First, the six basic solids
Before we join solids, we need each one alone. Here r = radius, h = height, l = slant height, a = side of a cube.
- Cube: surface 6a², volume a³.
- Cuboid (a box, sides l, b, h): surface 2(lb + bh + hl), volume lbh.
- Cylinder: curved surface 2πrh, total 2πr(r + h), volume πr²h.
- Cone: curved surface πrl, total πr(l + r), volume ⅓πr²h. Slant height l = √(r² + h²).
- Sphere: surface 4πr², volume ⁴⁄₃πr³.
- Hemisphere (half a ball): curved surface 2πr², total 3πr², volume ⅔πr³.
Tip: a cone holds exactly one third of a cylinder with the same base and height. Three cones of water fill one cylinder.
Surface area of a combined solid
Surface area means the outside skin. Imagine painting the object.
- Name each part (for example cylinder + cone).
- Take only the surfaces you can still see and touch.
- Leave out every face that is glued inside the joint.
- Add the visible parts.
Rocket (cylinder + cone, standing on its base): TSA = 2πrh (cylinder side) + πr² (bottom) + πrl (cone side).
So the total surface area of a combined solid is not the sum of the total surface areas of its parts. The hidden faces make it smaller.
Ready-made formulas for common combinations
| Object | Surface area | Volume |
|---|---|---|
| Cylinder + cone (rocket, tent with floor) | 2πrh + πr² + πrl | πr²h + ⅓πr²h₂ |
| Cylinder + hemisphere (one end) | 2πrh + πr² + 2πr² | πr²h + ⅔πr³ |
| Capsule (hemisphere at both ends) | 2πrh + 4πr² | πr²h + ⁴⁄₃πr³ |
| Cone + hemisphere (ice cream) | πrl + 2πr² | ⅓πr²h + ⅔πr³ |
| Cube + hemisphere on top | 6a² − πr² + 2πr² | a³ + ⅔πr³ |
Do not learn the table by heart. Build it each time from the rule: visible skin only.
Volume of a combined solid
Volume is the space inside. Water fills every part, and no part is counted twice. So:
Volume of combined solid = sum of the volumes of its parts.
If a part is cut out (like a hole or a scooped bowl), you subtract its volume instead.
Solids with a part scooped out
Suppose a hemisphere is scooped out of the top of a cube. The volume goes down: a³ − ⅔πr³. But the surface area goes up: you lose a flat circle πr² but gain a curved bowl 2πr². So TSA = 6a² − πr² + 2πr² = 6a² + πr².
Same idea for a cylinder with a cone dug out of it, or a pen stand with holes.
Board exam pattern
The Mensuration unit carries 10 marks in CBSE Class 10. Expect one 2-mark or 3-mark sum on combined solids and often a 4-mark case-study (a tent, toy, capsule or storage tank). Always write the formula first, then put the numbers, then the unit (cm² for area, cm³ for volume). Use π = 22/7 when the radius is a multiple of 7, else 3.14 unless the question says otherwise.
Try it at home
Take an empty ice-cream cone and a small ball (like a table tennis ball). Put the ball on the cone. Wrap paper around the outside and mark what you covered: the ball's top half and the cone's side. The circle where they touch never gets paper. That missing circle is the part we skip in surface area.
In the 3D above, go to the last step, choose Cube + hemisphere and make r bigger. Predict first: will the surface area go up or down?
Check your understanding
1. A cone sits on a hemisphere. Which circle is hidden? (The cone's base and the hemisphere's flat face, both at the joint.)
2. If you double the height of only the cylinder part, does the cone's volume change? (No. Only the cylinder part grows.)
Key formulas and definitions
- Cube: TSA = 6a², V = a³ · Cuboid: TSA = 2(lb + bh + hl), V = lbh
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h), V = πr²h
- Cone: l = √(r² + h²), CSA = πrl, V = ⅓πr²h
- Sphere: 4πr², V = ⁴⁄₃πr³ · Hemisphere: CSA = 2πr², TSA = 3πr², V = ⅔πr³
- Combined solid: TSA = sum of visible surfaces only (leave out the joint)
- Combined solid: V = sum of volumes of parts (subtract parts cut out)
Worked examples
1. Two cubes, each of side 5 cm, are joined face to face. Find the surface area and volume of the new cuboid.
The new cuboid is 10 cm × 5 cm × 5 cm. Surface area = 2(10×5 + 5×5 + 5×10) = 2(50 + 25 + 50) = 250 cm². Volume = 10 × 5 × 5 = 250 cm³. Check: two cubes had 2 × 150 = 300 cm², minus the 2 hidden faces (2 × 25 = 50) = 250 cm².
2. A capsule is a cylinder with a hemisphere at each end. Radius 3.5 mm, whole length 14 mm. Find its surface area (π = 22/7).
Cylinder part length = 14 − 3.5 − 3.5 = 7 mm. Curved surface of cylinder = 2πrh = 2 × 22/7 × 3.5 × 7 = 154 mm². Two hemispheres = 2 × 2πr² = 4 × 22/7 × 12.25 = 154 mm². Total = 154 + 154 = 308 mm².
3. A toy is a cone on a hemisphere of radius 3.5 cm. The whole toy is 15.5 cm tall. Find its total surface area (π = 22/7).
Cone height = 15.5 − 3.5 = 12 cm. Slant height l = √(12² + 3.5²) = √(144 + 12.25) = √156.25 = 12.5 cm. Cone surface = πrl = 22/7 × 3.5 × 12.5 = 137.5 cm². Hemisphere surface = 2πr² = 2 × 22/7 × 12.25 = 77 cm². TSA = 137.5 + 77 = 214.5 cm².
4. A rocket model is a cylinder (r = 3 cm, h = 10 cm) with a cone (height 4 cm) on top. Find its TSA and volume (π = 3.14). (This is the example animated in the 3D.)
l = √(3² + 4²) = 5 cm. Cylinder side = 2πrh = 188.4 cm². Base = πr² = 28.26 cm². Cone side = πrl = 47.1 cm². TSA = 188.4 + 28.26 + 47.1 = 263.76 cm². Volume = πr²h + ⅓πr²h₂ = 282.6 + 37.68 = 320.28 cm³.
5. A tent is a cylinder of radius 7 m and height 3 m with a cone roof of slant height 10 m. Canvas costs ₹50 per m². Find the cost of canvas (no floor).
Canvas = cylinder side + cone side = 2πrh + πrl = 2 × 22/7 × 7 × 3 + 22/7 × 7 × 10 = 132 + 220 = 352 m². Cost = 352 × 50 = ₹17,600.
6. From a 10 cm cube, a hemisphere of diameter 7 cm is scooped out of the top face. Find the new surface area and the volume left (π = 22/7).
r = 3.5 cm. Surface: lose the flat circle πr² = 38.5, gain the bowl 2πr² = 77. TSA = 600 − 38.5 + 77 = 638.5 cm². Volume left = 1000 − ⅔πr³ = 1000 − ⅔ × 22/7 × 42.875 = 1000 − 89.83 = 910.17 cm³.
7. A solid is a cylinder (r = 3 cm, h = 8 cm) with a cone (r = 3 cm, height 4 cm) on top. It is dropped into a cylindrical tub of radius 6 cm that is half full of water. By how much does the water rise?
Volume of solid = π × 9 × 8 + ⅓ × π × 9 × 4 = 72π + 12π = 84π cm³. Rise in water × area of tub = 84π, so rise = 84π ÷ (π × 36) = 84 ÷ 36 ≈ 2.33 cm.
Common mistakes
- Adding the full TSA of each part. The joined faces are hidden and must be left out.
- Using the vertical height h in πrl. The cone's curved surface needs the slant height l = √(r² + h²).
- Taking the whole length as the cylinder height in a capsule or toy. Subtract the radii of the hemispheres first.
- Mixing units: area is in cm², volume in cm³. Convert all lengths to the same unit before you start.