Mensuration 3D — CDS Elementary Mathematics

Formulas (3)

• Walls, edges and faces · Four walls of a room
  $$\text{walls} = 2(l + b)\,h$$
• Joining and cutting blocks · n cubes in a row
  $$\text{surface} = (4n + 2)\,a^2$$
• Open boxes and wall thickness · Box from a sheet
  $$V = x(L - 2x)(B - 2x)$$

Watch out for (2)

• Round the cans up, never down
  → Walls, edges and faces
• The open top has no wall
  → Open boxes and wall thickness

Formulas (3)

• The space diagonal · Space diagonal
  $$d = \sqrt{l^2 + b^2 + h^2}$$
• Sum of edges, surface area and diagonal · The square of the sum
  $$(l + b + h)^2 = d^2 + S$$
• Face areas, volume and reciprocals · Adjacent faces
  $$xyz = V^2, \qquad \frac1l + \frac1b + \frac1h = \frac{S}{2V}$$

Watch out for (1)

• The largest slice is not a face
  → The space diagonal

Formulas (3)

• Volume and surface of a cylinder · Cylinder
  $$V = \pi r^2 h, \quad \text{CSA} = 2\pi r h, \quad \text{TSA} = 2\pi r(r + h)$$
• Rolling a sheet into a cylinder · Sheet rolled along side a
  $$V = \frac{a^2 b}{4\pi}$$
• Hollow cylinders and pipes · Hollow cylinder
  $$V = \pi(R^2 - r^2)h = \pi(R + r)(R - r)h$$

Watch out for (1)

• Diameter or radius — again
  → Volume and surface of a cylinder

Formulas (3)

• Radius, height, slant height — one right triangle · Cone
  $$V = \tfrac13\pi r^2 h, \quad \text{CSA} = \pi r l, \quad l^2 = r^2 + h^2$$
• The vertical angle and the axial section · Semi-vertical angle α
  $$r = l\sin\alpha, \qquad h = l\cos\alpha$$
• Cones from a sector or a turning triangle · Sector rolled into a cone
  $$l = R, \qquad r = \frac{\theta}{360^\circ}\,R$$

Watch out for (2)

• Volume uses h, surface uses l
  → Radius, height, slant height — one right triangle
• Halve the vertical angle first
  → The vertical angle and the axial section

Formulas (3)

• Sphere and hemisphere · Sphere and hemisphere
  $$V_{\text{sphere}} = \tfrac43\pi r^3, \quad S = 4\pi r^2, \quad V_{\text{hemi}} = \tfrac23\pi r^3$$
• Hollow shells and their mass · Spherical shell
  $$V = \tfrac43\pi(R^3 - r^3)$$
• A plane cutting a sphere · Circle of a cut
  $$\rho^2 + d^2 = R^2$$

Formulas (3)

• The frustum · Frustum
  $$V = \tfrac13\pi h(R^2 + Rr + r^2), \qquad l = \sqrt{h^2 + (R - r)^2}$$
• A cone cut parallel to its base · Similar top cone
  $$\frac{V_{\text{top}}}{V} = k^3, \qquad \frac{S_{\text{top}}}{S} = k^2$$
• Solids joined together · Cylinder-and-cone tent
  $$\text{canvas} = 2\pi r h + \pi r l$$

Watch out for (3)

• Slant height uses the difference of the radii
  → The frustum
• Height from the base, or from the apex?
  → A cone cut parallel to its base
• The joining circle is not a surface
  → Solids joined together

Formulas (4)

• Volume is conserved · Recasting
  $$V_{\text{old}} = V_{\text{new}}$$
• How many pieces? · Same-shape pieces
  $$n = \left(\frac{R}{r}\right)^3$$
• Drawn into a wire · Sphere into wire
  $$\pi\rho^2 L = \tfrac43\pi R^3$$
• Surface area is not conserved · One sphere into n
  $$\frac{S_{\text{new}}}{S_{\text{old}}} = \sqrt[3]{n}$$

Formulas (2)

• Immersion raises the level · Rise in a cylinder
  $$\text{rise} = \frac{V_{\text{solid}}}{\pi R^2}$$
• Rainfall and flow — watch the units · Flow
  $$\text{volume per unit time} = \text{cross-section} \times \text{speed}$$

Watch out for (1)

• Litres, cubic metres and cubic centimetres differ by thousands
  → Rainfall and flow — watch the units

Formulas (3)

• Similar solids: lengths, squares, cubes · Scale factor k
  $$\frac{S_1}{S_2} = k^2, \qquad \frac{V_1}{V_2} = k^3$$
• Ratios of cones and cylinders · Cones or cylinders
  $$\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}$$
• Equal volume or equal surface · Cone : hemisphere : cylinder (radius r, height r)
  $$\tfrac13\pi r^3 : \tfrac23\pi r^3 : \pi r^3 = 1 : 2 : 3$$

Watch out for (1)

• An increase of p% is a factor of 1 + p/100
  → Similar solids: lengths, squares, cubes

Formulas (3)

• Cube, cylinder and sphere nested · Cube in a sphere
  $$a\sqrt3 = 2R$$
• A sphere inside a cone · Sphere in a cone
  $$\rho = \frac{rh}{r + l}$$
• Balls and cones that touch · Fourth ball on three
  $$\text{height of centre} = r + 2r\sqrt{\tfrac23}$$

Watch out for (1)

• Measure from the table, not from the lower centres
  → Balls and cones that touch