The chapter on Surface Areas and Volumes is part of the class 10 Maths syllabus, which introduces you to the different formulas through which you can find out the surface areas and volumes of various solids such as cones, cylinders, cuboids, and spheres. Being well versed in these formulas can not only help the students of the 10th standard ace their board exams, but are also essential for candidates appearing for different competitive exams. This blog will familiarise you with the different formulas used to derive the surface areas and volumes of different shapes along with examples which are part of the chapter on surface area and volume for class 10. Let’s take a look!
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Students must be familiar with real-life shapes such as cone, sphere, cylinder, and cuboid. Here is a simple presentation of a cylinder shape truck that uses multiple forms with a single cylinder and two hemispheres on both ends.
Similarly, a test tube that you use in a chemistry lab is a combination of a cylinder shape at the top and a hemisphere in the end.
For these shapes, you can’t get the exact surface areas and volumes using one formula. Here you have to combine formulas for surface areas and volumes, to get the desired results. This chapter on surface area and volume for class 10 will give you a comprehensive guide on finding surface areas and volumes for these complex objects.
Also Read: Maths Formulas for Class 10
Surface Area of a Combination of Solids
Starting on the chapter on surface area and volume for class 10, let’s look at a simple image of the following solid that has a cylindrical shape in the middle and circular on the ends.
For understanding these combinations of solids, you have to break down into simpler forms such, as a cylinder with a hemisphere on both ends. The above image gives you an exact idea of how three solid converge to form a combination of solid. Now to find the complete surface areas and volumes of this object, you have to add the surface area of all the three smaller objects. So the total surface area is the sum of the surface area of each part. Thus forming a simple equation for a formula like that;
Total Surface Area (TSA) of a new solid = Curved Surface Area (CSA) of first hemisphere + Curved Surface Area of cylinder + Curved Surface Area (CSA) of the second hemisphere
Similarly, all the combined shapes have multiple shapes in their fundamental design. You can rethink their structure to find complete surface areas of a complex solid.
There are several simple questions in exercise 13.1 class 10 that you must solve to practice the concepts of this topic and prepare for the board exams collectively.
Volume of a Combination of Solids
In this section of the chapter on surface area and volume for class 10, you will study to find formulas for volume with the combination of solids. While finding surface areas, some parts disappear as two shapes join together. Though for the volume aspect, all the parts remain constant. Here is a simple presentation:
The following shed has a cuboid shape with a half-cylindrical shaped mounted on top of that. Now to find the volume of air inside this solid, you have to assume its shape and combination. It will be the sum of the volume of the cuboid and half the volume of the cylinder above it. So total volume of this shape will be of the equation:
= Volume of the cuboid + 1/2 volume of the cylinder
So total volume of this shape will be of the equation
= Volume of the cuboid + 1/2 volume of the cylinder
= [ 15 X 7 X 8 + 1/2 X 22/7 X 7/2 X 7/2]
Conversion Of Solid From One Shape To Another
There are also situations when a shape takes the form of another. Here is a simple presentation of a Candle. Generally, they are cylindrical. But on special occasions, we find different candle shapes. So if the same amount of molten wax is available for both shapes, then how does it affect their formulas for surface areas and volumes?
Let’s consider a practical example to understand this conversion. We have a cone of height 24cm and a 6cm radius built up of clay. Now a child has reshaped this into a sphere. How can you find the radius of this sphere?
Let’s start the solution by finding the volume of the cone:
The volume of cone= 1/3×π×6×6×24cm
Assuming the radius of the sphere to be r, then the volume of the new cone is (4πr3 / 3)
As both have the same volume, so comparing them, we get the following
4×π×r3 = 1×π×6×6×24 33
i.e., r3 = 3×3×24=33 ×23 So, r=3×2=6
This section of the chapter on surface area and volume for class 10 offers step by step insights for the conversion of solids from one shape to another and find attributes for newer shapes.
Frustum of a Cone
In this section of the chapter on surface area and volume for class 10, we will study a different combination of objects. Let’s take a simple example of a drinking glass. To understand the shape of glass let us first have a look at the image below. Here we can see the following image is part of a cone shape after removing a lower portion of it. This proposition is known as a frustum of the cone.
So if we know the radius of smaller and larger surface areas then we can calculate the surface areas and volume of this part of the cone. Let’s take an example of a frustum of cone heights at h, slant height l, R, and r are radii at different ends (r1 > r2 ) for the frustum of a cone. Then formulas for volume and surface areas are:
- (i) Volume for this frustum of the cone =1/3 πH (R2 +Rr+r2 )
- (ii) Curved Surface Area (CSA) for this frustum of the cone =pi * l(R + r)
- (iii) Total Surface Area (TSA) of the frustum of the cone = pi * [l(R + r) + R^2 +r^2]
In this chapter on surface area and volume for class 10, we have discussed and studied the following understanding of formulas for surface areas and volumes.
- Determine the surface area for a combination of different solids such as cylinders, cones, cuboid, sphere, and hemisphere.
- Determine the volume of a combination of different forms of solids with cylinder, cuboid, cones, sphere, hemisphere, and more.
- Determine the volume of solids when they change shape from one to another.
- Find the volume, curved surface areas, and total surface areas of a frustum of a cone.
Also Read: Class 10 Quadratic Equations
Hope this blog was successful in familiarising you with the above concepts which are part of the chapter on surface area and volume for class 10. Students must practice these important lessons to stay on the right path for understanding the concepts of Surface Areas and Volumes. Confused about which stream to choose after 10th? Reach out to our experts at Leverage Edu for assistance in choosing the right stream as per your career goals. Sign up for a free session today!