All of us who study the chapters on surface area calculation have at least once wondered how to find the surface area of everyday objects like pencils, buckets, earthen pots and medicine capsules, isn’t it? Well, it isn’t as difficult as it seems- because these objects can be simplified as a combination of simple solid shapes. By the end of this article, you will thoroughly understand how to find the surface area of a combination of solids, right from the basics.
Table of Content
What is Surface Area?
Surface area is the total area occupied by the surface of a three-dimensional object. It is the sum of the areas of all the surfaces of the object.
Total Surface Area (TSA)
This term is typically used for objects that have both curved and flat surfaces, such as cylinders, cones, and spheres. TSA includes the area of all surfaces of the object, including both the curved and the flat surfaces.
Curved Surface Area (CSA)
This term usually refers to the area of only the curved surface of an object, excluding any flat surfaces. It’s often used in the context of cylinders, cones, and other similar shapes. For instance, the CSA of a cylinder is the area of its lateral surface, which is the curved part, while the bases (top and bottom) are excluded.
Surface Areas of Basic Solids
Before we move to finding the surface area of a combination of solids, let us look quickly revise the formulae of some basic solids that we know:
These formulae will be elementary, as we will break down the given solid to these basic shapes, whose formulae are listen above. Let’s understand this better in the next section.
Surface Area of Combinations of Solids
The total surface area (TSA) of a combination of solids refers to the sum of the surface areas of all the individual solids involved. The formula for calculating the TSA depends on the specific combination of solids.
If you have a combination of different solids (e.g., cylinder, sphere, cube), you would calculate the surface area of each solid individually using its respective formula, and then sum up all these individual surface areas to find the total surface area.
Combination of Two Solids
When you have a combination of two solids, such as a cylinder with a cone on top, or a cube with a hemisphere on one of its faces, calculating the total surface area involves finding the surface areas of each individual solid and then summing them up.
Cone and Hemisphere
This figure can be simplified into a cone and a hemisphere. But note that while calculating the curved surface area, we do not consider the base of the hemisphere and cone, as it is not exposed outside. Hence, we only add the curved surface area of the solids.
∴ S.A. of figure = C.S.A of cone + C.S.A of hemisphere
S.A. of figure = πrl + 2πr2
S.A. of figure = πr(l+2r)
Cylinder and Cone
For the total surface area of this solid, we need to take into account the curved surface area of the cylinder, the area of the base of the cylinder and the curved surface area of the cone (not TSA as the base of cone is inside the solid)
∴ S.A. of figure = CSA of cylinder + base area of cylinder + CSA of cone
S.A. of figure = 2πrh + πr2 + πrl
S.A. of figure = πr(2h + r + l)
Cylinder and Cube
Here, since there is an overlapped area of the top of the cylinder, here is how we calculate the surface area of this solid:
∴ S.A. of figure = (CSA of cylinder + area of base of cylinder + TSA of cube) – area of top of cylinder
S.A. of figure = (2πrh + πr2 + 6a2) – πr2
S.A. of figure = 2πrh + 6a2
Cone and Cube
Here, since there is an overlapped area of the top of the cone, here is how we calculate the surface area of this solid:
∴ SA of solid = (CSA of cone + TSA of cube) – area of top of cone
SA of solid = ( πrl + 6a2) – πr2
SA of solid = πr(l + r) + 6a2
Cube and Hemisphere
Here, since there is an overlapped area of the hemisphere, here is how we calculate the surface area of this solid:
∴ SA of solid = (SA of cube + SA of hollow hemisphere) – base area of hemisphere
SA of solid = (6a2 + 2πr2) – πr2
SA of solid = 6a2 + πr2
Combination of Three Solids
Now that we have seen the combination of two solids, let us look at a more complex topic: combination of three solids.
Cylinder and Two Cones
Here, let us consider that the cones are identical and base radius of the cone and the cylinder is the same.
∴ SA of solid = CSA of cylinder + 2 x (CSA of cone)
SA of solid = 2πrh + 2 x (πrl)
SA of solid = 2πr(h + l)
Cylinder and Two Hemispheres
Here, let us consider that the hemispheres are identical and base radius of the hemispheres and the cylinder is the same.
∴ SA of solid = CSA of cylinder + 2 x (CSA of hemisphere)
SA of solid = 2πrh + 2 x (2πr2)
SA of solid = 2πrh + 4πr2
SA of solid = 2πr(h + 2r)
Solved Problems on Surface Area of Combinations of Solids
Example 1: Find the total surface area of a cylindrical tin can with a hemispherical lid. The radius of the cylindrical part is 5 cm and the height is 12 cm.
Solution:
Surface area of the cylindrical part = 2𝜋𝑟ℎ = 2×𝜋×5×12 = 120𝜋 sq. cm
Surface area of the hemispherical lid = 2𝜋𝑟2 = 2×𝜋×52 = 50𝜋 sq. cm
Total surface area = Surface area of cylindrical part + Surface area of hemispherical lid
Total surface area = 120𝜋 + 50𝜋= 170𝜋 sq. cm
Example 2: A cone is placed over a hemisphere such that their bases coincide. If the radius of the hemisphere is 6 cm and the height of the cone is 8 cm, find the total surface area of the combination.
Solution:
Surface area of the hemisphere = 2𝜋𝑟2 = 2×𝜋×62 = 72𝜋 sq. cm
Slant height (𝑙) of the cone = 𝑟2+ℎ2 = 62+82 = 100 = 10 cm
Curved surface area of the cone = 𝜋𝑟𝑙 = 𝜋×6×10 = 60𝜋 sq. cm
Total surface area = Surface area of hemisphere + Curved surface area of cone
Total surface area = 72𝜋 + 60𝜋 = 132𝜋 sq. cm
Example 3: A solid iron pole consists of a cylinder of height 220 cm and base radius 9 cm, surmounted by another cylinder of height 60 cm and radius 7 cm. Find the total surface area of the pole.
Solution:
Surface area of the first cylinder = 2𝜋𝑟ℎ = 2×𝜋×9×220 = 3960𝜋 sq. cm
Surface area of the second cylinder = 2𝜋𝑟ℎ = 2×𝜋×7×60 = 840𝜋 sq. cm
Area of the circular end of the first cylinder = 𝜋𝑟2= 𝜋×92= 81𝜋 sq. cm
Area of the circular end of the second cylinder = 𝜋𝑟2=𝜋×72=49𝜋 sq. cm
Total surface area = Surface area of first cylinder + Surface area of second cylinder + Area of circular ends
Total surface area = 3960𝜋 + 840𝜋 + 81𝜋 + 49𝜋 = 4930𝜋 sq. cm
What about irregular shapes?
The above examples were simplified through the approach of breaking them down into basic solids. Now, what happens if the object cannot be broken down into simple solids? For instance a mango, rocks, or sculptures; how will we find the curved surface area of these solids? Here, we use the concept of integration, where we break down an object into tiny rectangles and then add them up.
Application in Real-Life Examples
Knowing the surface area of combination of solids can be very helpful in everyday-life as these combinations can be objects we use in real life as well. Some practical applications include:
- While constructing buildings or structures, civil engineers and architects need to know the surface areas of the complex structures to estimate the amount of paint required, to optimise designs to reduce costs. Etc.
- In designing efficient applications of thermal heat transfers such as refrigerators, cooling systems and radiators, designers need to calculate the surface areas of the surfaces to maximise heat transfer efficiency.
- Medication dosage to patients is given in proportion to the surface area of the individual for its uniform distribution over the body.
These are just a few applications that show us the importance of knowing the concept of surface area and how it is deeply related to our everyday life.
Conclusion
Surface area of a combination of solids is an important topic not just in mathematics, but also in many real-life situations. To recap, the easiest way to find the surface area of a combination of solids is to break down the object into fundamental solids, whose surface area is known to us. Here, we should be aware of which surfaces might be overlapping and redundant, and adjust our calculations accordingly. Many fields today use this fundamental concept to carry out the biggest of tasks and decisions.
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Frequently Asked Questions about Combined Solids’ Surface Area
What is the meaning of surface area of solid?
The surface area of an object is the sum of all the surfaces on its exterior.
What is a combination of solids?
A combination of solids is a solid that is formed by putting together individual objects/solids.
How to calculate the surface area of a combination of solids?
The surface area of a combination of solids can be calculated by adding the surface areas of individual parts that make the combined solid.
What are some real life examples of combinations of solids?
A few examples include, capsules, buckets, rockets, toy car, etc.
Why is it important to find the surface area of combined solids?
Finding the surface area of a combination of solids is helpful in real-life situations such as resource approximation, design optimisation, etc.