Mensuration 2D: SPI Mathematics

Formulas (3)

• Perimeter, area and cost of a rectangle · Rectangle and square
  $$P = 2(l + b), \quad A = lb, \quad P_{\text{square}} = 4a, \quad A_{\text{square}} = a^2$$
• Diagonals of rectangles and squares · Diagonals
  $$d = \sqrt{l^2 + b^2}, \qquad d_{\text{square}} = a\sqrt{2}, \qquad A_{\text{square}} = \frac{d^2}{2}$$
• Triangles and the regular hexagon · Triangle and hexagon
  $$A = \frac{1}{2}bh, \quad A_{\text{equilateral}} = \frac{\sqrt{3}}{4}a^2, \quad A_{\text{hexagon}} = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2$$

Watch out for (8)

• Perimeter where area is needed
  → Perimeter, area and cost of a rectangle
• Forgetting to halve the perimeter
  → Perimeter, area and cost of a rectangle
• Counting rectangles with a fixed perimeter
  → Perimeter, area and cost of a rectangle
• Mixing metres and centimetres
  → Diagonals of rectangles and squares
• Doubling the diagonal doubles the area
  → Diagonals of rectangles and squares
• Side or perimeter from an area
  → Diagonals of rectangles and squares
• Forgetting the half
  → Triangles and the regular hexagon
• Too few triangles in a hexagon
  → Triangles and the regular hexagon

Formulas (4)

• Circumference of a circle · Circumference
  $$C = 2\pi r = \pi d$$
• Wheels: distance and revolutions · Wheel
  $$\text{distance} = n \times 2\pi r, \qquad n = \frac{\text{distance}}{2\pi r}$$
• Area of a circle and a semicircle · Area of a circle
  $$A = \pi r^2, \qquad A_{\text{semicircle}} = \frac{1}{2}\pi r^2$$
• A wire bent from one shape into another · Same wire, new shape
  $$2\pi r = 4a, \qquad \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{4}{\pi} = \frac{14}{11}$$

Watch out for (8)

• The perimeter of a semicircle
  → Circumference of a circle
• Circumferences add by radius, areas do not
  → Circumference of a circle
• Kilometres against centimetres
  → Wheels: distance and revolutions
• Radius or diameter of the wheel
  → Wheels: distance and revolutions
• Squaring the diameter
  → Area of a circle and a semicircle
• A semicircle given the full circle's area
  → Area of a circle and a semicircle
• Keeping the area instead of the length
  → A wire bent from one shape into another
• Turning the area ratio round
  → A wire bent from one shape into another

Formulas (2)

• Area of a sector, the clock hand and segments · Sector and segment
  $$A_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2, \qquad A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}}$$
• Arc length and the perimeter of a sector · Arc and perimeter
  $$l = \frac{\theta}{360^\circ} \times 2\pi r, \qquad P = 2r + l, \qquad A = \frac{1}{2}lr$$

Watch out for (4)

• Minute hand or hour hand
  → Area of a sector, the clock hand and segments
• Giving the sector when the segment is asked
  → Area of a sector, the clock hand and segments
• Forgetting the second radius
  → Arc length and the perimeter of a sector
• Hunting for the angle
  → Arc length and the perimeter of a sector