Surface Areas and Volumes-Notes

•   Three measures of central tendency of ungrouped data are: (i) mean, (ii) median and (iii) mode.

•   Total surface area of a cuboid = 2[lb + bh + lh]

•   Total surface area of a cube = 6(side)²

•   Lateral surface area of a cuboid = Area of walls of a room = 2(l + b) × h

•   Lateral surface area of a cube = 4a²

•   Curved surface area of cylinder = 2πrh

•   Total surface area of a cylinder = 2πr(r + h)

•   Curved surface area of a cone = πrl

•   Total surface area of a cone = πr(r + l)

•   Surface area of a sphere = 4πr²

•   Curved surface area of a hemisphere = 2πr²

•   Total surface area of a hemisphere = 3πr²

•   Volume of a'cuboid = l × b × h

•   Volume of a cube = (side)³

•   Volume of a cylinder = πr²rh

The bodies occupying space are called solids, such as a cuboid, a cube, a cylinder, a cone, a sphere, etc.

These solids have plane or curved surfaces.

The outer surface of a cuboid is made up of six rectangles. If we take the length of the cuboid as l, breadth as b and the height as h, then

Surface area of a cuboid = 2[lb + bh + hl]

Let us recall that a cuboid whose length, breadth and height are equal is called a cube. Let each edge of the cube be �a', then

Surface area of a cube = 6a²

Note: I. Sometimes a cuboid does not have bottom and top faces. is called the lateral surface area.

              II. Lateral surface area of a cuboid is 2(l + b)h.

              III. Lateral surface area of a cube is 4a².

              IV. Surface area of a cuboid or cube means total surface area.

1.   What is the longest pole that can be put in a room of dimensions l = 10 cm, b = 10 cm and h = 5 cm?

2.   If the total surface area of a sphere is 154 cm². Find its total volume.

3.   If the radius of a sphere is 3r then what is its volume?

4.   If the total surface area of a cube is 216 cm², then find its volume.

5.   If the circumference of the base of a right circular cylinder is 110 cm, then find its base area.

6.   If the radius of a sphere is doubled, then find the ratio of their volumes.

7.   The radius of a cylinder is 7 cm. If its volume is 2002 cm³, then find its height and total surface area.

8.   The diameter of a road roller, 120 cm long is 84 cm. If it takes 500 complete revolutions to level a playground, find the cost of levelling it at Rs. 2 per square metre.

9.   A conical tent of radius 7 m and height 24 m is to be made. Find the cost of the 5 m wide cloth required at the rate of Rs. 50 per metre.

10.   How many lead balls, each of radius 1 cm, can be made from a sphere of radius 8 cm?