Surface Areas and Volumes
Real objects are often combinations of simple solids — a capsule is a cylinder with two hemispherical ends, a test tube is a cylinder with one hemispherical end. Break the shape into its simple parts, then add or subtract as needed.
1. Key Formulas
| Solid | Volume | Curved/Total Surface Area |
|---|---|---|
| Cylinder | πr²h | CSA=2πrh, TSA=2πr(r+h) |
| Cone | (1/3)πr²h | CSA=πrl, TSA=πr(r+l) |
| Sphere | (4/3)πr³ | 4πr² |
| Hemisphere | (2/3)πr³ | CSA=2πr², TSA=3πr² |
(l is the slant height of a cone; note that combined-solid problems only add curved surface areas at the joins, never the flat circular faces where two solids meet.)
Each hemisphere contributes 7 cm to the length, so the cylindrical part has height 31−14=17 cm.
Volume of cylinder = πr²h = (22/7)(49)(17) = 2618 cm³
The two hemispheres together make one full sphere: Volume = (4/3)πr³ = (4/3)(22/7)(343) ≈ 1437.33 cm³
Total volume ≈ 2618 + 1437.33 = 4055.33 cm³
Volume of cylinder = πr²h = (22/7)(25)(12) ≈ 942.86 cm³. Volume of cone = (1/3) of that ≈ 314.29 cm³.
Remaining volume ≈ 942.86 − 314.29 = 628.57 cm³
Volume is conserved: (1/3)πr²h(cone) = πR²H(cylinder) → (1/3)(36)(8) = (16)H → 96 = 16H
H = 6 cm
- Adding the flat circular face area where two solids join in a combined shape — that face is hidden inside the solid and isn’t part of the outer surface.
- Forgetting that a cone’s TSA needs the slant height l = √(r²+h²), not the vertical height h.
- Confusing surface area formulas (which have r² for sphere-type shapes) with volume formulas (which have r³).
Key Vocabulary Recap
Syllabus Reference
Practice questions
Surface Areas and Volumes — Practice
Covers: cylinders, cones, spheres, hemispheres, and their combinations. 10 questions. (Use π = 22/7 unless stated otherwise.)
A cylinder and a cone have the same base radius 7 cm and height 24 cm. Find the volume of the cone.
Find the total surface area of a hemisphere of radius 7 cm.
A solid is a cylinder with a hemisphere attached at each end, radius 7 cm, total length 31 cm. Find its volume.
A cone of radius 5 cm and height 12 cm is carved out of a solid cylinder of the same radius and height. Find the volume of the remaining solid.
Find the volume of a sphere of radius 10.5 cm.
A cube of side 4 cm is melted and recast into a sphere. Find the radius of the sphere.
Find the curved surface area of a cone with radius 7 cm and slant height 25 cm.
Find the total surface area of a right circular cylinder with radius 3.5 cm and height 10 cm.
A hemispherical bowl of radius 9 cm, full of liquid, is poured into cylindrical bottles of radius 3 cm and height 4 cm. How many bottles are needed?
A conical vessel of radius 6 cm and height 8 cm is full of water. The water is poured into a cylindrical vessel of radius 4 cm. Find the height to which the water rises.
More Practice
A cubical block of side 10 cm is surmounted by a hemisphere of the largest possible size. Find the surface area of the resulting solid.
A medicine capsule is a cylinder with two hemispheres stuck to each end, total length 14 mm, diameter 5 mm. Find its surface area.
A solid cone (radius 3 cm, height 3 cm) sits on a hemisphere (radius 3 cm), placed inside a cylinder of radius 3 cm and height 6 cm, full of water, touching the bottom. Find the volume of water left in the cylinder.
A tent is a cylinder surmounted by a cone. The cylindrical part has height 2.1 m and diameter 4 m; the cone’s slant height is 2.8 m. Find the area of canvas used.
A solid toy is a hemisphere surmounted by a cone. The cone’s height is 2 cm and the base diameter is 4 cm. Find the volume of the toy (use π=3.14).
A cylindrical pencil (radius 5 mm, length 210 mm) has a cone sharpened into one end, 24 mm high, with no wood wasted elsewhere. Find the volume of wood shavings produced.
An inverted conical vessel (height 8 cm, top radius 5 cm) is filled with water to the brim. Lead shots of radius 0.5 cm are dropped in until one-fourth of the water overflows. Find the number of lead shots dropped.
How many silver coins, 1.75 cm in diameter and 2 mm thick, must be melted to form a cuboid of dimensions 5.5 cm × 10 cm × 3.5 cm?
A metallic sphere of radius 4.2 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder.
A solid metallic sphere of diameter 21 cm is melted and recast into cones, each of diameter 3.5 cm and height 3 cm. Find the number of cones formed.
Worked solutions
Show worked solutions
Surface Areas and Volumes — Practice: Solutions
Cone r=7,h=24. Find volume.
Hemisphere TSA, r=7.
Cylinder+2 hemispheres, r=7, total length 31.
Cone (r=5,h=12) carved from cylinder.
Sphere r=10.5.
Cube side4 → sphere. Find radius.
Cone CSA, r=7, l=25.
Cylinder TSA, r=3.5,h=10.
Hemisphere bowl r=9 into bottles r=3,h=4.
Cone r=6,h=8 poured into cylinder r=4.
More Practice — Solutions
Cube side10 + largest hemisphere. Find surface area.
Capsule length14mm, diameter5mm. Find surface area.
Cone+hemisphere (r=3) inside cylinder r=3,h=6, water left.
Tent: cylinder h=2.1,d=4 + cone slant=2.8.
Toy: hemisphere+cone, h=2,d=4, π=3.14.
Pencil r=5mm, cone height24mm carved. Find shavings.
Cone h=8,r=5; lead shots r=0.5, ¼ overflows.
Silver coins 1.75cm diameter, 2mm thick, into cuboid.
Sphere r=4.2 into cylinder r=6. Find height.
Sphere diameter21 into cones diameter3.5,height3.
Book a free trial class
Tell us a little about your requirement.