Measurement applications: arcs and sectors

Task goals

• Apply the arc length formula (theta/360 x 2 pi r) to find the length of a curved edge in a real-world measurement scenario, given a radius and a sector angle.
• Apply the sector area formula (theta/360 x pi r squared) to find the area of a pie-slice-shaped region in a real-world measurement scenario, given a radius and a sector angle.
• Correctly identify and use the radius (not the diameter) in both the arc length and sector area formulas, staying consistent within a single problem.
• Use a sector's area or arc length as one component of a larger compound region's total, adding it to an adjoining polygon's area or length when both together make up a total.
• Use a sector's area or arc length to find a remaining area or length by subtracting it from a larger surrounding region, and correctly decide whether a problem requires adding or subtracting.
• Combine sector and polygon boundaries carefully, recognising when a straight edge is shared/internal (not part of the true outer boundary) rather than double-counting it.
• Use an arc length or sector area as an input to a further material-quantity or cost calculation (edging, planting, decking, glazing, paving), carrying full precision through the calculation and rounding only the final answer.
• Handle discrete-purchase rounding (buying material in whole units) once the required arc length or sector area is known.
• Check a calculated cost, value, or area against a stated budget, legal threshold, or design requirement, and state whether it complies.
• Combine two identical sector-based regions (two signs, two windows, two paths) into a single total-area and total-cost calculation.
• Convert between units of length before or after applying the arc length or sector area formula.
• Compare and justify a real-world measurement or purchasing decision using arc-length- or sector-area-derived numbers.