Let’s take every topic of Surface Area and Volume one-by-one

Important formulae related to surface areas and volumes

This is a formula based chapter and you should be thorough with the formulae of total SA, lateral surface area and volume of basic solid figures, namely, cube, cuboid, cylinder, cone, frustum of a cone, sphere and hemi sphere.
Let us have a look at these formulae and then solve a ques based on them:

Lets solve a formula based question which has come in boards:
Q1. Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Sol: Over here, note that whenever only SA is mentioned, we take total SA unless curved word is specified in the question.
Let the rad of hemisphere be ? units, then,
3??^2=2/3??^3
⇒?=9/2 units
∴ Diameter =2?=9 units

Surface area and Volume of combination of solids:

While calculating the SA and Volume of combination of solids, we need to modify the formulae for surface areas and volumes of basic solid figures to find the surface areas and volumes of a combination of these figures.

1. While calculating the surface area of a combination of solids, we need to exclude those surfaces that overlap each other on combining the solids
2. While calculating their volume, we either add or subtract the volumes of the two solids depending on the nature of the combination.

Lets solve a few questions that have been asked in boards previously based on this concept:
Q2. In figure, a tent is in the shape of a cylinder surmounted by a conical top of same diameter. If the height and diameter of cylindrical part are 2.1 ? and 3 ? respectively and the slant height of conical part is 2.8 ?, find the cost of canvas needed to make the tent if the canvas is available at the rate of ₹ 500/sq. metre. (Use ?=22/7)

Sol:
Radius, ?=3/2 ?
Canvas required = CSA of Cylindrical part + CSA of Conical top
=2??ℎ+???=?×[3/2][2×2.1+2.8]=33 ?^2
Cost of canvas =₹ 500×33=₹ 16,500

Q3. A solid is in the shape of a cone surmounted on a hemisphere, the radius of each of them being 3.5 ?? and the total height of solid is 9.5 ??. Find the volume of the solid. [Use ?=22/7].

Sol. Radius, ?=3.5 ??
Height of cone, ℎ=9.5−3.5=6 ??

Volume of solid = Volume of cone + Volume of hemisphere
=1/3??^2ℎ+2/3??^3=1/3??^2(ℎ+2?)
=1/3×22/7×(3.5)^2×(6+2×3.5)
=166.83 ??^3

Conversion of solids from one shape to another:

A lot of times questions are asked in which shape of solid changes: this can happen when a solid is melted and recast as another solid or a liquid is transferred from one container to another having a different shape. We need to keep in mind that in such
cases as the shape changes, the surface area will change. But the total volume will remain constant.

Lets solve a question based on this:
Q4. In a rain–water harvesting system, the rain–water from a roof of 22 ?×20 ? drains into a cylindrical tank having diameter of base 2 ? and height 3.5 ?. If the tank is full, find the rainfall in ??. Write your views on water conservation.

Sol. Let the roof be filled up to a height ℎ. Then,

Vol of water collected on the roof = Vol of water collected in cylindrical tank
⇒22 ×20×ℎ=?(1)^2×3.5
⇒ℎ=1/40?=2.5 ??

Let’s solve one of the questions: