Elementary Mensuration takes the flat shapes of Area and Perimeter into three dimensions: cubes, cuboids, cylinders, cones, spheres and their surface areas and volumes. The formulas are fixed, so AFCAT questions here are direct and scoring once you can recall the right one. This Cavalier guide organises every solid-figure formula, adds the scaling and melting-recasting tricks, and shows the worked patterns examiners repeat every cycle.
What Elementary Mensuration covers
Mensuration is the measurement of geometric figures. The ‘elementary’ tier tested in AFCAT deals with the everyday solids — boxes, cans, cones, balls and pipes — and asks for three things: volume (the space inside, a cubic measure), surface area (the skin around it, a square measure) and sometimes a derived length such as a diagonal or a slant height.
The single most useful habit is to track which measure a question wants. Volume is in cubic units (m3, cm3) and answers ‘how much can it hold’. Surface area is in square units and answers ‘how much material to cover it’. Length is in plain units. Matching the units to the question is half the battle.
At The Cavalier we tell defence aspirants that mensuration is pure formula recall plus careful unit work — there is no theory to derive under exam pressure. Candidates lose marks only by confusing curved with total surface area, or by forgetting to convert litres to cubic centimetres. Build a clean formula sheet, drill it, and these become some of the safest marks in the paper.
Volume is a cubic measure (length3), surface area is a squared measure (length2). If your units do not match the quantity asked, you have used the wrong formula.
Cube and cuboid
The cuboid (a rectangular box) and the cube (its equal-sided version) are the most common AFCAT solids, so their formulas must be instant.
- Cuboid (l, b, h): Volume = l × b × h; Total surface area = 2(lb + bh + hl)
- Cuboid diagonal = √(l2 + b2 + h2)
- Cube (side a): Volume = a3; Total surface area = 6a2; diagonal = a√3
A cuboidal tank is 3 m long, 2 m wide and 1.5 m high. Find its volume and total surface area.
1 cubic metre = 1000 litres. So the 9 cubic metre tank above holds 9000 litres — capacity questions hinge on this conversion.
Cylinder: curved and total surface area
The cylinder introduces the difference between curved (lateral) surface area and total surface area — a distinction AFCAT tests constantly. The curved surface is just the side; the total adds the two circular ends.
- Volume = π r2 h
- Curved surface area (CSA) = 2 π r h
- Total surface area (TSA) = 2 π r (h + r)
A cylindrical pillar has radius 7 cm and height 20 cm. Find its volume and curved surface area (take π = 22/7).
For painting only the outside of an open pipe or pillar, use the curved surface area, not the total. A closed tank needs the total surface area; an open one omits the missing end.
Cone and the slant height
The cone is built from a circular base and a slant side. Its formulas use the slant height l, which you usually have to find from the radius and the vertical height by Pythagoras.
- Slant height l = √(r2 + h2)
- Volume = ⅓ π r2 h (one-third of a cylinder of the same base and height)
- Curved surface area = π r l
- Total surface area = π r (l + r)
A cone has radius 6 cm and height 8 cm. Find its slant height, volume and curved surface area (take π = 3.14).
The slant height l is always the longest of the three (l > h and l > r) because it is the hypotenuse. Use the vertical height h for volume but the slant height l for surface area.
Sphere and hemisphere
The sphere and its half, the hemisphere, complete the standard solid set. Note that the hemisphere's total surface adds the flat circular face to the curved dome.
- Sphere: Volume = ⅔ π r3; Surface area = 4 π r2
- Hemisphere: Volume = ⅔ π r3; Curved surface area = 2 π r2
- Hemisphere total surface area = 3 π r2 (curved dome + flat circle)
Find the volume and surface area of a sphere of radius 3 cm (take π = 22/7).
A sphere has no edges or flat faces, so it has only one surface-area figure — there is no separate curved and total for a full sphere. The split only appears for the hemisphere.
The scaling rule for solids
This is the highest-value shortcut in mensuration. When every linear dimension of a solid is multiplied by a factor k, the surface area scales by k2 and the volume scales by k3. Surface area is a squared measure; volume is a cubed measure.
Multiply each dimension by k: Length → k, Surface area → k2, Volume → k3 times the original.
If the edge of a cube is doubled, how do its surface area and volume change?
So doubling the edge gives eight times the volume but only four times the surface. The same rule answers reverse questions: if a sphere's volume is to be multiplied by 27, its radius need only triple, since 33 = 27. Spotting the cube relationship lets you answer these instantly without plugging into the full formula.
Never apply the linear factor directly to volume. A 10% increase in radius raises volume by about 33% (1.13 = 1.331), not 10%.
Melting, recasting and the volume-conservation trick
A classic AFCAT theme: a solid is melted and recast into a different shape, or several small objects are made from one large one. The key insight is that volume is conserved — melting changes the shape but not the amount of material.
When one solid is recast into another, set Volume of old = Volume of new. For many small pieces from one big solid, Number = Volume of big ÷ Volume of one small piece.
A metal sphere of radius 6 cm is melted and recast into small spheres of radius 1 cm. How many small spheres are formed?
Because both shapes are spheres, the common ⅔π factor cancels and only the cube of the radii matters — a neat use of the scaling rule. The same idea handles a cylinder melted into a cone, a wire drawn from a metal block, or coins stacked from a sheet: equate the volumes and solve.
Surface area is not conserved during melting — only volume is. Splitting one large solid into many small ones always increases the total surface area.
Prisms, pyramids and combined solids
AFCAT occasionally moves beyond the basic five to prisms, pyramids and composite shapes such as a cone on top of a cylinder. The unifying rule keeps these manageable.
- Any prism: Volume = base area × height; Lateral surface = base perimeter × height
- Any pyramid: Volume = ⅓ × base area × height
- Combined solid: add the volumes, but add only the exposed surface areas
A solid is a cylinder of radius 7 cm and height 10 cm with a cone of the same radius and height 6 cm on top. Find the total volume (take π = 22/7).
When two solids are joined, the touching faces disappear. For surface area, count only what is on the outside — do not add the hidden circular join.
Capacity, units and conversions
Many mensuration questions are really unit-conversion questions in disguise — a tank's capacity in litres, a flow rate in cubic metres, or material cost per square metre. Keep the conversions instant.
1 cubic metre = 1000 litres; 1 litre = 1000 cubic cm; 1 cubic cm = 1 millilitre. So 1 cubic metre = 1,000,000 cubic cm.
A cylindrical water tank has radius 1 m and height 2 m. Find its capacity in litres (take π = 3.14).
Always convert all dimensions to one unit before applying a formula. Mixing metres and centimetres mid-calculation is the most frequent source of wrong answers in this topic.
Previous-year style question
Q. A solid metallic cylinder of radius 3 cm and height 8 cm is melted and recast into a cone of the same radius. What is the height of the cone formed?
Answer: Volume is conserved, so πr2Hcyl = ⅓πr2hcone. The radius is the same on both sides, so it cancels: Hcyl = ⅓ × hcone, giving hcone = 3 × Hcyl = 3 × 8 = 24 cm. A cone needs three times the height of a cylinder of equal base to hold the same volume.
Traps and time-savers to keep in mind
- Mixing curved surface area with total surface area — check whether the ends or base are included.
- Applying a linear percentage straight to volume instead of cubing the factor.
- Using the slant height for volume, or the vertical height for surface area, in a cone.
- Forgetting that joined faces vanish when two solids are combined.
Cone : Hemisphere : Cylinder of the same radius and height stand in the volume ratio 1 : 2 : 3. Memorise this and many comparison questions collapse to a single line.
Build a one-page formula sheet
Because mensuration is entirely formula-driven, the fastest gain is a single revised sheet listing volume, curved surface area and total surface area for each solid side by side. Drill it until recall is automatic, and always note next to each whether π appears, so you can choose 22/7 or 3.14 sensibly. On exam day the candidates who score full marks here are simply the ones who never hesitate over which formula to use.
Quick revision
- Volume is cubic, surface area is squared — match units to the question.
- Cube: V = a3, TSA = 6a2, diagonal = a√3; cuboid diagonal = √(l2+b2+h2).
- Cylinder: V = πr2h, CSA = 2πrh, TSA = 2πr(h+r).
- Cone: l = √(r2+h2), V = ⅓πr2h, CSA = πrl; Sphere: V = ⅔πr3, SA = 4πr2.
- Scaling: length ×k gives surface ×k2 and volume ×k3.
- Melting conserves volume — equate old and new volumes to recast.
Frequently asked questions
How important is Elementary Mensuration in AFCAT?
It reliably contributes one or two questions per paper and overlaps with Area and Perimeter. Because every question is formula-driven with no derivation needed, it is among the safest scoring topics if your recall is solid.
What is the difference between curved and total surface area?
Curved (lateral) surface area covers only the side of a solid, while total surface area also includes the flat ends or base. A closed can needs the total; painting only the side of a pillar needs the curved area.
Why does doubling the edge of a cube give eight times the volume?
Volume depends on the cube of the length. Multiplying the edge by 2 multiplies the volume by 2 cubed, which is 8. Surface area, being squared, multiplies only by 4.
How do melting and recasting problems work?
Volume is conserved when a solid is melted, so set the old volume equal to the new one and solve. For many small pieces from one big solid, divide the big volume by one small volume.
Should I use the slant height or the vertical height for a cone?
Use the vertical height for volume and the slant height for curved or total surface area. Find the slant height from radius and height using l = root of (r squared plus h squared).
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