Surface Area and Volume: AFCAT Numerical Ability

Formulas (4)

• Cube: volume, surface and diagonal · Cube of edge a
  $$V = a^3,\quad S = 6a^2,\quad \text{LSA} = 4a^2,\quad d = a\sqrt{3}$$
• Cuboid: volume, surface, diagonal and the open tank · Cuboid l × b × h
  $$V = lbh,\quad S = 2(lb + bh + hl),\quad d = \sqrt{l^2 + b^2 + h^2}$$
• Scaling a cube, and comparing a cube with a cuboid · Two cubes of edges a₁ and a₂
  $$\dfrac{S_1}{S_2} = \left(\dfrac{a_1}{a_2}\right)^2,\quad \dfrac{V_1}{V_2} = \left(\dfrac{a_1}{a_2}\right)^3$$
• Prisms and pyramids: base area times height · Base area A, height h
  $$V_{\text{prism}} = Ah,\quad V_{\text{pyramid}} = \tfrac13 Ah$$

Watch out for (8)

• Lateral surface is four faces, not six
  → Cube: volume, surface and diagonal
• Divide by 6 before the square root
  → Cube: volume, surface and diagonal
• Lateral surface leaves out top and bottom
  → Cuboid: volume, surface, diagonal and the open tank
• An open tank has no lid, and wet walls stop at the water
  → Cuboid: volume, surface, diagonal and the open tank
• Square for area, cube for volume
  → Scaling a cube, and comparing a cube with a cuboid
• A 10% longer edge is not 30% more volume
  → Scaling a cube, and comparing a cube with a cuboid
• Do not forget the third
  → Prisms and pyramids: base area times height
• Slant height for faces, vertical height for volume
  → Prisms and pyramids: base area times height

Formulas (4)

• Volume of a cylinder · Cylinder of radius r and height h
  $$V = \pi r^2 h$$
• Curved and total surface, and the stories that use them · Cylinder surfaces
  $$\text{CSA} = 2\pi rh,\quad \text{TSA} = 2\pi r(r + h)$$
• Hollow cylinders and pipes · Hollow cylinder
  $$V = \pi h (R^2 - r^2)$$
• Comparing two cylinders, and changing their size · Two cylinders
  $$\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2},\quad \dfrac{V}{\text{CSA}} = \dfrac{r}{2}$$

Watch out for (8)

• Which side it turns about
  → Volume of a cylinder
• Halve the diameter
  → Volume of a cylinder
• A wetted surface has one base, not two
  → Curved and total surface, and the stories that use them
• Turn centimetres into metres before a rate per m²
  → Curved and total surface, and the stories that use them
• R² − r² is not (R − r)²
  → Hollow cylinders and pipes
• A pipe's total surface has four parts
  → Hollow cylinders and pipes
• The radius counts twice in volume, once in surface
  → Comparing two cylinders, and changing their size
• A thinner wire grows by more than the cut
  → Comparing two cylinders, and changing their size

Formulas (4)

• Slant height and the surface of a cone · Cone of radius r, height h, slant l
  $$l = \sqrt{r^2 + h^2},\quad \text{CSA} = \pi r l,\quad \text{TSA} = \pi r (l + r)$$
• Volume of a cone · Cone
  $$V = \tfrac13 \pi r^2 h$$
• A cone cut parallel to its base · Small cone cut off the top
  $$\dfrac{V_{\text{top}}}{V} = k^3,\quad \dfrac{\text{CSA}_{\text{top}}}{\text{CSA}} = k^2$$
• The frustum: a bucket shape · Frustum of radii R and r, height h
  $$V = \dfrac{\pi h}{3}(R^2 + r^2 + Rr),\quad l = \sqrt{h^2 + (R - r)^2},\quad \text{CSA} = \pi (R + r) l$$

Watch out for (8)

• Curved surface uses the slant height
  → Slant height and the surface of a cone
• Cloth length is area divided by width
  → Slant height and the surface of a cone
• Doubling the height is a 100% rise
  → Volume of a cone
• The radius counts squared
  → Volume of a cone
• Measure from the right end
  → A cone cut parallel to its base
• Halving the height does not halve the volume
  → A cone cut parallel to its base
• The slant uses R − r, not R + r
  → The frustum: a bucket shape
• An open bucket has one circle
  → The frustum: a bucket shape

Formulas (2)

• Hemispheres: curved or total surface, solid or hollow · Hemisphere of radius r
  $$\text{CSA} = 2\pi r^2,\quad \text{TSA} = 3\pi r^2,\quad V = \tfrac23 \pi r^3$$
• Changing the radius of a sphere · Two spheres
  $$\dfrac{S_1}{S_2} = \left(\dfrac{r_1}{r_2}\right)^2,\quad \dfrac{V_1}{V_2} = \left(\dfrac{r_1}{r_2}\right)^3$$

Reference tables (1)

Watch out for (6)

• Four πr², not four thirds
  → Sphere and hemisphere formulas
• Halve the diameter
  → Sphere and hemisphere formulas
• A solid hemisphere's total surface counts the flat face
  → Hemispheres: curved or total surface, solid or hollow
• R³ − r³ is not (R − r)³
  → Hemispheres: curved or total surface, solid or hollow
• Becomes 27 times means up by 26 times
  → Changing the radius of a sphere
• Square for surface, cube for volume
  → Changing the radius of a sphere

Formulas (3)

• Solids on the same base, or with equal volumes · Same base, same height
  $$V_{\text{cone}} = \tfrac13 V_{\text{cylinder}},\quad V_{\text{cylinder}} : V_{\text{hemisphere}} : V_{\text{cone}} = 3 : 2 : 1 \;(h = r)$$
• The largest solid inside another · Carving one solid out of another
  $$V_{\text{left}} = V_{\text{outer}} - V_{\text{inner}}$$
• Solids joined together or hollowed out · Two solids joined on a face
  $$V = V_1 + V_2,\quad S = S_1 + S_2 - 2 \times (\text{area of the joined face})$$

Watch out for (6)

• Square the ratio of radii
  → Solids on the same base, or with equal volumes
• Equal volumes do not mean equal surfaces
  → Solids on the same base, or with equal volumes
• The edge is the diameter, not the radius
  → The largest solid inside another
• In a short cylinder the height decides
  → The largest solid inside another
• No floor under a tent
  → Solids joined together or hollowed out
• A cavity adds a surface
  → Solids joined together or hollowed out

Formulas (4)

• One solid melted into another · Volume is kept
  $$V_{\text{before}} = V_{\text{after}}$$
• Many small solids from one, or many into one · Spheres recast
  $$n = \left(\dfrac{R}{r}\right)^3,\quad R^3 = r_1^3 + r_2^3 + \dots$$
• A body sunk in water raises the level · Rise in level
  $$\text{rise} = \dfrac{V_{\text{sunk}}}{\text{base area of the vessel}}$$
• Earth dug and spread, water flowing and stored · Flow in a pipe
  $$\text{volume per hour} = \text{cross-section} \times \text{speed}$$

Watch out for (8)

• Melt only what is left
  → One solid melted into another
• Radius in, diameter out
  → One solid melted into another
• Radii do not add
  → Many small solids from one, or many into one
• Merging cuts the surface
  → Many small solids from one, or many into one
• Divide by the vessel's base, not the object's
  → A body sunk in water raises the level
• Match the units before dividing
  → A body sunk in water raises the level
• Spread over the field less the pit
  → Earth dug and spread, water flowing and stored
• cm² to m² is ten thousand, not a hundred
  → Earth dug and spread, water flowing and stored