We have learnt that for a closed plane figure, the perimeter is the distance around its boundary and its area is the region covered by it. We found the area and perimeter of various plane figures such as triangles, rectangles, circles etc. We have also learnt to find the area of pathways or borders in rectangular shapes. In this chapter, we will try to solve problems related to perimeter and area of other plane closed figures like quadrilaterals. We will also learn about surface area and volume of solids such as cube, cuboid and cylinder.
Let us take an example to review our previous knowledge. This is a figure of a rectangular park (Fig 11.1) whose length is 30 m and width is 20 m. (1) What is the total length of the fence surrounding it? To find the length of the fence we need to find the perimeter of this park, which is 100 m. (Check it) (2) How much land is occupied by the park? To find the land occupied by this park we need to find the area of this park which is 600 square meters (m2) (How?). (3) There is a path of one metre width running inside along the perimeter of the park that has to be cemented. If 1 bag of cement is required to cement 4 m2 area, how many bags of cement would be required to construct the cemented path?
Nazma owns a plot near a main road . Unlike some other rectangular plots in her neighbourhood, the plot has only one pair of parallel opposite sides. So, it is nearly a trapezium in shape. Can you find out its area? Let us name the vertices of this plot as shown in.
A general quadrilateral can be split into two triangles by drawing one of its diagonals. This “triangulation” helps us to find a formula for any general quadrilateral. Area of quadrilateral ABCD
We split a quadrilateral into triangles and find its area. Similar methods can be used to find the area of a polygon. Observe the following for a pentagon:
In your earlier classes you have studied that two dimensional figures can be identified as the faces of three dimensional shapes. Observe the solids which we have discussed so far Observe that some shapes have two or more than two identical (congruent) faces. Name them. Which solid has all congruent faces? All six faces are rectangular, and opposites faces are identical. So there are three pairs of identical faces.Now take one type of box at a time. Cut out all the faces it has. Observe the shape of each face and find the number of faces of the box that are identical by placing them on each other. Write down your observations.
Imran, Monica and Jaspal are painting a cuboidal, cubical and a cylindrical box respectively
of same height.They try to determine who has painted more area. Hari suggested that finding the
surface area of each box would help them find it out.
To find the total surface area, find the area of each face and then add. The surface
area of a solid is the sum of the areas of its faces. To clarify further, we take each shape
one by one.
Suppose you cut open a cuboidal box and lay it flat. We can see a net as shown below. Write the dimension of each side. You know that a cuboid has three pairs of identical faces. What expression can you use to find the area of each face?
Amount of space occupied by a three dimensional object is called its volume. Try to compare the volume of objects surrounding you. For example, volume of a room is greater than the volume of an almirah kept inside it. Similarly, volume of your pencil box is greater than the volume of the pen and the eraser kept inside it. Can you measure volume of either of these objects? Remember, we use square units to find the area of a region. Here we will use cubic units to find the volume of a solid, as cube is the most convenient solid shape (just as square is the most convenient shape to measure area of a region). For finding the area we divide the region into square units, similarly, to find the volume of a solid we need to divide it into cubical units. Observe that the volume of each of the adjoining solids is 8 cubic units
There is not much difference between these two words.
(a) Volume refers to the amount of space occupied by an object.
(b) Capacity refers to the quantity that a container holds.
A cylinder is formed by rolling a rectangle about its width. Hence the width of the paper becomes height and radius of the cylinder is 20 cm.
A rectangular piece of paper 11 cm × 4 cm is folded without overlapping to make a cylinder of height 4 cm. Find the volume of the cylinder.
Water is pouring into a cubiodal reservoir at the rate of 60 litres per minute. If the volume of reservoir is 108 m3, find the number of hours it will take to fill the reservoir.