c) from 4 to 8 (d) from 2 to 5 
Take five different shapes with corners. Name the corners. Examine them with your tester and tabulate your results for each case :
Corner Smaller than Larger than A ...... ...... B ...... ...... C ...... ......
Other names
An angle smaller than a right angle is called an acute angle. These are acute angles.
Roof Top  Sea saw  Opening book 
Do you see that each one of them is less than onefourth of a revolution?
Examine them with your RA tester.
If an angle is larger than a right angle, but less than a straight angle, it is called an obtuse angle. These are obtuse angles.
House  Book reading desk 
Do you see that each one of them is greater than onefourth of a revolution but less than half a revolution?
Your RA tester may help to examine.
Identify the obtuse angles in the previous examples too.
A reflex angle is larger than a straight angle. It looks like this. (See the angle mark)
Were there any reflex angles in the shapes you made earlier? How would you check for them?
 Look around you and identify edges meeting at corners to produce angles. List ten such situations.
 List ten situations where the angles made are acute.
 List ten situations where the angles made are right angles.
 Find five situations where obtuse angles are made.
 List five other situations where reflex angles may be seen.
5.5 Measuring Angles
The improvised ‘Rightangle tester’ we made is helpful to compare angles with a right angle. We were able to classify the angles as acute, obtuse or reflex.
But this does not give a precise comparison. It cannot find which one among the two obtuse angles is greater. So in order to be more precise in comparison, we need to ‘measure’ the angles. We can do it with a ‘protractor’.
The measure of angle
We call our measure, ‘degree measure’. One complete revolution is divided into 360 equal parts. Each part is a degree. We write 360° to say ‘three hundred sixty degrees’.
 Cut out a circular shape using a bangle or take a circular sheet of about the same size.
 Fold it twice to get a shape as shown. This is called a quadrant.
 Open it out. You will find a semicircle with a fold in the middle. Mark 90^{o} on the fold.
 Fold the semicircle to reach the quadrant. Now fold the quadrant once more as shown. The angle is half of 90^{o} i.e. 45^{o}.
 Open it out now. Two folds appear on each side. What is the angle upto the first new line? Write 45^{o } on the first fold to the left of the base line.
 The fold on the other side would be 90^{o} + 45^{o} = 135^{o}
 Fold the paper again upto 45° (half of the quadrant). Now make half of this. The first fold to the left of the base line now is half of 45° i.e. 22 ^{1o}/_{2}. The angle on the left of 135^{o} would be 157^{1o}/_{2}.
You have got a ready device to measure angles. This is an approximate protractor.
The Protractor
You can find a readymade protractor in your ‘instrument box’. The curved edge is divided into 180 equal parts. Each part is equal to a ‘degree’. The markings start from 0° on the right side and ends with 180° on the left side, and viceversa.
Suppose you want to measure an angle ABC.
Given ∠ABC  Measuring∠ABC 
1. Place the protractor so that the mid point (M in the figure) of its straight edge lies on the vertex B of the angle.
2. Adjust the protractor so that is along the straightedge of the protractor.
3. There are two ‘scales’ on the protractor : read that scale which has the 0° mark coinciding with the straightedge (i.e. with ray
).
4. The mark shown by on the curved edge gives the degree measure of
the angle.
We write m∠ABC= 40°, or simply ∠ABC= 40°.
5.6Perpendicular Lines
When two lines intersect and the angle between them is a right angle, then the lines are said to be perpendicular. If a line AB is perpendicular to CD, we write AB ⊥ CD.
Perpendiculars around us!
You can give plenty of examples from things around you for perpendicular lines (or line segments). The English alphabet T is one. Is there any other alphabet which illustrates perpendicularity?
Consider the edges of a post card. Are the edges perpendicular?
Let be a line segment. Mark its mid point as M. Let MN be a line perpendicular to through M.
Does MN divide into two equal parts?
MN bisects (that is, divides into two equal parts) and is also perpendicular to
So we say MN is the perpendicular bisector of .
You will learn to construct it later.
5.7Classification of Triangles
Do you remember a polygon with the least number of sides? That is a triangle. Let us see the different types of triangle we can get
Using a protractor and a ruler find the measures of the sides and angles of the given triangles. Fill the measures in the given table
(a)  (b)  (c) 
(d)  (e)  (f) 
(g)  (h) 
The measure of the angles of the triangle  What can you sayabout the angles?  Measures of the sides 

(a)...60^{0}..., ....60^{0}.., ....60^{0}.....,  All angles are equal  .. 
(b)......, ......, .........,  ....... angles .......,  .. 
(c)......, ......, .........,  ....... angles .......,  .. 
(d)......, ......, .........,  ....... angles .......,  .. 
(e)......, ......, .........,  ....... angles .......,  .. 
(f)......, ......, .........,  ....... angles .......,  .. 
(g)......, ......, .........,  ....... angles .......,  .. 
(h)......, ......, .........,  ....... angles .......,  .. 
Observe the angles and the triangles as well as the measures of the sides carefully. Is there anything special about them?
What do you find?
 Triangles in which all the angles are equal. If all the angles in a triangle are equal, then its sides are also ..............
 Triangles in which all the three sides are equal. If all the sides in a triangle are equal, then its angles are............. .
 Triangle which have two equal angles and two equal sides. If two sides of a triangle are equal, it has .............. equal angles. and if two angles of a triangle are equal, it has ................ equal sides.
 Triangles in which no two sides are equal. If none of the angles of a triangle are equal then none of the sides are equal. If the three sides of a triangle are unequal then, the three angles are also............. .
Take some more triangles and verify these. For this we will again have to measure all the sides and angles of the triangles.
(a)  (b)  (c) 
The triangles have been divided into categories and given special names. Let us see what they are.
(d)  (e)  (f) 
Naming triangles based on sides
A triangle having all three unequal sides is called a Scalene Triangle [(c), (e)].
A triangle having two equal sides is called an Isosceles Triangle [(b), (f)].
A triangle having three equal sides is calledan Equilateral Triangle [(a), (d)].
Classify all the triangles whose sides you measured earlier, using these definitions.
Naming triangles based on angles
If each angle is less than 90°, then the triangle is called an acute angled triangle. If any one angle is a right angle then the triangle is called a right angled triangle. If any one angle is greater than 90°, then the triangle is called an obtuse angled triangle.
Acute Angle Triangle  Right Angle Triangle  Obtuse Angle Triangle 
 Try to draw rough sketches of
 a scalene acute angled triangle.
 an obtuse angled isosceles triangle.
 a right angled isosceles triangle.
 a scalene right angled triangle.
 Do you think it is possible to sketch
 an obtuse angled equilateral triangle ?
 a right angled equilateral triangle ?
 a triangle with two right angles?
5.8 QUADRILATERALS
 Place a pair of unequal sticks such that they have their end points joined at one end. Now place another such pair meeting the free ends of the first pair.
What is the figure enclosed?
It is a quadrilateral, like the one you see here.
There are 4 angles for this quadrilateral.
They are given by ∠BAD, ∠ADC, ∠DCB and _____.
BD is one diagonal. What is the other?
Measure the length of the sides and the diagonals.
Measure all the angles also. 
Using four unequal sticks, as you did in the above activity, see if you can form a quadrilateral such that
(a) all the four angles are acute.
(b) one of the angles is obtuse.
(c) one of the angles is right angled.
(d) two of the angles are obtuse.
(e) two of the angles are right angled.
(f) the diagonals are perpendicular to one another.
You have two setsquares in your instrument box. One is 30° – 60° – 90° setsquare, the other is 45°– 45°– 90° set square. You and your friend can jointly do this.
(a) Both of you will have a pair of 30°– 60°– 90° setsquares. Place them as shown in the figure. Can you name the quadrilateral described? What is the measure of each of its angles? This quadrilateral is a rectangle.
(b) If you use a pair of 45°– 45°–90° setsquares, you get another quadrilateral this time. It is a square. Are you able to see that all the sides are of equal length? What can you say about the angles and the diagonals? Try to find a few more properties of the square.
Do you notice that the opposite sides are parallel? Are the opposite sides equal? Are the diagonals equal?
Quadrilateral  Opposite sides  All sides Equal  Opposite Angles Equal  Diagonals  

Parallel  Equal  Equal  Perpendicular  
Parallelogram  Yes  Yes  No  Yes  No  No 
Rectangle  ..  ..  No  ..  ..  .. 
Square  ..  ..  ..  ..  ..  Yes 
Rhombus  ..  ..  No  Yes  ..  .. 
Trapezium  ..  ..  No  ..  ..  .. 
5.9 Polygons
So far you studied polygons of 3 or 4 sides (known as triangles and quardrilaterals respectively). We now try to extend the idea of polygon to figures with more number of sides. We may classify polygons according to the number of their sides.
Number of sides  Name  Illustration 

3  Triangle  
4  Quadrilateral  
5  Pentagon  
6  Hexagon  
8  Octagon 
You can find many of these shapes in everyday life. Windows, doors, walls, almirahs, blackboards, notebooks are all usually rectanglular in shape. Floor tiles are rectangles. The sturdy nature of a triangle makes it the most useful shape in engineering constructions.
The triangle finds application in constructions. 
The bee knows the usefulness of a hexagonal shape in building its house. 
Look around and see where you can find all these shapes.
5.10 Three Dimensional Shapes
Here are a few shapes you see in your daytoday life. Each shape is a solid. It is not a‘flat’ shape.
The ball is a sphere.  The icecream is in the form of a cone.  This can is a cylinder. 
The box is a cuboid.  The playing die is a cube.  This is the shape of a pyramid. 
Name any five things which resemble a sphere.
Name any five things which resemble a cone.
Faces, edges and vertices
In case of many three dimensional shapes we can distinctly identify their faces, edges and vertices. What do we mean by these terms: Face, Edge and Vertex? (Note ‘Vertices’ is the plural form of ‘vertex’).
Consider a cube, for example.
Each side of the cube is a flat surface called a flat face (or simply a face). Two faces meet at a line segment called an edge. Three edges meet at a point called a vertex.
Here is a diagram of a prism.
Have you seen it in the laboratory? One of its faces is a triangle. So it is called a triangular prism.
The triangular face is also known as its base.
A prism has two identical bases; the other faces are rectangles.
If the prism has a rectangular base, it is a rectangular prism. Can you recall another name for a rectangular prism?
A pyramid is a shape with a single base; the other faces are triangles.
Here is a square pyramid. Its base is a square. Can you imagine a triangular pyramid? Attempt a rough sketch of it.
The cylinder, the cone and the sphere have no straight edges. What is the base of a cone? Is it a circle? The cylinder has two bases. What shapes are they? Of course, a sphere has no flat faces! Think about it.
 A cuboid looks like a rectangular box. It has 6 faces. Each face has 4 edges.
 A cube is a cuboid whose edges are all of equal length. It has ______ faces.
 A triangular pyramid has a triangle as its base. It is also known as a tetrahedron. Faces : _______
 A square pyramid has a square as its base. Faces : _______
 A triangular prism looks like the shape of a Kaleidoscope. It has triangles as its bases. Faces : _______
Each face has 4 corners (called vertices).
Each face has ______ edges.
Each face has ______ vertices.
Edges : _______
Corners : _______
Edges : _______
Corners : _______
Edges : _______
Corners : _______