Notes for SSC CGL examinations — unfold surfaces, fill volumes, compare capacities and conserve matter with diagram-first clarity.
Surface area measures the exposed covering of a solid and uses square units. Volume measures occupied space and uses cubic units. Capacity is the volume a container can hold.
Each metric step multiplies length by \(10\), area by \(10^2\), and volume by \(10^3\).
| Solid | Lateral / curved surface | Total surface | Volume | Key length |
|---|---|---|---|---|
| Right prism, base area \(B\), base perimeter \(p\) | \(ph\) | \(ph+2B\) | \(Bh\) | perpendicular height \(h\) |
| Cuboid \(l,b,h\) | \(2h(l+b)\) | \(2(lb+bh+hl)\) | \(lbh\) | space diagonal \(d=\sqrt{l^2+b^2+h^2}\) |
| Cube, side \(a\) | \(4a^2\) | \(6a^2\) | \(a^3\) | face diagonal \(a\sqrt2\), body diagonal \(a\sqrt3\) |
| Right circular cylinder | \(2\pi rh\) | \(2\pi r(h+r)\) | \(\pi r^2h\) | axis height \(h\) |
| Hollow cylinder, radii \(R,r\) | material volume not a surface | depends on open/closed ends | \(\pi h(R^2-r^2)\) | thickness \(t=R-r\) |
| Right circular cone | \(\pi rl\) | \(\pi r(l+r)\) | \(\frac13\pi r^2h\) | \(l=\sqrt{r^2+h^2}\) |
| Sphere | not separated | \(4\pi r^2\) | \(\frac43\pi r^3\) | diameter \(d=2r\) |
| Hemisphere | \(2\pi r^2\) | \(3\pi r^2\) | \(\frac23\pi r^3\) | TSA includes circular base \(\pi r^2\) |
| Regular right pyramid, base area \(B\), perimeter \(p\) | \(\frac12p\ell\) | \(B+\frac12p\ell\) | \(\frac13Bh\) | face slant height \(\ell\) |
A right prism has two congruent parallel bases; lateral edges are perpendicular to them. Every cross-section parallel to the base has the same area.
For a triangular prism, \(B\) is the triangle area and \(p\) its perimeter. For any polygonal prism, only the base formula changes.
There are \(12\) edges with total length \(4(l+b+h)\). A cuboid’s longest rod is its space diagonal:
If an open box lacks the top, area \(=lb+2h(l+b)\); do not use full TSA.
A cube cut into pieces of side \(x\) gives \((a/x)^3\) small cubes when \(x\) divides \(a\). New surface area rises because hidden cut faces become exposed.
For an \(n\times n\times n\) cube made of unit cubes and painted outside:
Use these only for \(n\ge2\), and interpret \(n=2\) carefully.
Unroll the curved surface: it becomes a rectangle of length \(2\pi r\) and breadth \(h\).
Open at one end: surface \(=2\pi rh+\pi r^2\). Open at both ends: surface \(=2\pi rh\).
Material volume equals outer cylinder minus inner cylinder. If thickness \(t\) is given, inner radius is \(r=R-t\).
For a hollow pipe open at both ends, total surface includes outer curved, inner curved and two annular end faces:
The perpendicular height \(h\), radius \(r\), and slant height \(l\) form a right triangle.
A cone with the same base and height as a cylinder has one-third its volume. Its curved-surface net is a circular sector, not a triangle.
Every point on the surface is at distance \(r\) from the centre. A sphere has no edge, face or separate curved/base area.
If radius changes by factor \(k\), surface changes by \(k^2\) and volume by \(k^3\).
Half a sphere has a circular base. Curved area excludes it; total area includes it.
For a thick hollow sphere with outer radius \(R\) and inner radius \(r\):
Total boundary area, if both inner and outer surfaces are exposed, is \(4\pi(R^2+r^2)\).
The apex lies directly above the centre of a regular base. Do not confuse perpendicular height \(h\) with face slant height \(\ell\).
For square base side \(a\): \(B=a^2\), \(p=4a\), and \(\ell^2=h^2+(a/2)^2\). For an equilateral triangular base, calculate its base area separately.
For similar solids with linear scale factor \(k\):
Thus volume ratio \(m:n\) gives corresponding length ratio \(\sqrt[3]{m}:\sqrt[3]{n}\). If radius doubles, a sphere’s area becomes \(4\) times and volume \(8\) times.
When a hemisphere is mounted on a cylinder, the shared circular face is internal. Volume adds, but surface area excludes the shared circle twice.
When material is melted and recast with no loss, volume is conserved; surface area generally is not.
If \(N\) equal small spheres of radius \(r\) form one sphere of radius \(R\):
Number of small solids \(=V_{large}/V_{small}\), provided there is no wastage.
Capacity questions use inner dimensions. If a tank is filled to fraction \(f\), liquid volume is \(fV_{tank}\). Time to fill at net rate \(q\) is volume divided by rate.
A fully submerged object displaces liquid equal to its own volume. Rise in water level in a uniform vessel:
Same base and same perpendicular height:
So one prism/cylinder fills exactly three matching pyramids/cones.
When pipe thickness \(t\) is given, avoid two squares:
This is especially fast when \(R+r\) is a multiple of \(7\).
For similar solids, take cube root of the volume ratio before comparing radii, heights or edges.
Do not compare lengths directly as \(64:125\).
For spheres, common constants cancel:
Eight equal small spheres fuse into one sphere of double radius because \(\sqrt[3]{8}=2\).
Start from total surface and subtract missing faces. It is safer than memorising many cases.
Cuboid without lid: \(2(lb+bh+hl)-lb=lb+2h(l+b)\).
Use \(l^2=r^2+h^2\). Common right-triangle triples save time:
Scale the entire triple when dimensions share a factor.
For a vessel of uniform cross-section:
For a cylindrical vessel, \(\Delta h=V_{object}/(\pi R^2)\). Cancel \(\pi\) if the object is spherical or cylindrical.
For identical shape families, constants cancel. One sphere recast into balls:
One cylinder recast into equal cylinders: \(N=R^2H/(r^2h)\).
“Paint the curved wall” excludes circular ends; “paint the closed tank” includes them. Exposed surfaces—not the solid’s name—decide.
Volume uses perpendicular \(h\); curved surface uses slant \(l\). TCS often supplies both and offers swapped-formula options.
A bowl’s inner curved area may be \(2\pi r^2\), but a solid hemisphere’s total surface includes the flat base.
Pipe outer diameter \(D\) gives \(R=D/2\). Thickness subtracts from radius once: \(r=R-t\), not \(R-2t\).
Wall thickness means outside dimensions cannot be used for storage. Subtract twice the thickness from each affected full dimension.
Height loses \(t\) only when the top is open and only the bottom has thickness.
Melting preserves material volume, not external surface. Many small spheres have more total surface than one large sphere.
Since \(1\,\mathrm{m}=100\,\mathrm{cm}\), the volume multiplier is \(100^3\), not \(100\) or \(100^2\).
A shared circular face is counted once on each separate solid, but becomes invisible after joining. Subtract it twice from the separate TSAs.
In a square pyramid, face slant height runs from apex to midpoint of a base side—not to a base vertex. Apex-to-vertex is a lateral edge.
A prism carries the same base straight through height \(h\), so every slice is \(B\).
A pyramid or cone narrows to a point, so it holds one-third of its matching prism or cylinder.
Unwrap the jacket: belt length is circumference \(2\pi r\), jacket height is \(h\).
Height is vertical inside; slant height lies on the outer cloth. Their right triangle includes radius.
Surface has four radius-squares; volume has four-thirds radius-cubes.
The hemisphere curved cover is \(2\pi r^2\); closing the base adds one more \(\pi r^2\).
The exponent follows dimension: surface spreads in two directions, volume fills three.
Recasting changes appearance and surface, but the amount of material—volume—stays fixed.