ICSE Class 10 Maths Chapter 20 deals with finding surface area and cylindrical volume, cones and spheres, solid conversion, solid mixture, and other miscellaneous issues. For the purpose of clarifying the doubts of students, the ICSE Class 10 Maths Ch 20 Selina Solutions for Class 10 Mathematics is prepared by our experts which are built after deep research. It also provides students with instructions to confidently solve problems. ICSE Class 10 Maths Chapter 20 solutions are available on our website. We will cover the important formulas involved in solving problems related to cylinder cone and sphere Class 10. Chapter 20 Maths ICSE Class 10 is one of the important topics as this will be applicable in higher classes as well as during engineering majors.
A cylinder, one of the most basic of curvilinear geometric forms, has historically been a three-dimensional solid. It is the idealized representation of a physical solid tin that can have top and bottom lids. In elementary geometry treatments, this conventional view is still used, but the advanced mathematical point of view has changed to the infinite curvilinear surface and this is how a cylinder is now described in various modern branches of geometry and topology.
Some difficulty with terminology has been generated by the change in the basic sense (solid versus surface). It is usually hoped that the interpretation is made clear by context. Typically, both points of view are described and separated by referring to solid cylinders and cylindrical surfaces, but the unadorned term cylinder could refer in the literature to any of these or to the right circular cylinder, and even more specialized entity.
Understand the Volume for a Cylinder: There are three components in the formula for the volume of a cylinder: the radius of the cylinder, the height of the cylinder, and the ratio of the circumference of the circle to its pi-diameter. You add pi by the height of the cylinder and by the square of its radius to find the volume of a cylinder. If your calculator does not have a pi key, the pi is approximately 3.14159 and can be rounded down to 3.14. Here is the formula in mathematical terms:
V = πr^{2}
The radius of cylinder: You need to rearrange the formula for solving the term r, which is the radius because you want to find the radius of the cylinder. Divide both sides by pi and h, first of all. On the right side of the equation, these terms will be cancelled, leaving only r2. Now take the root of the square on both sides to get rid of the radius of the square.
\[\text{Radius }= \sqrt{\frac{V}{\pi \times h}}\]
Curved Surface Area: The curved surface area is defined as the only curved surface area, leaving the top and base circular. A cylinder's curved surface is equal to a rectangle whose length is 2πr and h is the width. Where: r = circular face radius and h = cylinder height.
Curved Surface Area= 2πrh
Total Surface Area: The total cylinder surface area is the area of the circles in the centre, plus the area of the round part. The surface area (SA) is found using:
Total Surface Area= 2πr(h + r)
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One that is empty from the inside is a hollow cylinder and has some difference between the internal and external radius. Tubes, circular houses, straws are examples of a hollow cylinder. A 'hollow cylinder' is called an entity bounded by two co-axial cylinders of the same height and various radii. Let R and r be the cylinder's outer and inner radii. Let its height be h.
Curved Surface Area: Hollow cylinder C.S.A. = outer cylinder C.S.A. + inner cylinder C.S.A.
= 2πRh + 2πrh
= 2πh(R + r)
Total Surface Area: Hollow cylinder T.S.A. = C.S.A. + Area at the top and bottom of two rings.
= 2π(R + r)h + 2π(R^{2} − r^{2})
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A cone is a geometric, three-dimensional shape that smoothly tapers from a flat base to a point called the apex or vertex. A cone consists of a sequence of line segments, half-lines, or lines connecting the apex, a common point, to all the points on the base of a plane that does not include the apex. The base may be limited to a circle, any one-dimensional quadratic shape in the plane, any closed one-dimensional figure, or any of the above plus all of the enclosed points, depending on the author. A cone is a solid object if the enclosed points are included in the base. Otherwise, in three-dimensional space, it is a two-dimensional object.
Volume: The volume V of any conical solid is one-third of pi, radius r, and the height h.
\[V = \frac{1}{3} \pi r^{2}h\]
Mass Centre: One-quarter of the way from the middle of the base to the vertex, on the straight line joining the two, is the centre of the mass of a conical solid of uniform density.
Curved Surface Area: The curved surface area is also known as the lateral area. Multiply the base radius of the cone by π to find the curved surface area of any cone. Multiply your response by the length of the cone's side now. If you want the total area of the rim, remember to add the base area of the cone.
The curved surface area of a cone = πrl
Total Surface Area: The combination of the curved surface, as well as the base region of a cone, is the total surface area of a cone. The formula for measuring the cone's total surface area is:
TSA of the cone = πr^{2} + πrl
= square units πr(l + r).
Area of the Curved Surface: Total Cone Surface Area = Circular base area + Curved surface area.
= ½ × l × 2πr
= πrl
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A sphere is a geometric structure that is the surface of a ball in three-dimensional space (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk"). A sphere is mathematically defined, like a circle in a two-dimensional space, as a set of points all at the same distance r from a given point in a three-dimensional space.
Diameter: The radius is equal to 2 times the diameter of a circle or sphere.
Diameter = 2 × Radius
Surface Area of Sphere: The 3-D shape is a sphere where the curved surface area equals the figure's total surface area. The area in which only the area of the curved portion is covered is the curved surface area. The circular base is not taken into consideration by the formula. On the other hand, the total surface area is a variation of the curved area together with the base area.
Total Surface Area = Curved Surface Area: 4πr^{2}
The Volume of a Sphere: The number of cubic units required to fill a sphere is known as the volume of the sphere. The sphere-volume formula is given by:
\[\text{Volume }= \frac{4}{3} \pi r^{3}\] volume (Cubic Units).
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In geometry, the generalization of an annulus to three dimensions is a spherical shell. It is the area of a ball of different radii between two concentric spheres.
Surface Area: 4π(R^{2} - r^{2}) sq. units.
Volume: A spherical shell's volume is the difference between the outer sphere's enclosed volume and the inner sphere's enclosed volume:
\[V = \frac{4}{3}(R^{3} - r^{3}) cubic units.
where the radius of the inner sphere is r and the radius of the outer sphere is R.
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1. The largest sphere is to be carved out of the right circular cylinder with a radius of 7 cm and a height of 14 cm. Find the sphere's volume. (Answer to the nearest integer correct)
Solution: The radius of the largest sphere which can be formed inside the cylinder should be equal to the radius of the cylinder.
The radius of the biggest sphere = 7 cm
Sphere Volume = \[\frac{4}{3} \pi r^{3}\]
= \[\frac{4}{3} \pi (7)^{3}\]
=\[\frac{4}{3} \pi \times 343\]
= \[\frac{4312 \pi}{3}\]
=1437cm^{2}
2. The height of the circular cylinder is 20 cm and the radius of the base is 7 cm. Find out:
(i) The volume
(ii) The total surface area.
Solution:
Height of the cylinder (h) = 20 cm
and radius of the base (r) = 7 cm
(î) Volume = \[\pi h r^{2}\]
= \[\pi (20) (7)^{2}\]
= \[\pi (20)(49)\]
= 3080 cm^{3}
(ii) Total Surface Area = 2πr(h + r)
= 2 × 22/7 × 7(20 + 7)
= 22/7 × 7(27)
= 44 × 27
= 1188 cm^{2}
1. What is a Cone's Volume and Surface Area?
Solution: A cone is a type of shape that is geometric. Different kinds of cones exist. On one side, they all have a flat surface that tapers to a point on the other side. The surface area of the cone is the area outside the cone plus the area at the end of the circle. There's a special formula used to figure this out. The first step in finding the surface area of the cone is to measure the radius of the cone part of the circle. The next step is to find a circle or base area. The area of the circle is 3.14 times the radius of the square (πr^{2}).
Surface Area = πr^{2} + πrl
The number of cubic units that will fill a cone exactly is known as the volume of the cone. The volume enclosed by the cone shall be determined by the formula
Volume = V = ⅓ π r^{2}h
Where “r” is the radius of the circular base of the cone, and “h” is the height of the cone.
2. Could There be a Cone with a Vertex?
Solution: A cone is a shape formed by a set of line segments or lines connecting a common point, called an apex or a vertex, to all points of a circular base (which does not contain the apex). The height of the cone is the distance from the vertex of the cone to the base. The circular base has a radius value measured. The cone has one face, but there are no edges or vertices. His face is in the shape of a circle. As the circle is a flat, flat shape, it's a face. But as it is round outside, there are no edges or vertices.
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