Measurement applications: compound 3D shapes

Task goals

• Calculate the volume or capacity of a compound 3D solid assembled from two familiar solids (a cylinder and a cone, a cylinder and a hemisphere, or a cuboid and a pyramid), using the correct perpendicular height in each volume formula.
• Recognise which part of a compound solid actually holds material or liquid (its internal volume/capacity) versus which parts of the solid are exposed and need a material, coating, or cover (its exposed surface area), and never confuse the two.
• Calculate the exposed surface area of a compound solid from its curved, flat, and sloped faces, using a given slant height rather than deriving one.
• Convert between units of length, area, or volume before calculating, and recognise that a volume conversion factor is cubed.
• Calculate the mass of the contents of a compound solid from its volume and a given density, converting between grams, kilograms, and tonnes as needed.
• Use a calculated volume or exposed area, a material's coverage rate or unit size, and a price to find a total quantity and cost.
• Round a discrete purchase (tins, sheets, bags) UP to the next whole item even when only slightly over, but round a quantity that must physically fit inside a fixed space DOWN to the next whole item, and calculate the total cost to the nearest cent.
• Compare two real material, insulation, or supplier options by total cost and/or a non-cost factor, check a stated constraint such as a budget or a certification requirement, and write a short, numbers-based justification for a real-world measurement decision.