In this blog, we will discuss the surface area and volume of a combination of solids. We have already discussed how to find the surface area of solids made up of a combination of two basic solids.
Here, we shall see how to calculate their volumes. The volume of the solid formed by joining two basic solids will actually be the sum of the volumes of the constituents.
Combination of Solids: Combination of solids is also known as composite solids or composite shapes. It is obtained from combining two different solids. Two or more shapes are combined to form what we call as composite solids. In order to find the surface area and volume of such combination of solids you must know the surface and volume formula for each solid for examples generally combination of solids are obtained from mixing general shapes such as cone, cube, cylinder, sphere etc.
Let us see in this example.
1. Shanta runs an industry in a shed which is in the shape of a cuboid surmounted by a half-cylinder. If the base of the shed is of dimension 7 m × 15 m, and the height of the cuboidal portion is 8 m, find the volume of air that the shed can hold. Further, suppose the machinery in the shed occupies a total space of 300 m3, and there are 20 workers, each of whom occupy about 0.08 m3 space on an average. Then, how much air is in the shed?
2. A juice seller was serving his customers using glasses. The inner diameter of the cylindrical glass was 5 cm, but the bottom of the glass had a hemispherical raised portion which reduced the capacity of the glass. If the height of a glass was 10 cm, find the apparent capacity of the glass and its actual capacity. So the glass is the combination of a cylinder and a hemisphere.
Solution: The inner diameter of the glass = 5 cm, height = 10 cm,
The capacity of the glass = πr2h
= 3.14 × 2.5 × 2.5 × 10 cm3 = 196.25 cm3.
The volume of the hemisphere at the base of the glass = 2/3πr3 = 2/3 x 22/7 x 2.5 x 2.5 x 2.5 = 32.71cm3
So, the actual capacity of the glass = apparent capacity of glass π volume of the hemisphere = (196.25 – 32.71) cm3
= 163.54 cm3
The surface area of a combination of solids
In this section, we will introduce the surface area of combination of solids. In our day-to-day life, we come across a number of solids made up of combinations of two or more of the basic solids.
The truck with a container fitted on its back carrying oil or water from one place to another. Capsule, pencil, top, letterbox. Can you guess the solids combined in this tank and capsule? These both are made of a cylinder with two hemispheres as its ends. Now guess for pencil. Yes, the pencil is a combination of cone and cylinder. And this top is a combination of cone and hemisphere. And this letterbox has a hemisphere on a cylinder.
How do we find the surface area of such a solid? Let us take this capsule and we can see that this solid is made up of a cylinder with two hemispheres stuck at either end. If we consider the surface of the newly formed object, we would be able to see only the curved surfaces of the two hemispheres and the curved surface of the cylinder.
So, the total surface area of the new solid is the sum of the curved surface areas of each of the individual parts. This gives, TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of another hemisphere where TSA, CSA stand for ‘Total Surface Area’ and ‘Curved Surface Area’ respectively.
Now let us consider this top. To know the surface area of the toy, which consists of the CSA of the hemisphere and the CSA of the cone?
So, we can say: Total surface area of the toy = CSA of hemisphere + CSA of cone where CSA stands for curved surface area.
1. Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted byhemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour.
So, the height of the cone = height of the top – height (radius) of the hemispherical part. That is 5 minus 3.5 by 2 which is equal to 3.25 cm.
2. The decorative block shown is made of two solids — a cube and a hemisphere. The base of the block is a cube with an edge of 5 cm, and the hemisphere fixed on the top has a diameter of 4.2 cm. Find the total surface area of the block.
Solution: The total surface area of the cube = 6 × (edge)2
= 6 × 5 × 5 cm2
= 150 cm2
The surface area of the block = TSA of cube – base area of hemisphere + CSA of hemisphere
= 150 – πr2 + 2πr2
= 150 + πr2
= 150 + (22 b/ 7 x 4.2 /2 x 4.2 /2) cm2
= 150 +13.86 cm2
= 163.86 cm2