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CDS Mathematics · Formula sheet

Mensuration 3D formulas

30 formulas and 12 common traps for CDS Mathematics Mensuration 3D, grouped by subtopic.

Full notes with worked examples

Cubes & Cuboids

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Walls, edges and faces

Four walls of a room

walls=2(l+b) h\text{walls} = 2(l + b)\,h

Joining and cutting blocks

n cubes in a row

surface=(4n+2) a2\text{surface} = (4n + 2)\,a^2

Open boxes and wall thickness

Box from a sheet

V=x(L−2x)(B−2x)V = x(L - 2x)(B - 2x)

Common traps

Round the cans up, never down

12.512.5 litres of paint means 1313 one-litre cans. The rounded-down value is always one of the options.

The open top has no wall

An open box adds thickness to the height only once, at the bottom. Adding it twice, as for a closed box, overstates the material.

Diagonals & Cuboid Identities

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The space diagonal

Space diagonal

d=l2+b2+h2d = \sqrt{l^2 + b^2 + h^2}

Sum of edges, surface area and diagonal

The square of the sum

(l+b+h)2=d2+S(l + b + h)^2 = d^2 + S

Face areas, volume and reciprocals

Adjacent faces

xyz=V2,1l+1b+1h=S2Vxyz = V^2, \qquad \frac1l + \frac1b + \frac1h = \frac{S}{2V}

Common traps

The largest slice is not a face

A plane through two opposite edges of a cube cuts a rectangle a×a2a\times a\sqrt2, larger than a face (a2a^2) and larger than the hexagonal section through the centre.

Cylinders

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Volume and surface of a cylinder

Cylinder

V=πr2h,CSA=2πrh,TSA=2πr(r+h)V = \pi r^2 h, \quad \text{CSA} = 2\pi r h, \quad \text{TSA} = 2\pi r(r + h)

Rolling a sheet into a cylinder

Sheet rolled along side a

V=a2b4πV = \frac{a^2 b}{4\pi}

Hollow cylinders and pipes

Hollow cylinder

V=π(R2−r2)h=π(R+r)(R−r)hV = \pi(R^2 - r^2)h = \pi(R + r)(R - r)h

Common traps

Diameter or radius — again

Stems give the diameter of a well, a box or a pipe. Halve it before squaring; the option built on the full diameter is four times too big.

Cones

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Radius, height, slant height — one right triangle

Cone

V=13πr2h,CSA=πrl,l2=r2+h2V = \tfrac13\pi r^2 h, \quad \text{CSA} = \pi r l, \quad l^2 = r^2 + h^2

The vertical angle and the axial section

Semi-vertical angle α

r=lsin⁡α,h=lcos⁡αr = l\sin\alpha, \qquad h = l\cos\alpha

Cones from a sector or a turning triangle

Sector rolled into a cone

l=R,r=θ360∘ Rl = R, \qquad r = \frac{\theta}{360^\circ}\,R

Common traps

Volume uses h, surface uses l

Putting the slant height into 13πr2h\dfrac13\pi r^2 h is the planted error — with r=9r = 9 and l=15l = 15 it gives 405π405\pi instead of 324π324\pi.

Halve the vertical angle first

The trigonometry uses the half-angle at the apex. Taking sin⁡120∘\sin 120^\circ instead of sin⁡60∘\sin 60^\circ happens to give the same value, which hides the slip until a 90∘90^\circ or 60∘60^\circ cone exposes it.

Spheres, Hemispheres & Shells

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Sphere and hemisphere

Vsphere=43πr3,S=4πr2,Vhemi=23πr3V_{\text{sphere}} = \tfrac43\pi r^3, \quad S = 4\pi r^2, \quad V_{\text{hemi}} = \tfrac23\pi r^3

Hollow shells and their mass

Spherical shell

V=43π(R3−r3)V = \tfrac43\pi(R^3 - r^3)

A plane cutting a sphere

Circle of a cut

ρ2+d2=R2\rho^2 + d^2 = R^2

Frustums & Combined Solids

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The frustum

Frustum

V=13πh(R2+Rr+r2),l=h2+(R−r)2V = \tfrac13\pi h(R^2 + Rr + r^2), \qquad l = \sqrt{h^2 + (R - r)^2}

A cone cut parallel to its base

Similar top cone

VtopV=k3,StopS=k2\frac{V_{\text{top}}}{V} = k^3, \qquad \frac{S_{\text{top}}}{S} = k^2

Solids joined together

Cylinder-and-cone tent

canvas=2πrh+πrl\text{canvas} = 2\pi r h + \pi r l

Common traps

Slant height uses the difference of the radii

l=h2+(R−r)2l = \sqrt{h^2 + (R - r)^2}, not h2+R2\sqrt{h^2 + R^2}. Using the full radius gives the slant of the whole cone, which is longer.

Height from the base, or from the apex?

kk measures the top cone from the APEX. A question asking for the height of the cut above the base wants 1−k1 - k of the height, and the kk value is the planted option.

The joining circle is not a surface

Where the cone meets the hemisphere, or the dome meets the cylinder, the circle is inside the solid. Adding πr2\pi r^2 for it is the commonest overcount.

Melting & Recasting

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Volume is conserved

Recasting

Vold=VnewV_{\text{old}} = V_{\text{new}}

How many pieces?

Same-shape pieces

n=(Rr)3n = \left(\frac{R}{r}\right)^3

Drawn into a wire

Sphere into wire

πρ2L=43πR3\pi\rho^2 L = \tfrac43\pi R^3

Surface area is not conserved

One sphere into n

SnewSold=n3\frac{S_{\text{new}}}{S_{\text{old}}} = \sqrt[3]{n}

Water — Immersion, Flow & Rainfall

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Immersion raises the level

Rise in a cylinder

rise=VsolidπR2\text{rise} = \frac{V_{\text{solid}}}{\pi R^2}

Rainfall and flow — watch the units

Flow

volume per unit time=cross-section×speed\text{volume per unit time} = \text{cross-section} \times \text{speed}

Common traps

Litres, cubic metres and cubic centimetres differ by thousands

The options on these questions are usually the same digits with different numbers of zeros. Write the unit next to every number until the last line.

Scaling & Comparing Solids

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Similar solids: lengths, squares, cubes

Scale factor k

S1S2=k2,V1V2=k3\frac{S_1}{S_2} = k^2, \qquad \frac{V_1}{V_2} = k^3

Ratios of cones and cylinders

Cones or cylinders

V1V2=r12h1r22h2\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}

Equal volume or equal surface

Cone : hemisphere : cylinder (radius r, height r)

13πr3:23πr3:πr3=1:2:3\tfrac13\pi r^3 : \tfrac23\pi r^3 : \pi r^3 = 1 : 2 : 3

Common traps

An increase of p% is a factor of 1 + p/100

300%300\% more is 44 times, not 33. Convert the percentage to a factor before taking roots.

Solids Inside Solids

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Cube, cylinder and sphere nested

Cube in a sphere

a3=2Ra\sqrt3 = 2R

A sphere inside a cone

Sphere in a cone

ρ=rhr+l\rho = \frac{rh}{r + l}

Balls and cones that touch

Fourth ball on three

height of centre=r+2r23\text{height of centre} = r + 2r\sqrt{\tfrac23}

Common traps

Measure from the table, not from the lower centres

The fourth ball's centre is 2r232r\sqrt{\tfrac23} above the other CENTRES. The question usually asks for its height above the plane, which adds one more rr.

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