We come across many objects with basic shapes like cubes, cuboids or spheres and with the known formulae we can very easily calculate their surface area and volume. But have you ever wondered what if these basic shapes conjoin and form a shape different from the original? How then shall we calculate the capacity of the new shape? The chapter below throws light on the same.
Surface Area and Volume of Combination of SolidsWe come across a lot of solids which are a combination of one or more basic shapes. Tents, capsules, and ice-cream cones are the most common examples. You might have also seen trucks with capsule-shaped containers carrying petrol or Liquefied Petroleum Gas. Are they similar to the basic shapes, or a combination of different shapes?
These are certainly a combination of two or more shapes. So in a nutshell, the combination of solids include shapes formed from the fusion of two or solid shapes, which together form a new shape. The shapes formed by the combination of different shapes are called composite shapes.
Learn more about Cuboid and Cube here in detail.
While calculating the surface area and volume of these shapes we need to first observe the number of solid shapes that form these shapes. As we already know that, solid shapes are three-dimensional structures of a one-dimensional shape, for example, a cube is formed when six square-shaped cards are assembled adjacent to each other.
When we measure the surface area and volume of these solid shapes we consider all the three dimensions: length, breadth, and height. Now, when we combine these solid shapes to form a new shape, we end up calculating these at a different level. Calculating the surface area and volume of the composite shapes takes us to a new level of thinking.
For this, our understanding towards shapes and structures must be accurate. When we calculate the surface area and volume of composite shapes, we break the shape into its constituting shapes. This process of calculation is exactly similar to breaking a bigger problem into a smaller problem, to reach an accurate solution.
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Surface Area of Composite Shapes
The surface area of any solid shape is the sum of the areas of all faces in that solid shape. For example, when finding the surface area of a cuboid we add the area of each rectangle constituting the cuboid. Similarly, when finding the surface area of composite shapes, we add the area of all the surfaces of structures constituting that composite structure.
Learn more about Sphere here in detail.
As said earlier we first break the composite structure into its smaller constituents and then add all the solutions to get the major solution to our problem. Here to understand the subject, we start with a simple composite structure of an Ice-cream cone:
An ice-cream filled cone constitutes a cone and a hemisphere-shaped ice-cream. So when we need to find the surface area covered by the whole structure we add the individual surface areas. So, the total surface area of the cone shall be the sum of individual surface areas of the constituting shape. In case of an ice-cream filled cone:
Total surface area of the Ice-cream cone = Curved Surface Area of Hemisphere + Curved Surface Area of the Cone. Curved Surface area of a hemisphere = 2πr2 and Curved Surface area of the cone = πrl. So,
The Total Surface Area of Ice-cream Cone = 2πr2 + πrl
Let’s take another example of a tent. In a tent we see two structures, one is the cone while the other is a cylinder. To Calculate the Total Surface Area of the tent we calculate the individual surface areas of shapes constituting the tent’s structure. Hence, Total Surface Area of a tent house = Curved Surface Area of the cone + Curved Surface Area of the cylinder = πrl + 2πrh.
Volumes of Composite Shapes
When we want to find the volume of a container we intend to calculate the capacity it can hold. While finding the volume of composite shapes we calculate the capacity of that structure if its hollow. But if its a concrete structure then the calculation for volume is done just to get an idea of the density of that structure.
For finding the volume of combined solids we follow the same level of calculation as we did in the surface area of a composite solid shape. So let’s find the volume of a capsule container on a truck carrying petroleum.
The capsule-shaped container loaded on a truck is a combination of a cylinder with adjoined hemispheres on both sides. Now, when we find the volume of this capsule-shaped container we add the individual volumes of all the constituting shapes. So, in this case, we add the individual volumes of 2 hemispheres and one cylinder.
The volume of a capsule container = Volume of hemisphere + Volume of Cylinder + Volume of Hemisphere = 2/3πr3 + πr2h + 2/3πr3 . This method of finding volumes of combined shapes is used in all types of composite shapes.
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.
Learn more about Frustum of Cone here in detail.
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: s3, s 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.
When calculating the surface area and volume of combined solid shapes all we have to remember is the constituting shapes and their formulae. As already said, measuring surface area and volume of combine shapes is the next level of measuring capacities and areas, and this needs thorough practice and precision.
Solved Examples for You
Question 1: Shreya made a cylindrical bird-bath with a hemispherical depression at the upper end. The radius of the circular shaped top is 30cm and height of the cylinder is 1.45 m. Find the total surface area of the bird bath?
Answer : The radius of both cylinder and hemisphere are common, hence taken as r = 30cm = 0.3m. Height (h) of the cylinder = 1.45 m.
TSA of the Bird-Bath = CSA of Cylinder + CSA of the Hemisphere
= 2 πrh + 2πr2
= 2 π r(h + r)
= 2 × (22/7) × 0.30 (1.45 + 0.30) m
= 3.3 m2
Question 2: Explain the volume of a cone?
Answer: The volume of any kind of cone can be ascertained with the formula V = ⅓ A∙h. In this formula, A represents the area of the base and h represents the height.
Question 3: Differentiate between the surface area and volume?
Answer: Surface area refers to the sum of the areas belonging to all the faces of the solid figure. In contrast, volume refers to the number of cubic units that comprise a solid figure.
Question 4: How can one find the volume of a cube?
Answer: Volume of a cube is equal to the side time’s side times side. Furthermore, each side of a square is exactly the same. Therefore, it can be the length of one side cubed. Suppose a square has one side that is of 4 inches, the volume will be equal to 4 inches times 4 inches times 4 inches.
Question 5: What is the formula to find the surface area of a rectangle?
Answer: The formula of surface area regarding a rectangular prism is A = 2wl + 2lh + 2hw, where w represents the width, the l represents the length, and the h is the representative of height.