Ppt On Surface Area And Volume For Class 9 Free Download
Angie Troia
NCERT solutions for class 10 maths chapter 13 Surface Area and Volumes helps students revise several solid shapes like cube, cone, cylinder, cuboid, etc., along with the formulas of surface area and volume of these solid figures that they have learned in the previous classes. These concepts are a part of our everyday life; hence it is important to understand these solids and their properties. Class 10 maths NCERT solutions chapter 13 Surface Area and Volumes will further illustrate the surface area and volume of these solids in combination. This means that through this chapter, students will learn how to calculate the surface area and volume of two or more solid shapes combined together.
NCERT solutions class 10 maths chapter 13 will help students explore several topics such as surface area and volume of combination solids, conversion of solid from one shape to another, and frustum of a cone. This chapter has a variety of questions based on different shapes and their combinations. The questions are curated by experts; hence each and every sum will benefit students by helping them explore a lot of distinct concepts. We will study this chapter further with the help of this article and also you can find some of these in the exercises given below.
Topics Covered: The important topics covered in class 10 maths NCERT solutions Chapter 13 are the surface area of solid shapes, combination solids, the volume of solid shapes, combination solids, conversion of solid from one shape to another, and frustum of a cone.
NCERT Solutions Class 10 Maths Chapter 13 starts with a run-through of previous concepts such as surface area of cuboid, the surface area of the hemisphere, cylinder, cone, etc. Further, these topics cover a combination of solids, conversion of one shape of solid to another, Frustum of a cone, etc.
NCERT Solutions Class 10 Maths Chapter 13 has a long list of important formulas based on surface area and volume of different shapes like cube, cuboid, cylinder, hemisphere, cone, etc. All the questions in this chapter are based on formulas; hence it is important to remember all of them. Moreover, students are advised to understand the derivation of each of these formulas. This will help them understand the application and solve all the questions with ease.
Ex 13.1 Class 10 Maths Question 2.
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Solution:
Ex 13.1 Class 10 Maths Question 3.
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Solution:
Ex 13.1 Class 10 Maths Question 4.
A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
Solution:
Ex 13.1 Class 10 Maths Question 5.
A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter d of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Solution:
Ex 13.1 Class 10 Maths Question 6.
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Solution:
Ex 13.1 Class 10 Maths Question 8.
From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm2.
Solution:
Ex 13.1 Class 10 Maths Question 9.
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.
Solution:
The surface area of a solid which is a combination of two or more solids is calculated by adding the surface areas of the individual solids which are visible in the new solid formed.
For Example:
If we consider the surface of the newly formed object as given in the figure above, we would be able to see only the curved surfaces of the two hemispheres and the curved surface of the cylinder.
So, the total surface area of the new solid is the sum of the curved surface areas of each of the individual parts. This gives, TSA of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere
The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies.[1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.
Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of finitely many pieces that can be represented in the parametric form
One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the Schwarz lantern.[2][3]
Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of fractals. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory. A specific example of such an extension is the Minkowski content of the surface.
Surface area is important in chemical kinetics. Increasing the surface area of a substance generally increases the rate of a chemical reaction. For example, iron in a fine powder will combust,[5] while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired.
The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion.[7] Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion.[8] The epithelial tissue lining the digestive tract contains microvilli, greatly increasing the area available for absorption.[9] Elephants have large ears, allowing them to regulate their own body temperature.[10] In other instances, animals will need to minimize surface area;[11] for example, people will fold their arms over their chest when cold to minimize heat loss.
Objectives: Among children with β-thalassaemia, skeletal changes and abnormalities, such as decreased volume or obliteration of the sinus, result primarily from hypertrophy and expansion of the erythroid marrow due to ineffective erythropoiesis. This study evaluated the volumes and surface areas of the maxillary sinuses of children with β-thalassaemia using cone beam computed tomography (CBCT), and compared these findings with corresponding measurements in age- and sex-matched control children.
Methods: CBCT images were retrospectively evaluated for 16 children with β-thalassaemia, 19 children with a class I skeletal pattern (class I group), and 18 children with a class II skeletal pattern (class II group). After three-dimensional analyses and segmentation of each maxillary sinus, the volumes and surface areas were calculated.
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