Mensuration (Area and Volume)

In the AAI Junior Executive (Civil) syllabus under Quantitative Aptitude · 2 parts

Mensuration — 2D Figures · Mensuration — 3D Solids

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📑 Contents (13 sections)

Part 1 of 2

Mensuration — 2D Figures

Last reviewed 22 Sept 2026 · 5 min read

FormulaAreas and perimeters

FormulaFour ways to an area

Base and height: $\frac{1}{2} bh$
Three sides (Heron): $s = \frac{a + b + c}{2}$ , area $= s (s - a) (s - b) (s - c)$
Equilateral of side $a$ : $\frac{3}{4} a^{2}$ , height $\frac{3}{2} a$
Right-angled: $\frac{1}{2} \times$ product of the two legs

FormulaCircle formulas

For radius $r$ and an angle $θ$ in degrees:

Area = π r^{2}, Circumference = 2 π r

Arc = \frac{θ}{360} \times 2 π r, Sector area = \frac{θ}{360} \times π r^{2} = \frac{arc \times r}{2}

Minor segment = sector - triangle, Ring (annulus) = π (R^{2} - r^{2})

Use $π = \frac{22}{7}$ when the radius is a multiple of 7, and $3.14$ otherwise.

A path of width $w$ outside a rectangle $l \times b$ : outer size $(l + 2 w) \times (b + 2 w)$ ; path area $=$ outer $-$ inner.
A path inside: inner size $(l - 2 w) \times (b - 2 w)$ .
Two crossing paths through the middle of a field: area $= l w + b w - w^{2}$ — the overlap is counted once.

Part 2 of 2

Last reviewed 22 Sept 2026 · 5 min read

FormulaVolumes and surfaces

Solid	Volume	Curved / lateral surface	Total surface
Cuboid	$l bh$	$2 h (l + b)$	$2 (l b + bh + h l)$
Cube	$a^{3}$	$4 a^{2}$	$6 a^{2}$
Cylinder	$π r^{2} h$	$2 π r h$	$2 π r (r + h)$
Cone	$\frac{1}{3} π r^{2} h$	$π r l$	$π r (r + l)$
Sphere	$\frac{4}{3} π r^{3}$	—	$4 π r^{2}$
Hemisphere	$\frac{2}{3} π r^{3}$	$2 π r^{2}$	$3 π r^{2}$

Diagonal of a cuboid $= l^{2} + b^{2} + h^{2}$ ; of a cube $= a 3$ .

FormulaCone, frustum, prism and pyramid

Cone slant height: $l = r^{2} + h^{2}$
Frustum of a cone (radii $R$ and $r$ , height $h$ ): $V = \frac{π h}{3} (R^{2} + R r + r^{2})$ , slant $l = h^{2} + (R - r)^{2}$ , curved surface $= π l (R + r)$
Hollow cylinder (outer $R$ , inner $r$ ): $V = π h (R^{2} - r^{2})$
Prism: $V =$ base area $\times$ height; lateral surface $=$ base perimeter $\times$ height
Pyramid: $V = \frac{1}{3} \times$ base area $\times$ height; lateral surface $= \frac{1}{2} \times$ base perimeter $\times$ slant height

1 m^{3} = 1000 litres, 1 litre = 1000 cm^{3}

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A timed chapter test from the AAI Junior Executive (Civil) series, on exactly this chapter.