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Worksheet
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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