Have you seen a capsule? It is a combination of two hemispheres and a cylinder. Shapes that are a Combination of Solids are a common sight in the day to day life. These shapes may be hollow or solids and they can be studied as a Combination of Solids whose volume and surface areas are known.
Combination of Solids
Solid shapes are three-dimensional structures of otherwise planar shapes. A square becomes a cube, a rectangle is a cuboid and a triangle becomes a cone when changed to a 3-d structure.For planar shapes, our measurement is limited to just the area of that shape.
But when we need to measure 3-d shapes we intend to scale their volume, surface area or curved surface area. Solid shapes as already said are 3-d counterparts of their planar shape. But what if these solid shapes combine to form a different shape altogether? Combination of Solids leads to a different level of measurement.
In our daily lives, we see many shapes that are a blend of different shapes. From Huts to tents, capsules and an ice-cream filled cone, seeing such shapes is a common occurrence. So what exactly is a combination of solid? A combination of a solid is that figure which is formed by combining two or more different solids. Two cubes may combine to form a cuboid, while a cone over a cylinder might fuse to form a tent.
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Examples of Combinations of Solids
A Circus Tent or a Hut
A circus tent is a combination of a cylinder and a cone. Some circus tents also constitute a cuboid and a cone. A hut is a kutcha house and has a tent-like structure.
An Ice Cream Cone
An ice cream cone is a combination of a cone and a hemisphere.
A Dome on any Solid Shape
A dome is generally built on buildings or tents. A dome is the upper half of the hemisphere. Now if a building has a dome structure then we combine the solid shape of that building with the dome shape.
Surface Area and Volume of Combined Solid Structures
While dealing with calculations of solid structures we should be extra cautious about our measurements. Finding the volume and surface area for the combination of solids is a matter of logic and knowledge. The first thing to do in such calculations is to know what shapes have combined to form that structure.
As soon as you figure out the constituent shapes, finding the surface area or volume for that structure becomes easy and fast. For finding the surface area of a solid structure formed by combining two solids, you have to add the surface areas of the constituting structures.
For example in a circus tent, you add the surface areas of the cone and cylinder to measure the surface area. In a tent which is formed from a cone and cylinder, we first calculate the surface area of a cone and cylinder individually and then add them.
The Volume of Solid Structures
When calculating the capacity or volume of a solid structure formed from the combination of solids, we first figure out the solid shapes involved in the structure. After that, we calculate the individual volumes of these shapes and then add them to get the volume of the solid structure.
In an Ice cream, we first find the volume of the cone and the volume of the hemisphere individually and then add these quantities to get the whole volume of the cone.
Formulae for Various Shapes
The surface area of various solid shapes are given below:
- Cuboid: 2(lb+bh+hl), where l, b, and h are the length, breadth and height of a cuboid.
- Cube: 6a2, a is the side of the cube.
- Cylinder: 2πr (r+h), r is the radius of circular base and h is the height of the cylinder.
- Cone: πr(l+r), r is the radius of the circular base, l is the slant height of the cone.
- Sphere: 4πr2, r is the radius of the sphere.
- Hemisphere: 3πr2, r is the radius of the hemisphere.
Volume is the capacity of any solid shape. The formulae for volumes of various shapes are:
- Cuboid: l*b*h, where l, b and h are the length, breadth and height of a cuboid.
- Cube: a3, a is the side of the cube.
- Cylinder: πr2 h, r is the radius of circular base and h is the height of the cylinder.
- Cone: 1/3 πr 2h, r is the radius of the circular base, l is the slant height of the cone.
- Sphere: 4/3 πr3, r is the radius of the sphere.
- Hemisphere: 2/3 πr3, r is the radius of the hemisphere.
The formulae given above, form the basis of our calculation of surface areas or volumes of solid structures formed from the combination of solids. For accuracy, it is a prerequisite that you understand the basic concept of solid structures. A little logic and a little practice will eventually help in a better calculation of such tricky solids.
Solved Questions for You
Question 1:A container in the form of a cylinder mounted by a hollow hemisphere is used to store wheat. If the diameter of the hemisphere is 16 cm. If the total height of the vessel is 15 cm find the volume of wheat stored in it?
Answer : The radius of the cylindrical vessel= d/2 = 16cm/2 =8cm.
Height of the total vessel: 14 cm
Height of the cylinder: 15-8 = 7cm
Volume of wheat stored in the container = volume of hemisphere+ volume of cylindrical vessel =2/3 πr3 + πr2h
= (2/3*22/7* 8*8*8 ) + (22/7*8*8*7)
=(2/3*22/7*512) +1408 =1072.76 + 1408
The volume of wheat stored in the container is 2480.76 cm3 or 2kg 480 gm.
Question 2: What’s a volume?
Answer: Volume refers to the quantity of three-dimensional space which a closed surface encloses. For instance, the space occupied by a shape or a substance (solid, liquid, gas, or plasma). We often quantify volume numerically by making use of the SI derived unit, the cubic metre.
Question 3: How do we find the volume of an irregular object?
Answer: You can measure the mass on a balance or a scale as the volume refers to the amount of space the object is occupying. Moreover, you can find out the volume of an irregular object by plunging it in water in a beaker or some other container that has volume markings. Finally, you just need to note how much the level is going up.
Question 4: How to find the volume of solid structures?
Answer: When you want to calculate the capacity or volume of a solid structure that forms from the combination of solids, you have to first find out the solid shapes that involve in the structure. Later, you will have to calculate the individual volumes of them shapes and do the addition of them to find out the volume of the solid structure.
Question 5: What are solid shapes?
Answer: Solid shapes refer to three-dimensional structures that have otherwise planar shapes. A square will become a cube, a rectangle into a cuboid and a triangle will become a cone when we change it into a 3D structure. For planar shapes, our measurement limits to merely the area of that shape.