Area and Perimeter: AFCAT Numerical Ability

Formulas (3)

• Area, perimeter and diagonal of a rectangle and a square · Rectangle and square
  $$A = lb,\quad P = 2(l + b),\quad d = \sqrt{l^2 + b^2};\qquad A_{\text{square}} = a^2 = \dfrac{d^2}{2}$$
• Finding the sides from two facts · Two facts, two sides
  $$l + b = \dfrac{P}{2},\qquad 2lb = (l + b)^2 - d^2$$
• Fencing, flooring, tiles and paths · Path outside a rectangle
  $$\text{path} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2$$

Watch out for (7)

• A square from its diagonal
  → Area, perimeter and diagonal of a rectangle and a square
• Perimeter is a length
  → Area, perimeter and diagonal of a rectangle and a square
• Halve the perimeter before splitting it
  → Finding the sides from two facts
• Shape is not size
  → Finding the sides from two facts
• A path adds its width on both sides
  → Fencing, flooring, tiles and paths
• Same units before dividing
  → Fencing, flooring, tiles and paths
• Throw out the root that cannot fit
  → Fencing, flooring, tiles and paths

Formulas (3)

• Half the base times the height · Area of a triangle
  $$A = \tfrac12 \times b \times h,\qquad h = \dfrac{2A}{b}$$
• Right triangles and Pythagoras · Pythagoras
  $$c^2 = a^2 + b^2,\qquad (a + b)^2 = (a - b)^2 + 4ab$$
• Heron's formula and the isosceles triangle · Heron's formula
  $$A = \sqrt{s(s - a)(s - b)(s - c)},\quad s = \dfrac{a + b + c}{2}$$

Watch out for (6)

• Keep the half
  → Half the base times the height
• The height is not the slant side
  → Half the base times the height
• The hypotenuse faces the right angle
  → Right triangles and Pythagoras
• Legs, not the hypotenuse, in the isosceles case
  → Right triangles and Pythagoras
• s is half the perimeter
  → Heron's formula and the isosceles triangle
• Halve the base for the height
  → Heron's formula and the isosceles triangle

Formulas (4)

• Rhombus and kite: working from the diagonals · Rhombus
  $$A = \tfrac12 d_1 d_2,\qquad \text{side} = \sqrt{\left(\tfrac{d_1}{2}\right)^2 + \left(\tfrac{d_2}{2}\right)^2}$$
• Trapezium, parallelogram and any quadrilateral · Trapezium and quadrilateral
  $$A_{\text{trap}} = \tfrac12(a + b)h,\qquad A_{\text{quad}} = \tfrac12 d(h_1 + h_2)$$
• Equilateral triangles and the regular hexagon · Equilateral triangle and hexagon
  $$A_{\triangle} = \dfrac{\sqrt 3}{4}a^2,\qquad A_{\text{hex}} = 6 \times \dfrac{\sqrt 3}{4}a^2 = \dfrac{3\sqrt 3}{2}a^2$$
• Angles and perimeter of a regular polygon · Polygon angles
  $$\text{sum} = (n - 2) \times 180°,\qquad \text{exterior} = \dfrac{360°}{n}$$

Watch out for (8)

• Halve the diagonals first
  → Rhombus and kite: working from the diagonals
• Half the product
  → Rhombus and kite: working from the diagonals
• Each height goes with its own sides
  → Trapezium, parallelogram and any quadrilateral
• The slant side is not the height
  → Trapezium, parallelogram and any quadrilateral
• Height has a 2 under it, area a 4
  → Equilateral triangles and the regular hexagon
• Six triangles, not three
  → Equilateral triangles and the regular hexagon
• Sum or each
  → Angles and perimeter of a regular polygon
• Exterior angles always total 360°
  → Angles and perimeter of a regular polygon

Formulas (4)

• Circumference and area · Circle
  $$C = 2\pi r = \pi d,\qquad A = \pi r^2$$
• Rings, tracks and two circles · Area of a ring
  $$A = \pi(R^2 - r^2) = \pi(R + r)(R - r)$$
• Semicircles, arcs, sectors and clock hands · Arc and sector
  $$\text{arc} = \dfrac{\theta}{360} \times 2\pi r,\qquad \text{sector} = \dfrac{\theta}{360} \times \pi r^2$$
• Animals tied with a rope · Grazed area
  $$\text{area} = \dfrac{\theta}{360} \times \pi\,(\text{rope})^2$$

Watch out for (8)

• Radius or diameter
  → Circumference and area
• Convert the distance first
  → Circumference and area
• Difference of squares, not square of the difference
  → Rings, tracks and two circles
• Width adds to the radius
  → Rings, tracks and two circles
• A semicircle's edge includes its diameter
  → Semicircles, arcs, sectors and clock hands
• Minutes into degrees
  → Semicircles, arcs, sectors and clock hands
• Only the angle at the peg
  → Animals tied with a rope
• Check the rope against the walls
  → Animals tied with a rope

Formulas (3)

• Figures inscribed in one another · Square and circle
  $$\text{circle in square: } r = \tfrac a2;\qquad \text{square in circle: } d = 2r,\ A = 2r^2$$
• Shaded regions by subtraction · Shaded area
  $$\text{shaded} = \text{outer} - \text{removed},\qquad \text{segment} = \text{sector} - \triangle$$
• Building a figure from pieces · Rectangle with semicircular ends
  $$A = lb + \pi\left(\tfrac b2\right)^2,\qquad P = 2l + \pi b$$

Watch out for (6)

• Which length is shared
  → Figures inscribed in one another
• The circumradius is not half the side
  → Figures inscribed in one another
• A segment is not a sector
  → Shaded regions by subtraction
• Read what is shaded
  → Shaded regions by subtraction
• Two semicircles, one circle
  → Building a figure from pieces
• Do not count the overlap twice
  → Building a figure from pieces

Formulas (2)

• Scaling a figure and comparing areas · Scale factor k
  $$\dfrac{P_2}{P_1} = k,\qquad \dfrac{A_2}{A_1} = k^2$$
• Percentage changes in the sides · Two percentage changes
  $$\text{area change} = a + b + \dfrac{ab}{100}\ \%$$

Watch out for (4)

• Square the ratio for area
  → Scaling a figure and comparing areas
• Square roots on the way back
  → Scaling a figure and comparing areas
• Percentages multiply, they do not add
  → Percentage changes in the sides
• Undoing a rise is a smaller fall
  → Percentage changes in the sides