Mensuration 3D: SPI Mathematics

Formulas (3)

• Surface area of a cuboid and a cube · Surface of a cuboid and a cube
  $$S = 2(lb + bh + hl), \quad \text{walls} = 2(l + b)h, \quad S_{\text{cube}} = 6a^2, \quad \text{walls}_{\text{cube}} = 4a^2$$
• Volume of a cuboid, litres and how many blocks fit · Volume and units
  $$V = lbh, \quad V_{\text{cube}} = a^3, \quad 1\ \text{litre} = 1000\ \text{cm}^3, \quad 1\ \text{m}^3 = 1000\ \text{litres}, \quad \text{rise} = \frac{\text{volume}}{\text{area}}$$
• The longest rod: the diagonal of a cuboid · Diagonal of a cuboid
  $$d = \sqrt{l^2 + b^2 + h^2}, \qquad d_{\text{cube}} = a\sqrt{3}$$

Watch out for (9)

• Total surface or four walls
  → Surface area of a cuboid and a cube
• Converting an area by the length factor
  → Surface area of a cuboid and a cube
• Counting the hidden faces
  → Surface area of a cuboid and a cube
• Mixing units inside one product
  → Volume of a cuboid, litres and how many blocks fit
• Leaving the pit in the area
  → Volume of a cuboid, litres and how many blocks fit
• Forgetting the mortar
  → Volume of a cuboid, litres and how many blocks fit
• Dividing volumes when the blocks cannot fit
  → Volume of a cuboid, litres and how many blocks fit
• Stopping at the floor diagonal
  → The longest rod: the diagonal of a cuboid
• Leaving the root unsimplified
  → The longest rod: the diagonal of a cuboid

Formulas (3)

• Volume and surface of a cylinder · Cylinder
  $$V = \pi r^2 h, \quad \text{CSA} = 2\pi rh, \quad \text{TSA} = 2\pi r(r + h)$$
• Volume and surface of a cone · Cone
  $$l^2 = r^2 + h^2, \quad V = \frac{1}{3}\pi r^2 h, \quad \text{CSA} = \pi rl, \quad \text{TSA} = \pi r(l + r)$$
• One third: comparing cones and cylinders · Cone against cylinder
  $$V_{\text{cone}} = \frac{1}{3}V_{\text{cylinder}} \ (\text{same } r, h), \qquad \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}$$

Reference tables (1)

Watch out for (9)

• Curved or total surface
  → Volume and surface of a cylinder
• Halving the radius does not halve the volume
  → Volume and surface of a cylinder
• Height where the slant height belongs
  → Volume and surface of a cone
• Adding a floor to the tent
  → Volume and surface of a cone
• Keeping the negative root
  → Volume and surface of a cone
• Ratio of radii is not ratio of volumes
  → One third: comparing cones and cylinders
• Turning the ratio the wrong way
  → One third: comparing cones and cylinders
• Naming the piece instead of the section
  → Solids made by turning or cutting a shape
• Turning about the hypotenuse
  → Solids made by turning or cutting a shape

Formulas (3)

• Surface area and volume of a sphere · Sphere
  $$S = 4\pi r^2, \qquad V = \frac{4}{3}\pi r^3$$
• Surface area and volume of a hemisphere · Hemisphere
  $$\text{CSA} = 2\pi r^2, \quad \text{TSA} = 3\pi r^2, \quad V = \frac{2}{3}\pi r^3$$
• Ratios and percent changes: square for area, cube for volume · Scaling rule
  $$\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2, \qquad \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3$$

Watch out for (6)

• Using the diameter as the radius
  → Surface area and volume of a sphere
• Answering with the radius when the diameter is asked
  → Surface area and volume of a sphere
• Bowl or solid
  → Surface area and volume of a hemisphere
• A hemisphere is not half the sphere's surface
  → Surface area and volume of a hemisphere
• Using the volume ratio as the radius ratio
  → Ratios and percent changes: square for area, cube for volume
• Copying the percent of the length to the area
  → Ratios and percent changes: square for area, cube for volume

Formulas (3)

• Solids joined together · Cone on a hemisphere
  $$V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3, \qquad S = \pi rl + 2\pi r^2$$
• The frustum: buckets and glasses · Frustum
  $$l = \sqrt{h^2 + (R - r)^2}, \quad \text{CSA} = \pi(R + r)l, \quad V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)$$
• A cone cut parallel to its base · Similar top cone
  $$\frac{h}{H} = \frac{r}{R} = k, \qquad \frac{V_{\text{top}}}{V} = k^3$$

Watch out for (7)

• Counting the glued faces
  → Solids joined together
• A right vertical angle is the whole angle
  → Solids joined together
• Adding the radii in the slant height
  → The frustum: buckets and glasses
• Closing the top of a bucket
  → The frustum: buckets and glasses
• Leaving the capacity in cubic cm
  → The frustum: buckets and glasses
• Measuring from the wrong end
  → A cone cut parallel to its base
• Square root for a volume fraction
  → A cone cut parallel to its base

Formulas (2)

• Melting and recasting: how many pieces · Recasting
  $$V_{\text{old}} = n \times V_{\text{piece}}, \qquad n = \frac{V_{\text{old}}}{V_{\text{piece}}}$$
• Rise in water level · Water rise in a cylinder
  $$\pi R^2 \times \text{rise} = V_{\text{solid}}$$

Watch out for (4)

• Using the coin's diameter as its radius
  → Melting and recasting: how many pieces
• Same shape, smaller size: cube the ratio
  → Melting and recasting: how many pieces
• Dividing by the wrong circle
  → Rise in water level
• Worrying about the starting depth
  → Rise in water level