{"slug":"cbse-class-10-maths-surface-areas-and-volumes","title":"Surface Areas and Volumes","description":"Surface areas and volumes of solids made by combining cuboids, cylinders, cones, spheres and hemispheres, weighted towards the cone-and-hemisphere toy and cone-volume questions set across the 2026 papers.","board":"CBSE","grade":"Class 10","subject":"Maths","chapter":"Surface Areas and Volumes","url":"https://www.flipnlearn.app/deck/cbse-class-10-maths-surface-areas-and-volumes","studyUrl":"https://www.flipnlearn.app/deck/cbse-class-10-maths-surface-areas-and-volumes/study","cards":[{"id":"surface-area-of-a-combination-of-solids","kind":"concept-question","front":"Why is the total surface area of a combined solid usually not the sum of the total surface areas of its parts?","back":"When two solids are joined, the faces pressed together are hidden and no longer part of the outside surface. Only the exposed surfaces count, so a top made of a cone on a hemisphere has area CSA of hemisphere + CSA of cone."},{"id":"volume-of-a-combination-of-solids","kind":"concept-question","front":"Why is the volume of a combined solid simply the sum of the volumes of its parts?","back":"Joining solids hides some surface, but no space inside either solid is lost. So, unlike surface area, the volume of the new solid is the total of the constituent volumes."},{"id":"curved-surface-area-of-a-hemisphere","kind":"formula","front":"Curved surface area of a hemisphere","back":"2πr², which is half of the sphere's surface area 4πr². If the flat circular face is exposed too, add its area πr²."},{"id":"slant-height-of-a-cone","kind":"formula","front":"Slant height of a cone","back":"l = √(r² + h²), where r is the base radius and h the vertical height. The radius, height and slant height form a right triangle."},{"id":"curved-surface-area-of-a-cone","kind":"formula","front":"Curved surface area of a cone","back":"πrl, where r is the base radius and l the slant height."},{"id":"curved-surface-area-of-a-cylinder","kind":"formula","front":"Curved surface area of a cylinder","back":"2πrh, where r is the base radius and h the height. Add πr² for each circular end that is part of the surface."},{"id":"surface-area-of-a-cube","kind":"formula","front":"Surface area of a cube","back":"6 × (edge)², since a cube has six equal square faces."},{"id":"volume-of-a-cylinder","kind":"formula","front":"Volume of a cylinder","back":"πr²h, where r is the base radius and h the height. Half a cylinder has volume ½πr²h."},{"id":"volume-of-a-cone","kind":"formula","front":"Volume of a cone","back":"⅓πr²h, one-third of the volume of a cylinder with the same base radius and height."},{"id":"volume-of-a-hemisphere","kind":"formula","front":"Volume of a hemisphere","back":"⅔πr³, half the volume of a sphere of the same radius."},{"id":"how-to-find-the-surface-area-of-a-top","kind":"worked-step","front":"A top is a cone on a hemisphere, 5 cm tall overall with diameter 3.5 cm. How do you find the area to colour?","back":"The area is CSA of hemisphere + CSA of cone. The hemisphere's radius 1.75 cm is part of the height, so the cone is 5 − 1.75 = 3.25 cm tall and its slant height is about 3.7 cm. Then 2πr² + πrl ≈ 39.6 cm²."},{"id":"how-to-find-the-surface-area-of-a-cube-with-a-hemisphere","kind":"worked-step","front":"A hemisphere of diameter 4.2 cm sits on a cube of edge 5 cm. How do you find the block's total surface area?","back":"Start with the cube's 150 cm², remove the circle covered by the hemisphere (πr²) and add the hemisphere's curved surface (2πr²). The net change is +πr², giving 150 + 13.86 = 163.86 cm²."},{"id":"how-to-find-painted-areas-of-a-toy-rocket","kind":"worked-step","front":"A rocket is a cone of base radius 2.5 cm on a narrower cylinder of radius 1.5 cm. How do you work out the cone's painted area?","back":"Paint the cone's curved surface plus the ring of its base left uncovered by the cylinder: πrl + πr² − πr′². With slant height 6.5 cm this is 3.14 × 20.25 ≈ 63.585 cm²."},{"id":"how-to-find-the-surface-area-of-a-bird-bath","kind":"worked-step","front":"A bird-bath is a cylinder of height 1.45 m and radius 30 cm with a hemispherical hollow at the top. How do you find its total surface area?","back":"Only the cylinder's curved surface and the inside of the hemispherical hollow are counted: 2πrh + 2πr² = 2πr(h + r). With r = 30 cm and h = 145 cm this is 33000 cm², or 3.3 m²."},{"id":"how-to-find-the-air-in-a-shed","kind":"worked-step","front":"A shed is a 15 m × 7 m × 8 m cuboid topped by a half cylinder. How do you find the air it holds with machinery and workers inside?","back":"Add the cuboid's volume to half the cylinder's (diameter 7 m, length 15 m) to get 1128.75 m³. Then subtract the space taken by the machinery (300 m³) and the workers (1.6 m³), leaving 827.15 m³."},{"id":"how-to-find-the-actual-capacity-of-a-glass","kind":"worked-step","front":"A cylindrical glass has a raised hemispherical bottom. How do you find its actual capacity?","back":"Its apparent capacity is the cylinder's volume, πr²h, here 196.25 cm³. The raised hemisphere takes up ⅔πr³, about 32.71 cm³, so the actual capacity is the difference, 163.54 cm³."},{"id":"how-to-compare-a-toy-with-its-circumscribing-cylinder","kind":"worked-step","front":"A toy is a cone of height 2 cm on a hemisphere of radius 2 cm. How do you find the space between it and a cylinder that just encloses it?","back":"Toy volume = ⅔πr³ + ⅓πr²h = 25.12 cm³. The enclosing cylinder has radius 2 cm and height 2 + 2 = 4 cm, so its volume is 3.14 × 4 × 4 = 50.24 cm³. The difference is 25.12 cm³."},{"id":"why-two-joined-cubes-lose-surface-area","kind":"concept-question","front":"Two equal cubes of edge a are joined face to face into a cuboid. Why is its surface area 10a² rather than 12a²?","back":"Each cube has 6 faces of area a², 12 in all, but the two faces pressed together are hidden inside the cuboid. That leaves 10 exposed faces, so the surface area is 10a²."}]}