Explore the concepts of calculating surface areas and volumes of various geometric shapes, including cubes, cylinders, cones, and spheres, to solve real-world problems.
Surface Areas and Volumes - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Surface Areas and Volumes from Mathematics for Class X (Mathematics).
Basic comprehension exercises
Strengthen your understanding with fundamental questions about the chapter.
Questions
Explain how to find the total surface area of a combination of a cylinder and two hemispheres attached at its ends, like a water tanker. Provide a real-life example where this calculation is useful.
Remember that the total surface area includes only the curved parts of the hemispheres and the cylinder, not the base areas since they are attached and not exposed.
Solution
To find the total surface area of a combination of a cylinder and two hemispheres, first calculate the curved surface area (CSA) of the cylinder and the CSA of the two hemispheres. The CSA of the cylinder is given by 2πrh, where r is the radius and h is the height of the cylinder. The CSA of one hemisphere is 2πr², so for two hemispheres, it's 4πr². The total surface area (TSA) is the sum of the CSA of the cylinder and the two hemispheres: TSA = 2πrh + 4πr². This calculation is useful in real life for determining the amount of material needed to manufacture a water tanker or to paint its surface.
Describe the steps to calculate the volume of a toy shaped like a cone surmounted on a hemisphere. Include the formulas used and explain each step.
The radius of the cone and the hemisphere must be equal for the toy to be properly formed.
Solution
To calculate the volume of a toy shaped like a cone surmounted on a hemisphere, first find the volume of the hemisphere and the cone separately. The volume of a hemisphere is (2/3)πr³, and the volume of a cone is (1/3)πr²h, where r is the radius and h is the height of the cone. Add these two volumes to get the total volume of the toy: Total Volume = (2/3)πr³ + (1/3)πr²h. Ensure that the radius of the cone and the hemisphere are the same for the toy to have a smooth surface.
A decorative block is made of a cube with a hemisphere attached on top. Explain how to find its total surface area, considering the hemisphere's base is not part of the surface.
The base of the hemisphere replaces part of the cube's surface, so it's subtracted from the cube's total surface area.
Solution
The total surface area of the decorative block is the sum of the surface area of the cube minus the area where the hemisphere is attached, plus the curved surface area of the hemisphere. First, calculate the surface area of the cube: 6 × (edge)². Subtract the base area of the hemisphere (πr²) from the cube's surface area since it's covered by the hemisphere. Then, add the curved surface area of the hemisphere (2πr²). The total surface area is: TSA = 6 × (edge)² - πr² + 2πr² = 6 × (edge)² + πr².
How would you determine the volume of air inside a shed shaped like a cuboid surmounted by a half-cylinder? Include the steps and formulas used.
The half-cylinder's diameter is equal to the breadth or length of the cuboid, depending on how it's attached.
Solution
To determine the volume of air inside the shed, calculate the volume of the cuboid and the half-cylinder separately and then add them. The volume of the cuboid is length × breadth × height. The volume of a full cylinder is πr²h, so the volume of a half-cylinder is (1/2)πr²h. The total volume of air inside the shed is the sum of the cuboid's volume and the half-cylinder's volume: Total Volume = (length × breadth × height) + (1/2)πr²h. Ensure that the dimensions are consistent and the radius of the half-cylinder matches the cuboid's breadth or length where it's attached.
A medicine capsule is shaped like a cylinder with two hemispheres at each end. Explain how to find its total surface area, including the formulas and steps.
The total surface area only includes the curved parts of the cylinder and hemispheres.
Solution
The total surface area of the medicine capsule includes the curved surface area of the cylinder and the curved surface areas of the two hemispheres. The CSA of the cylinder is 2πrh, and the CSA of one hemisphere is 2πr², so for two hemispheres, it's 4πr². The total surface area is the sum of these: TSA = 2πrh + 4πr². There's no need to add the base areas of the hemispheres because they are attached to the cylinder and not exposed.
Describe how to calculate the volume of a solid formed by hollowing out a cone from a cylinder of the same height and diameter. What is the remaining solid's volume?
The cone and cylinder share the same radius and height in this scenario.
Solution
To calculate the volume of the remaining solid after hollowing out a cone from a cylinder, first find the volume of the cylinder and the cone. The volume of the cylinder is πr²h, and the volume of the cone is (1/3)πr²h. Subtract the volume of the cone from the volume of the cylinder to get the remaining volume: Remaining Volume = πr²h - (1/3)πr²h = (2/3)πr²h. This gives the volume of the solid left after removing the cone.
Explain the process to find the total surface area of a wooden article made by scooping out a hemisphere from each end of a solid cylinder. Include all necessary formulas.
Only the curved surfaces contribute to the total surface area after scooping out the hemispheres.
Solution
To find the total surface area of the wooden article, calculate the curved surface area of the cylinder and the curved surface areas of the two hemispheres. The CSA of the cylinder is 2πrh, and the CSA of one hemisphere is 2πr², so for two hemispheres, it's 4πr². The total surface area is the sum of these: TSA = 2πrh + 4πr². The base areas of the hemispheres are not included because they are scooped out from the cylinder's ends and are not part of the external surface.
A gulab jamun is shaped like a cylinder with two hemispherical ends. How would you calculate its total volume? Provide the steps and formulas.
The cylinder and hemispheres must have the same radius for the gulab jamun to be properly formed.
Solution
To calculate the total volume of the gulab jamun, find the volume of the cylindrical part and the two hemispherical ends separately and then add them. The volume of the cylinder is πr²h, and the volume of one hemisphere is (2/3)πr³, so for two hemispheres, it's (4/3)πr³. The total volume is the sum of the cylinder's volume and the two hemispheres' volumes: Total Volume = πr²h + (4/3)πr³. Ensure that the radius of the cylinder and the hemispheres are the same for a smooth shape.
A tent is shaped like a cylinder surmounted by a cone. Describe how to find the area of the canvas needed to make the tent, excluding the base.
The slant height of the cone can be found using the Pythagorean theorem if the height and radius are known.
Solution
To find the area of the canvas needed for the tent, calculate the curved surface area of the cylindrical part and the conical part. The CSA of the cylinder is 2πrh, and the CSA of the cone is πrl, where l is the slant height of the cone. The total canvas area is the sum of these two areas: Total Canvas Area = 2πrh + πrl. The base of the tent is not covered with canvas, so it's excluded from the calculation.
A solid toy is in the form of a hemisphere surmounted by a cone. Explain how to find its volume and the difference in volumes if a cylinder circumscribes the toy.
The height of the cylinder is equal to the sum of the cone's height and the hemisphere's radius.
Solution
To find the volume of the toy, calculate the volume of the hemisphere and the cone separately and add them. The volume of the hemisphere is (2/3)πr³, and the volume of the cone is (1/3)πr²h. The total volume of the toy is (2/3)πr³ + (1/3)πr²h. If a cylinder circumscribes the toy, its volume is πr²H, where H is the height of the cylinder (sum of the cone's height and the hemisphere's radius). The difference in volumes is the cylinder's volume minus the toy's volume: Difference = πr²H - [(2/3)πr³ + (1/3)πr²h].
Surface Areas and Volumes - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Surface Areas and Volumes to prepare for higher-weightage questions in Class X Mathematics.
Intermediate analysis exercises
Deepen your understanding with analytical questions about themes and characters.
Questions
A toy is made by mounting a cone on a hemisphere of the same radius. The total height of the toy is 15.5 cm and the radius of the hemisphere is 3.5 cm. Calculate the total surface area of the toy.
Remember to subtract the base area of the cone from the total surface area since it's not part of the external surface.
Solution
The total surface area of the toy is the sum of the curved surface area of the hemisphere and the curved surface area of the cone. First, calculate the slant height of the cone using the Pythagorean theorem. Then, calculate the CSA of the hemisphere and the cone separately and add them together.
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.
The hemisphere's radius is half of its diameter. The cylinder's height is the total height minus the hemisphere's radius.
Solution
The inner surface area includes the curved surface area of the hemisphere and the curved surface area of the cylinder. Calculate the height of the cylinder by subtracting the radius of the hemisphere from the total height. Then, calculate the CSA of both parts and add them together.
A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.
Volume of hemisphere = (2/3)πr³, Volume of cone = (1/3)πr²h. Here, r = h = 1 cm.
Solution
The volume of the solid is the sum of the volume of the hemisphere and the volume of the cone. Use the formulas for the volume of a hemisphere and a cone, then add them together.
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.
The total length includes the lengths of both hemispheres and the cylinder. The height of the cylinder is total length minus twice the radius of the hemispheres.
Solution
The surface area of the capsule is the sum of the curved surface areas of the two hemispheres and the curved surface area of the cylinder. First, find the radius and the height of the cylindrical part, then calculate the areas accordingly.
A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. 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.
The total surface area is the CSA of the cylinder plus twice the CSA of a hemisphere (since there are two hemispheres).
Solution
The total surface area includes the curved surface area of the cylinder and the curved surface areas of the two hemispheres. Since the hemispheres are scooped out from the ends, their curved surface areas are added to the cylinder's CSA.
A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8g mass.
Volume of a cylinder = πr²h. Convert diameters to radii before calculation.
Solution
First, calculate the volumes of both cylinders. Then, add them to get the total volume. Multiply the total volume by the mass per cm³ to find the total mass.
A gulab jamun contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.
The total length includes the cylinder and both hemispheres. The height of the cylinder is total length minus twice the radius of the hemispheres.
Solution
First, calculate the volume of one gulab jamun by adding the volumes of the two hemispheres and the cylinder. Then, find 30% of this volume for the syrup in one gulab jamun. Multiply by 45 to get the total syrup volume.
A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.
Volume of cuboid = length × breadth × height. Subtract the total volume of the four cones from this to get the volume of wood.
Solution
Calculate the volume of the cuboid and subtract the volumes of the four conical depressions. The volume of each cone is (1/3)πr²h.
A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.
Volume of cone = (1/3)πr²h. Volume of sphere = (4/3)πr³. Number of lead shots = (Volume of displaced water) / (Volume of one lead shot).
Solution
First, calculate the volume of the cone. Then, find one-fourth of this volume, which is the volume of water displaced by the lead shots. Calculate the volume of one lead shot and divide the displaced volume by this to find the number of lead shots.
A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.
Volume of cylinder = πr²h. Volume of cone = (1/3)πr²h. Volume of hemisphere = (2/3)πr³. Remaining water = Volume of cylinder - (Volume of cone + Volume of hemisphere).
Solution
First, calculate the volume of the cylinder. Then, calculate the volumes of the cone and the hemisphere. Subtract the sum of these volumes from the cylinder's volume to find the remaining water volume.
Surface Areas and Volumes - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Surface Areas and Volumes in Class X.
Advanced critical thinking
Test your mastery with complex questions that require critical analysis and reflection.
Questions
A toy is made by mounting a cone on a hemisphere of the same radius. The total height of the toy is 15.5 cm and the radius is 3.5 cm. Calculate the total surface area of the toy.
Remember to calculate the slant height of the cone using the given height and radius.
Solution
The total surface area of the toy includes the curved surface area of the cone and the hemisphere. The slant height of the cone can be found using the Pythagorean theorem. The curved surface area of the hemisphere is half of the sphere's surface area. Sum these areas to get the total surface area.
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.
Calculate the height of the cylindrical part first.
Solution
The inner surface area includes the curved surface area of the hemisphere and the cylinder. The height of the cylinder can be found by subtracting the radius of the hemisphere from the total height.
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.
The hemisphere sits on top of the cube, so its diameter matches the cube's side length.
Solution
The greatest diameter of the hemisphere is equal to the side of the cube. The surface area of the solid is the total surface area of the cube minus the base area of the hemisphere plus the curved surface area of the hemisphere.
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 is 5 mm. Find its surface area.
Calculate the length of the cylindrical part by subtracting the sum of the radii of the hemispheres from the total length.
Solution
The surface area includes the curved surface areas of the two hemispheres and the cylinder. The length of the cylindrical part is the total length minus the diameters of the two hemispheres.
A tent is in the shape of a cylinder surmounted by a conical top. The height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m. Find the area of the canvas used for making the tent.
The base of the tent is not covered with canvas, so do not include the base area in your calculations.
Solution
The canvas area includes the curved surface area of the cylinder and the cone. The radius of the cone is the same as the cylinder's. Use the slant height to find the cone's curved surface area.
From a solid cylinder of height 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and diameter is hollowed out. Find the total surface area of the remaining solid.
Consider both the outer and inner surfaces in your calculations.
Solution
The remaining solid's surface area includes the outer surface of the cylinder, the inner surface of the conical cavity, and the base area of the cylinder minus the base area of the cone.
A wooden article is made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm and its base radius is 3.5 cm, find the total surface area of the article.
The hemispheres are scooped out from the ends, so their curved surfaces are part of the article's surface.
Solution
The total surface area includes the curved surface area of the cylinder and the two hemispheres. The base areas of the hemispheres are not part of the external surface.
A juice seller serves customers using glasses shaped like a cylinder with a hemispherical depression at the bottom. The inner diameter is 5 cm and the height is 10 cm. Find the apparent and actual capacity of the glass.
The actual capacity is the volume of the cylinder minus the volume of the hemisphere.
Solution
The apparent capacity is the volume of the cylinder. The actual capacity is less due to the hemispherical depression. Subtract the hemisphere's volume from the cylinder's volume to find the actual capacity.
A solid toy is in the form of a hemisphere surmounted by a cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy.
The radius of the hemisphere and the cone is the same.
Solution
The volume of the toy is the sum of the volumes of the hemisphere and the cone. Use the given dimensions to calculate each volume separately.
A gulab jamun is shaped like a cylinder with two hemispherical ends. Each gulab jamun has a length of 5 cm and a diameter of 2.8 cm. Approximately how much syrup would be found in 45 gulab jamuns if they contain syrup up to 30% of their volume?
First find the volume of the entire gulab jamun, then calculate 30% of that volume for the syrup.
Solution
Calculate the total volume of one gulab jamun, including both hemispherical ends and the cylindrical part. Then find 30% of this volume for the syrup content per gulab jamun. Multiply by 45 for the total syrup volume.
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