NCERT Solutions -

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AREAS OF PARALLELOGRAMS AND TRIANGLES

In Chapter 5, you have seen that the study of Geometry, originated with the
measurement of earth (lands) in the process of recasting boundaries of the fields and
dividing them into appropriate parts. For example, a farmer Budhia had a triangular
field and she wanted to divide it equally among her two daughters and one son. Without
actually calculating the area of the field, she just divided one side of the triangular field
into three equal parts and joined the two points of division to the opposite vertex. In
this way, the field was divided into three parts and she gave one part to each of her
children. Do you think that all the three parts so obtained by her were, in fact, equal in
area? To get answers to this type of questions and other related problems, there is a
need to have a relook at areas of plane figures, which you have already studied in
earlier classes.

**Chapter - 9 - Exercise - ****9.1**** ****9.2**** ****9.3**

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HERON’S FORMULA

12.1 Introduction
You have studied in earlier classes about figures of different shapes such as squares,
rectangles, triangles and quadrilaterals. You have also calculated perimeters and the
areas of some of these figures like rectangle, square etc. For instance, you can find
the area and the perimeter of the floor of your classroom.
Let us take a walk around the floor along its sides once; the distance we walk is its
perimeter. The size of the floor of the room is its area.
So, if your classroom is rectangular with length 10 m and width 8 m, its perimeter
would be 2(10 m + 8 m) = 36 m and its area would be 10 m × 8 m, i.e., 80 m

^{2} .
Unit of measurement for length or breadth is taken as metre (m) or centimetre
(cm) etc.
Unit of measurement for area of any plane figure is taken as square metre (m

^{2} ) or
square centimetre (cm

^{2} ) etc.
Suppose that you are sitting in a triangular garden. How would you find its area?
From Chapter 9 and from your earlier classes, you know that:

**Chapter -12 - Exercise - ****12.1** **12.2**

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SURFACE AREAS AND VOLUMES

13.1 Introduction
Wherever we look, usually we see solids. So far, in all our study, we have been dealing
with figures that can be easily drawn on our notebooks or blackboards. These are
called plane figures. We have understood what rectangles, squares and circles are,
what we mean by their perimeters and areas, and how we can find them. We have
learnt these in earlier classes. It would be interesting to see what happens if we cut
out many of these plane figures of the same shape and size from cardboard sheet and
stack them up in a vertical pile. By this process, we shall obtain some solid figures
(briefly called solids) such as a cuboid, a cylinder, etc. In the earlier classes, you have
also learnt to find the surface areas and volumes of cuboids, cubes and cylinders. We
shall now learn to find the surface areas and volumes of cuboids and cylinders in
details and extend this study to some other solids such as cones and spheres.

**Chapter - 13 - Exercise - ****13.1** **13.2 13.3 13.4 13.5 13.6 13.7 13.8**