Which surfaces count in a combination
When you stack a ball on a box, the face where they touch is hidden from the paint, so it is not painted.
In a solid made by joining solids, the total surface area (TSA) is the sum of the outer surfaces only. The area where the solids are joined is not counted at all. Volumes just add, but surface areas do not.
Formula: TSA = sum of outer surfaces
In the figure: Cube 14 with a hemisphere r = 7 on top: 1176 − 154 + 308 = 1330.
Rule 1: Remove the joined part — Subtract the touching area from each solid.
- Easy: Cube 14 + hemisphere r = 7: the cube loses πr² = 154.
- Medium: The hemisphere adds only its curved part 2πr² = 308.
- Exam-type: TSA = 1176 − 154 + 308 = 1330.
Rule 2: Pick the right piece of each solid — Curved surface, flat surface or total, depending on what is visible.
- Easy: A hemisphere on a surface shows only its curved part 2πr².
- Medium: A cone on a cylinder shows the cone’s curved surface πrl.
- Exam-type: An open vessel has no top lid, so no top face.
Before adding, list every outer surface and tick only those you can see.
Adding the total surface area of each solid, including the hidden joined part.
"See it, count it; hidden, skip it."
Practice questions
- 1A capsule has radius r = 15 cm and, where relevant, cylindrical height h = 8 cm and slant height l. Which expression gives its outer surface area?
- 2A test tube (cylinder with a hemispherical bottom, open at the top) has radius r = 20 cm and, where relevant, cylindrical height h = 9 cm and slant height l. Which expression gives its outer surface area?
- 3A test tube (cylinder with a hemispherical bottom, open at the top) has radius r = 3 cm and, where relevant, cylindrical height h = 8 cm and slant height l. Which expression gives its outer surface area?
- 4A test tube (cylinder with a hemispherical bottom, open at the top) has radius r = 15 cm and, where relevant, cylindrical height h = 19 cm and slant height l. Which expression gives its outer surface area?
- 5A tent (cone on a cylinder) has radius r = 15 cm and, where relevant, cylindrical height h = 25 cm and slant height l. Which expression gives its outer surface area?
- 6A cone has base radius 8 cm and height 15 cm. Find its slant height (in cm).
- 7A cone has base radius 20 cm and height 15 cm. Find its slant height (in cm).
- 8A cone has base radius 48 cm and slant height 50 cm. Find its height (in cm).
Show answers (try them yourself first)
- 12πrh + 4πr²
- 22πrh + 2πr²
- 32πrh + 2πr²
- 42πrh + 2πr²
- 52πrh + πrl
- 617
- 725
- 814
Cylinder with hemispheres (capsule, vessel)
A medicine capsule is a cylinder with a hemisphere on each end; a round-bottomed pot is a cylinder with one hemisphere.
The cylinder shows only its curved surface 2πrh, because its circular ends are joined to hemispheres. Each hemisphere gives its curved surface 2πr². The total length of a capsule is h + 2r.
Formula: TSA = 2πrh + 2 × 2πr²
In the figure: Capsule of length 24 and r = 7: h = 10, TSA = 440 + 616 = 1056.
Cylinder with hemispheres (capsule, vessel)
In the app you can move this figure with your finger.
Open in the appRule 1: Capsule — Subtract the two radii from the total length to get h.
- Easy: Length 24, r = 7: h = 24 − 14 = 10.
- Medium: Curved part of cylinder 2 × 22/7 × 7 × 10 = 440.
- Exam-type: Two hemispheres 2 × 308 = 616; TSA = 1056 cm².
Rule 2: Open vessel with a hemispherical bottom — Only one hemisphere and no top lid.
- Easy: r = 7, h = 10: outer surface = 440 + 308 = 748 cm².
- Medium: Tinning at ₹0.5 per cm²: 748 × 0.5 = ₹374.
- Exam-type: Top rim is not covered, so πr² is not added.
Check the capsule: curved parts only, never the two circular joins.
Taking the total length 24 as the cylinder height instead of 24 − 2r.
"Cylinder side plus two caps."
Practice questions
- 1A capsule is a cylinder of radius 9 cm and height 7 cm with a hemisphere of the same radius attached at each end. Find its surface area (in cm², in terms of π).
- 2A capsule is a cylinder of radius 2 cm and height 25 cm with a hemisphere of the same radius attached at each end. Find its surface area (in cm², in terms of π).
- 3A capsule is a cylinder of radius 4 cm and height 20 cm with a hemisphere of the same radius attached at each end. Find its surface area (in cm², in terms of π).
- 4A capsule is a cylinder of radius 1 cm and height 11 cm with a hemisphere of the same radius attached at each end. Find its surface area (in cm², in terms of π).
- 5A capsule is a cylinder of radius 4 cm and height 14 cm with a hemisphere of the same radius attached at each end. Find its surface area (in cm², in terms of π).
- 6A capsule is made of a cylinder with a hemisphere at each end. Its total length is 36 mm and its diameter is 6 mm. Find its surface area (in mm², in terms of π).
- 7A capsule is made of a cylinder with a hemisphere at each end. Its total length is 15 mm and its diameter is 12 mm. Find its surface area (in mm², in terms of π).
- 8A capsule is made of a cylinder with a hemisphere at each end. Its total length is 48 mm and its diameter is 18 mm. Find its surface area (in mm², in terms of π).
Show answers (try them yourself first)
- 1450π
- 2116π
- 3224π
- 426π
- 5176π
- 6216π
- 7180π
- 8864π
Cone with hemisphere or cylinder (toy, tent)
A spinning top is a cone sitting on a hemisphere; a circus tent is a cone on a cylinder.
The cone shows its curved surface πrl where l = √(r² + h²) is the slant height. A hemisphere below shows 2πr². A cylinder below shows 2πrh, and a tent has no floor.
Formula: l = √(r² + h²); TSA = πrl + 2πr²
In the figure: Cone h = 24, r = 7: l = 25. Toy TSA = 550 + 308 = 858.
Cone with hemisphere or cylinder (toy, tent)
In the app you can move this figure with your finger.
Open in the appRule 1: Toy: cone on a hemisphere — Find l first by Pythagoras.
- Easy: r = 7, h = 24: l = √(49 + 576) = 25.
- Medium: Cone πrl = 22/7 × 7 × 25 = 550.
- Exam-type: Hemisphere 2πr² = 308; TSA = 858 cm².
Rule 2: Tent: cone on a cylinder — Canvas needed is the curved surfaces only.
- Easy: Cylinder r = 7, h = 10: 2πrh = 440.
- Medium: Cone r = 7, h = 24: πrl = 550.
- Exam-type: Canvas = 440 + 550 = 990 m².
Use the slant height l in πrl, never the straight height h.
Using πrh for the cone’s curved surface.
"Cone side uses the slant."
Practice questions
- 1A toy is a cone of base radius 24 cm and height 10 cm placed on a hemisphere of the same radius. Find its total surface area (in cm², in terms of π).
- 2A toy is a cone of base radius 12 cm and height 5 cm placed on a hemisphere of the same radius. Find its total surface area (in cm², in terms of π).
- 3A toy is a cone of base radius 12 cm and height 16 cm placed on a hemisphere of the same radius. Find its total surface area (in cm², in terms of π).
- 4A toy is a cone of base radius 16 cm and height 30 cm placed on a hemisphere of the same radius. Find its total surface area (in cm², in terms of π).
- 5A toy is a cone surmounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm. Find its total surface area (in cm², in terms of π).
- 6A toy is a cone surmounted on a hemisphere of radius 15 cm. The total height of the toy is 51 cm. Find its total surface area (in cm², in terms of π).
- 7A toy is a cone surmounted on a hemisphere of radius 21 cm. The total height of the toy is 41 cm. Find its total surface area (in cm², in terms of π).
- 8A toy is a cone of base radius 20 cm and height 15 cm on a hemisphere of radius 20 cm. Find the ratio of the curved surface area of the cone to that of the hemisphere.
Show answers (try them yourself first)
- 11776π
- 2444π
- 3528π
- 41056π
- 5273π
- 61035π
- 71491π
- 85 : 8
Cube or cuboid with a hemisphere
A paperweight is a block with a dome on top; a hemispherical dent in a block is a scoop.
A hemisphere of radius r sitting on a flat face hides a circle πr² of the block and adds its curved surface 2πr². The net gain is πr². A hemispherical dent removes πr² and adds 2πr², so it also gains πr².
Formula: TSA = original + πr²
In the figure: A bump and a dent of the same size give the same surface area.
Rule 1: Cube with a hemisphere — Original surface plus πr².
- Easy: Cube 10, r = 5: 600 + 78.5 = 678.5 cm² (π = 3.14).
- Medium: Cube 14, r = 7: 1176 + 154 = 1330 cm².
- Exam-type: The hemisphere’s diameter cannot exceed the side.
Rule 2: Cuboid with a hemisphere — Find the cuboid’s 2(lb + bh + lh) first.
- Easy: Cuboid 15 × 10 × 5: 2(150 + 50 + 75) = 550.
- Medium: Hemisphere r = 5 on top: add πr² = 78.5.
- Exam-type: TSA = 628.5 cm².
Do the cube alone first, then add the net πr².
Adding 2πr² without removing the πr² of the block that is covered.
"Cover one circle, add a dome: net one circle."
Practice questions
- 1A hemisphere of the greatest possible diameter is placed on a cube of side 10 cm. Find the surface area of the solid (in cm², in terms of π).
- 2Two cubes, each of volume 64 cm³, are joined end to end to form a cuboid. Find the surface area of the cuboid (in cm²).
- 3Two cubes, each of volume 1728 cm³, are joined end to end to form a cuboid. Find the surface area of the cuboid (in cm²).
- 4A hemispherical depression of radius 11 cm is cut out from one face of a cubical block of side 26 cm. Find the surface area of the remaining block (in cm², in terms of π).
- 5A solid is a cuboid with a square base of side 6 cm and height 12 cm, with a hemisphere of diameter 6 cm on its top face. Find its total surface area (in cm², in terms of π).
- 6A hemispherical depression of radius 5 cm is cut out from one face of a cubical block of side 24 cm. Find the surface area of the remaining block (in cm², in terms of π).
- 7A hemispherical depression of radius 7 cm is cut out from one face of a cubical block of side 18 cm. Find the surface area of the remaining block (in cm², in terms of π).
- 8A solid is a cuboid with a square base of side 4 cm and height 20 cm, with a hemisphere of diameter 4 cm on its top face. Find its total surface area (in cm², in terms of π).
Show answers (try them yourself first)
- 125π + 600
- 2160
- 31440
- 4121π + 4056
- 59π + 360
- 625π + 3456
- 749π + 1944
- 84π + 352
Cavities and applications
A wooden toy with a scooped hollow has both the outside and the inside of the hollow to polish.
When a hemisphere is scooped out of a solid, the old flat top disappears and the inner curved surface 2πr² appears. Painting cost = area × rate per unit area.
Formula: cost = area × rate
In the figure: Cylinder r = 7, h = 14 with a hemispherical hollow: 616 + 154 + 308 = 1078 cm².
Rule 1: Scooped cylinder — Curved side, flat bottom and inner hollow.
- Easy: Outer side 2πrh = 616.
- Medium: Bottom πr² = 154 and hollow 2πr² = 308.
- Exam-type: Total = 1078 cm²; the top ring is zero since r equals the radius.
Rule 2: Cost of painting — Multiply the total area by the rate.
- Easy: 1078 cm² at ₹0.5 per cm² costs ₹539.
- Medium: Use the same unit for area and rate.
- Exam-type: A plain cylinder 616 + 2 × 154 = 924 has less area than the scooped one.
List the surfaces: which are outside, which are inside the hollow, which are gone?
Forgetting the inner surface of the hollow.
"Gone on top, new inside."
Practice questions
- 1From a solid cylinder of height 30 cm and base radius 16 cm, a conical cavity of the same height and the same base radius is hollowed out. Find the total surface area of the remaining solid (in cm², in terms of π).
- 2A test tube is a cylinder of radius 3 cm and height 3 cm with a hemispherical bottom of the same radius, open at the top. Find the outer surface area (in cm², in terms of π).
- 3A test tube is a cylinder of radius 1 cm and height 4 cm with a hemispherical bottom of the same radius, open at the top. Find the outer surface area (in cm², in terms of π).
- 4From a solid cylinder of height 15 cm and base radius 8 cm, a conical cavity of the same height and the same base radius is hollowed out. Find the total surface area of the remaining solid (in cm², in terms of π).
- 5A test tube is a cylinder of radius 6 cm and height 7 cm with a hemispherical bottom of the same radius, open at the top. Find the outer surface area (in cm², in terms of π).
- 6A solid cylinder has base radius 16 cm and height 30 cm. A conical cavity of the same base and height is hollowed out. By how much does the total surface area of the solid change (increase) in cm², in terms of π?
- 7A wooden article is made by scooping out a hemisphere of radius 2 cm from each end of a solid cylinder of height 12 cm and base radius 2 cm. Find its total surface area (in cm², in terms of π).
- 8A solid cylinder has base radius 14 cm and height 48 cm. A conical cavity of the same base and height is hollowed out. By how much does the total surface area of the solid change (increase) in cm², in terms of π?
Show answers (try them yourself first)
- 11760π
- 236π
- 310π
- 4440π
- 5156π
- 6288π
- 764π
- 8504π