CBSE Class 10th Exam Date:17 Feb' 26 - 17 Feb' 26
Imagine you're designing a water tank for your home. You need to calculate how much water it can hold (volume) and how much material will be needed to paint the outside of this water tank(surface area). The answer to these questions lies in understanding surface areas and volumes. This chapter helps us to understand how to calculate the area related to covering a solid object and the space it occupies. It also deals with calculating the surface area and volume of different solids like cubes, cylinders, cones, and spheres, and the formulas for surface area and volume. NCERT Solutions for Class 10 can help the students understand these concepts and will make them more efficient in solving problems involving Surface Areas and Volumes.
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These NCERT Solutions for class 10 Maths are designed by our experienced subject matter experts to offer a structured approach and step-by-step solutions for students to prepare for the board exams. These Solutions cover all exercises in the NCERT textbook and are easily available for download. The step-by-step explanations for different types of problems help students gain a strong understanding of surface area and volume. Many toppers rely on NCERT Solutions since they are designed as per the latest syllabus. Find everything in one place – NCERT Books, Solutions, Syllabus, and Exemplar Problems with Solutions – in this NCERT article.
The NCERT Solutions for Class 10 Maths Chapter 12 have been prepared by Careers360 experts to make learning simpler and to help you score better in exams. A downloadable PDF is available — click the link below to access it.
NCERT Class 10 Maths Chapter 12 Surface Areas and Volumes question answers with detailed explanations are provided below.
Surface Areas and Volumes Class 10 Question Answers Exercise: 12.1 Page number: 166-167 Total questions: 9 |
Q1. 2 cubes each of volume 64 cm3 are joined end to end. Find the surface area of the resulting cuboid.
Answer:
We are given that the volume of the cube $=\ 64\ cm^3$
Also, the volume of a cube is given by $=\ a^3$ ( here $a$ is the edge of the cube)
Thus : $a^3\ =\ 64$
$a\ =\ 4\ cm$
Now, according to the question, we have combined the two cubes.
The edge lengths of the formed cuboid are 4 cm, 4 cm, and 8 cm.
The surface area of a cuboid is : $=\ 2\left ( lb\ +\ bh\ +\ hl \right )$
$=\ 2\left ( 8\times 4\ +4\times 4\ +\ 4\times 8 \right )$
$=\ 2\left ( 80 \right )$
$=\ 160\ cm^2$
Thus, the area of the formed cuboid is 160 cm2.
Answer:
Since the vessel consists of a hemisphere and a cylinder, its area is given by :
Area of vessel = Inner area of the cylinder(curved) + Inner area of the hemisphere
The inner surface area of the hemisphere $=\ 2\pi r^2$
$=\ 2\times \left ( \frac{22}{7} \right ) \times 7^2$
$=\ 308\ cm^2$
The surface area of the cylinder $=\ 2\pi rh$
$=\ 2\times \frac{22}{7}\times 7\times 6$
$=\ 264\ cm^2$
Thus, the inner surface area of the vessel is $= 308 + 264 = 572 \:cm^2.$
Answer:
The required surface area of the toy is given by :
Area of the toy = Surface area of the hemisphere + Surface area of the cone
Firstly, consider the hemisphere :
The surface area of a hemisphere $=\ 2 \pi r^2$
$=\ 2 \times \frac{22}{7}\times \left ( 3.5 \right )^2$
$=\ 77\ cm^2$
Now for cone, we have :
The surface area of a cone $=\ \pi rl$
Thus, we need to calculate the slant of the cone.
We know that :
$l^2\ =\ h^2\ +\ r^2$
$=\ 12^2\ +\ 3.5^2$
$=\ \frac{625}{4}$
$l\ =\ \frac{25}{2}\ =\ 12.5\ cm$
Thus, the surface area of a cone $=\ \pi rl$
$=\ \frac{22}{7}\times 3.5\times 12.5$
$=\ 137.5\ cm^2$
Hence, the total surface area of the toy $=\ 77\ +\ 137.5\ =\ 214.5\ cm^2.$
Answer:
It is given that the hemisphere is mounted on the cuboid; thus, the hemisphere can take on complete as its diameter (which is maximum).
Thus, the greatest diameter of the hemisphere is 7 cm.
Now, the total surface area of the solid = Surface area of the cube + Surface area of the hemisphere - Area of the base of a hemisphere (as this is counted on one side of the cube)
The surface area of the cube $=\ 6a^3$
$=\ 6\times 7^3\ =\ 294\ cm^2$
Now, the area of a hemisphere $=\ 2\pi r^2$
$=\ 2\times \frac{22}{7}\times \left ( \frac{7}{2} \right )^2\ =\ 77\ cm^2$
The area of the base of a hemisphere $=\ \pi r^2\ =\ \frac{22}{7}\times \left ( \frac{7}{2} \right )^2\ =\ 38.5\ cm^2$
Hence, the surface area of the solid is $= 294 + 77 - 38.5 = 332.5 \:cm^2$.
Answer:
It is given that the diameter of the hemisphere is equal to the edge length of the cube.
The total surface area of the solid is given by :
The surface area of a solid = Surface area of the cube + Surface area of the hemisphere - Area of the base of the hemisphere
The surface area of the cube $=\ 6l^2$
The surface area of the hemisphere:
$=\ 2\pi r^2\ =\ 2\pi \left ( \frac{l}{2} \right )^2$
Area of the base of the hemisphere:
$=\ \pi r^2\ =\ \pi \left ( \frac{l}{2} \right )^2$
Thus, the area of the solid is:
$=\ 6l^2\ +\ \pi \left ( \frac{l}{2} \right )^2\ unit^2.$
Answer:
It is clear from the figure that the capsule has a hemisphere and cylinder structure.
The surface area of the capsule = 2 (Area of the hemisphere) + Area of the cylindrical part
Area of hemisphere $=\ 2\pi r^2$
$=\ 2\pi\times \left ( \frac{5}{2} \right )^2$
$=\ \frac{25}{2} \pi\ mm^2$
The area of the cylinder $=\ 2\pi rh$
$=\ 2\pi \times \frac{5}{2}\times 9$
$=\ 45 \pi\ mm^2$
Thus, the area of the solid $=\ 2\left ( \frac{25}{2} \right )\pi\ +\ 45 \pi$
$=\ 25\pi\ +\ 45 \pi$
$=\ 70 \pi$
$=\ 220\ mm^2.$
Answer:
The canvas will cover the cylindrical part as well as the conical part.
So, the area of the canvas = Area of the cylindrical part (curved) + Area of the conical part
Now, the area of the cylindrical part $=\ 2\pi rh$
$=\ 2\pi \times 2\times 2.1$
$=\ 8.4 \pi\ m^2$
The area of the cone is $=\ \pi rl$
$=\ \pi \times 2\times 2.8$
$=\ 5.6\pi\ m^2$
Thus, the area of the canvas $=\ 8.4\pi\ +\ 5.6\pi$
$=\ 14\pi\ =\ 44\ m^2$
Further, it is given that the rate of canvas per m2 is Rs. 500.
Thus, the required money is $=500 \times 44=$ Rs. 22,000.
Answer:
Firstly, we need to calculate the slant height of the cone :
$l^2\ =\ r^2\ +\ h^2$
$=\ (0.7)^2\ +\ (2.4)^2$
or $l^2\ =\ 6.25$
or $l\ =\ 2.5\ cm$
Now, the total surface area of a solid can be calculated as :
The surface area of the solid = Surface area of the cylinder + Surface area of the cone + Area of the base of the cylinder
The surface area of the cylinder is $=\ 2 \pi rh$
$=\ 2 \pi \times 0.7\times 2.4$
$=\ 10.56\ cm^2$
Now, the surface area of a cone $=\ \pi rl$
$=\ \pi \times 0.7\times 2.5$
$=\ 5.50\ cm^2$
And the area of the base of the cylinder is $=\ \pi r^2$
$=\ \pi \times 0.7\times 0.7$
$=\ 1.54\ cm^2$
Thus, the required area of the solid = 10. 56 + 5.50 + 1.54 = 17.60 cm2.
Thus, the total surface area of the remaining solid to the nearest cm2 is 18 cm2.
Answer:
The required surface area is given by :
The surface area of the article = Surface area of the cylindrical part + 2 (Surface area of the hemisphere)
Now, the area of the cylinder $=\ 2\pi rh$
$=\ 2\pi \times 3.5\times 10$
$=\ 70\pi \ cm^2$
The surface area of the hemisphere $=\ 2\pi r^2$
$=\ 2\pi \times 3.5\times 3.5$
$=\ 24.5\pi\ cm^2$
Thus the required area $=70 \pi +2\times 24.5 \pi = 374 cm^2.$
Surface Areas and Volumes Class 10 Question Answers Exercise: 12.2 Page number: 169-170 Total questions: 8 |
Answer:
The volume of the solid is given by :
The volume of the solid = Volume of the cone + Volume of a hemisphere
The volume of the cone $=\ \frac{1}{3} \pi r^2h$
$=\ \frac{1}{3} \pi \times 1^2\times 1$
$=\ \frac{\pi}{3}\ cm^3$
The volume of the hemisphere $=\ \frac{2}{3}\pi r^3$
$=\ \frac{2}{3}\pi \times 1^3$
$=\ \frac{2\pi}{3}\ cm^3$
Hence, the volume of so lithe d is :
$=\ \frac{\pi}{3}\ +\ \frac{2\pi}{3}\ =\ \pi\ cm^3.$
Answer:
The volume of air present = Volume of cylinder + 2 (Volume of a cone)
Now, the volume of a cylinder: $=\ \pi r^2h$
$=\ \pi \left ( \frac{3}{2} \right )^2\times 8$
$=\ 18\pi \ cm^3$
The volume of a cone is $=\ \frac{1}{3} \pi r^2h$
$=\ \frac{1}{3} \pi \times \left ( \frac{3}{2} \right )^2\times 2$
$=\ \frac{3}{2} \pi \ cm^3$
Thus, the volume of air is $=\ 18 \pi\ +\ 2\times \frac{3}{2} \pi \ =\ 21\pi$
$=\ 66\ cm^3$
Answer:
It is clear from the figure that Gulab Jamun has one cylindrical part and two hemispherical parts.
Thus, the volume of the gulab jamun is = Volume of cylindrical part + 2 (Volume of the hemisphere )
Now, the volume of the cylinder is $=\ \pi r^2h$
$=\ \pi\times 1.4^2\times 2.2$
$=\ 13.55\ cm^3$
The volume of a hemisphere is :
$\\=\ \frac{2}{3}\pi r^3\\\\=\ \frac{2}{3}\times \pi \times (1.4)^3\\\\=\ 5.75\ cm^3$
Thus, the volume of 1 gulab jamun is $= 13.55 + 2 (5.75) = 25.05 cm^3.$
Hence, the volume of 45 gulab jamun $=\ 45(25.05)\ cm^3\ =\ 1127.25\ cm^3$
Further, it is given that one gulab jamun contains sugar syrup up to 30%.
So, the total volume of sugar present $=\ \frac{30}{100}\times 1127.25\ =\ 338\ cm^3.$
Answer:
The volume of wood is given by = volume of the cuboid - the volume of four cones.
Firstly, the volume of the cuboid $=\ lbh$
$=\ 15\times 10\times 3.5$
$=\ 525\ cm^3$
And the volume of the cone $=\frac13 \pi r^2h=\frac13 \pi \times 0.5^2 \times 1.4= 0.3665 \ cm^3$
Thus, the volume of wood is $= 525 - 4 (0.3665) = 523.53 \:cm^3.$
Answer:
According to the question :
Water spilt from the container = Volume of lead balls.
Let us assume the number of lead balls to be n.
Thus, the equation becomes :
$\frac{1}{4}\times Volume_{cone}\ =\ n\times \frac{4}{3} \pi r^3$
or $\frac{1}{4}\times \frac{1}{3}\pi\times5^2\times 8\ =\ n\times \frac{4}{3} \pi\times 0.5^3$
or $n\ =\ \frac{25\times 8}{16\times \left ( \frac{1}{2} \right )^3}$
or $n\ =\ 100$
Hence, the number of lead shots dropped is 100.
Answer:
The pole can be divided into one large cylinder and one small cylinder.
Thus, the volume of the pole = volume of a large cylinder + volume of a small cylinder
$=\ \pi r_l^2h_l\ +\ \pi r_s^2h_s$
$=\ \pi \times 12^2\times 220\ +\ \pi \times 8^2\times 60$
$=\ \pi \times \left ( 144\times 220\ +\ 64\times 60 \right )$
$=\ 3.14\times 35520$
$=\ 111532.5\ cm^3$
Now, according to the question, the mass of the pole is :
$=\ 8\times 111532.5$
$=\ 892262.4\ g\ =\ 892.262\ Kg.$
Answer:
It is clear from the question that the required volume is :
The volume of water (left) =Volume of a cylinder - Volume of a solid
Now the volume of the cylinder is $=\ \pi r^2h$
or $=\ \pi\times (60)^2\times 180\ cm^3$
The volume of the solid is :
$\\=\ \frac{1}{3} \pi r^2h\ +\ \frac{2}{3}\pi r^3\\\\=\ \frac{1}{3} \times \pi \times (60)^2\times 120\ +\ \frac{2}{3}\times \pi \times (60)^3\\\\=\ \pi (60)^2\times 80\ cm^3$
Thus, the volume of water left :
$\\=\ \pi (60)^2\times 180\ -\ \pi (60)^2\times 80\\\\=\ \pi (60)^2\times 100\\\\=\ 1131428.57\ cm^3\\\\=\ 1.131\ m^3.$
Answer:
The volume of the vessel is given by :
The volume of the vessel = Volume of the sphere + Volume of the cylindrical part
Now, the volume of the sphere is :
$\\=\ \frac{4}{3}\pi r^3\\\\=\ \frac{4}{3}\pi \left ( \frac{8.5}{2} \right )^3\\\\=\ 321.55\ cm^3$
The volume of the cylinder is:-
$\\=\ \pi r^2h\\\\=\ \pi \times (1)^2\times 8\\\\=\ 25.13\ cm^3$
Thus, the volume of the vessel is $= 321.55 + 25.13 = 346.68\: cm^3.$
Exercise-wise NCERT Solutions of Surface Areas and Volumes Class 10 Maths Chapter 12 are provided in the links below.
Question:
A cylinder of height 8 cm and radius 6 cm is melted and converted into three cones of the same radius and height as the cylinder. Determine the total curved surface area of cones.
Answer:
The radius ($r$) of each cone = radius of the cylinder = 6 cm
The height ($h$) of each cone = height of the cylinder = 8 cm
The curved surface area of a cone, where $l$ is the slant height of the cone $= \pi r l$
Now, $l = \sqrt{r^2 + h^2}= \sqrt{(6)^2 + (8)^2} = 10 \text{ cm}$
The curved surface area of a cone, where $l$ is the slant height of the cone $= \pi r l$
So, the curved surface area of each cone,
$= \pi (6) (10) = 60\pi \text{ cm}^2$
Since there are 3 cones, the total curved surface area,
$= 3 \times 60\pi = 180\pi \text{ cm}^2$
Hence, the correct answer is $180\pi \text{ cm}^2$.
Total Surface Area, denoted as TSA, signifies the entire expanse covered by an object's surface. This encompasses the collective area of all its external facets. Here are some prominent geometrical figures and their corresponding TSA formulas.
The concept of Curved Surface Area (CSA) or Lateral Surface Area (LSA) comes into focus, particularly in forms such as cylinders, cones, and pyramids. CSA represents the area of the curved component or sides, excluding the top and bottom facets. LSA, on the other hand, pertains to the lateral area of various shapes. The following are some key examples of these surface area measures:
Volume emerges as a fundamental attribute, denoting the spatial extent occupied by an object or substance, quantified in cubic units. The following volume formulas unravel the essence of these geometric measurements:
Topics you will learn in NCERT Class 10 Maths Chapter 12 Surface Areas and Volumes include:
Access all NCERT Class 10 Maths solutions from one place using the links below.
Also, read,
Given below are the subject-wise exemplar solutions of class 10 NCERT:
Here are some useful links for NCERT books and the NCERT syllabus for class 10:
Frequently Asked Questions (FAQs)
The concepts of surface areas and volumes are applied widely in real life. In construction, architects and engineers use these principles to calculate the amount of material needed for building structures, such as the walls or roofs of buildings. In the packaging industry, companies use surface area formulas to design packaging materials that minimise waste while ensuring the container fits the product properly.
To find the surface area of a combination of solids, we require breaking the complex solid into simpler shapes. Like, if we have a cone placed on top of a hemisphere, we first calculate the surface area of the cone(excluding its base) and then calculate the surface area of the hemisphere. Once we have the surface areas of the individual solids, we can simply add them together to find the total surface area. We have to carefully exclude any overlapping areas, like the base of the cone that touches the hemisphere, to avoid double-counting.
Important formulas for this chapter are:
Surface Area and Volume of Sphere, Hemisphere, Cone, Cylinder, and Frustum.
A combination of solids (like a cone on a cylinder).
Surface Areas and Volumes is a high-scoring chapter in board exams. Many application-based and word problems are asked in exams from this chapter, making it crucial for both conceptual understanding and practical use.
In board exams, questions from Chapter 12 typically focus on calculating the surface area and volume of solids. These can include both direct calculation problems, where you simply apply the formulas, and more complex word problems that involve real-life situations, such as finding the amount of material needed to cover a tank or determining the volume of water a container can hold. There may also be questions involving the combination of solids, where you need to calculate the surface area or volume of an object made up of more than one solid. These questions test both your conceptual understanding and problem-solving skills.
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