Mensuration 2D — CDS Elementary Mathematics

Formulas (6)

• Base × height, or two sides and the angle between them · Two sides and the included angle
  $$\text{Area} = \tfrac12 bh = \tfrac12\,ab\sin C$$
• Heron's formula — three sides only · Heron's formula
  $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \tfrac{a+b+c}{2}$$
• A right triangle from its perimeter · Area from the sum of the legs
  $$\text{Area} = \frac{ab}{2} = \frac{(a+b)^2 - c^2}{4}$$
• The equilateral triangle · Equilateral triangle, side a
  $$h = \tfrac{\sqrt3}{2}a, \qquad \text{Area} = \tfrac{\sqrt3}{4}a^2 = \tfrac{h^2}{\sqrt3}$$
• Scale the sides, square the area · Similar figures
  $$\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2$$
• Recover the sides first · Sides from altitudes
  $$a : b : c = \frac{1}{h_a} : \frac{1}{h_b} : \frac{1}{h_c}$$

Watch out for (4)

• The angle must be the one between the two sides
  → Base × height, or two sides and the angle between them
• Two sides plus the perimeter is three sides
  → Heron's formula — three sides only
• Don't solve for the legs unless you must
  → A right triangle from its perimeter
• Altitudes and sides go the opposite way
  → Recover the sides first

Formulas (5)

• Rectangles and squares as a pair of equations · Square from its diagonal
  $$\text{Area} = s^2 = \frac{d^2}{2}$$
• Borders, paths and percentage changes · Path of width w outside
  $$\text{Path} = (l + 2w)(b + 2w) - lb$$
• The rhombus — half the product of the diagonals · Rhombus
  $$\text{Area} = \tfrac12 d_1 d_2, \qquad 4a^2 = d_1^2 + d_2^2$$
• Parallelogram and trapezium · Parallelogram and trapezium
  $$A_{\text{par}} = ab\sin\theta, \qquad A_{\text{trap}} = \tfrac12 (p + q)\,h$$
• Cut it into triangles · Quadrilateral split along a diagonal
  $$[ABCD] = [ABC] + [ACD]$$

Watch out for (3)

• The diagonal is an option on purpose
  → Rectangles and squares as a pair of equations
• Width counts twice
  → Borders, paths and percentage changes
• An unequal-leg trapezium needs two right triangles
  → Parallelogram and trapezium

Formulas (3)

• The wire keeps its length · Length is conserved
  $$\text{perimeter of the old shape} = \text{perimeter of the new shape}$$
• Same perimeter: the rounder shape wins · Equal perimeter
  $$\frac{\text{circle}}{\text{square}} = \frac{4}{\pi}, \qquad \frac{\text{triangle}}{\text{square}} = \frac{4\sqrt3}{9}$$
• Same perimeter: the square beats every rectangle · Square minus rectangle
  $$s^2 - lb = \left(\frac{l-b}{2}\right)^2$$

Watch out for (2)

• A semicircle's boundary includes the diameter
  → The wire keeps its length
• Equal perimeter and equal area flip the answer
  → Same perimeter: the rounder shape wins

Formulas (3)

• Circumference and wheel revolutions · Wheel revolutions
  $$n = \frac{\text{distance}}{2\pi r}$$
• Area grows as the square of the radius · Circle
  $$A = \pi r^2, \qquad \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2$$
• The ring between two circles · Ring from a tangent chord
  $$\text{Ring} = \pi(R^2 - r^2) = \pi\left(\tfrac{\text{chord}}{2}\right)^2$$

Watch out for (3)

• Radius or diameter?
  → Circumference and wheel revolutions
• Equal-area rings get thinner outward
  → Area grows as the square of the radius
• "Cannot be determined" is the bait
  → The ring between two circles

Formulas (3)

• Arc length — radius times angle in radians · Arc length
  $$l = r\theta = \frac{\theta^\circ}{360^\circ}\times 2\pi r$$
• Sector area — a slice of the disc · Sector
  $$A = \frac{\theta}{360^\circ}\,\pi r^2 = \tfrac12 r l$$
• The segment — sector minus triangle · Minor segment
  $$\text{Segment} = \tfrac12 r^2(\theta - \sin\theta)$$

Watch out for (2)

• Never put degrees into rθ
  → Arc length — radius times angle in radians
• Sector is not segment
  → The segment — sector minus triangle

Formulas (5)

• Square in a circle, circle in a square · In a circle of radius r
  $$\text{square} = 2r^2, \qquad \text{equilateral triangle} = \tfrac{3\sqrt3}{4}r^2$$
• A triangle's incircle and circumcircle · Inradius and circumradius
  $$r = \frac{\Delta}{s}, \qquad R = \frac{abc}{4\Delta}$$
• Figures in a semicircle or a quarter circle · Square in a semicircle
  $$s^2 = \tfrac45 r^2$$
• A square or rectangle wedged into a corner · Square in the right angle
  $$s = \frac{ab}{a + b}$$
• Regular polygons — hexagon and octagon · Hexagon and octagon
  $$A_{\text{hex}} = \tfrac{3\sqrt3}{2}a^2, \qquad \text{octagon side} = a(\sqrt2 - 1)$$

Watch out for (2)

• The inradius and the circumradius swap easily
  → A triangle's incircle and circumcircle
• Side or perimeter?
  → A square or rectangle wedged into a corner

Formulas (2)

• Join the centres · Distance between centres
  $$O_1O_2 = r_1 + r_2 \;(\text{outside}), \qquad O_1O_2 = R - r \;(\text{inside})$$
• The gap between touching circles · Three equal touching circles
  $$\text{gap} = \frac{\sqrt3}{4}(2r)^2 - 3\cdot\frac{60^\circ}{360^\circ}\pi r^2 = \frac{r^2}{2}(2\sqrt3 - \pi)$$

Watch out for (2)

• Reject the root that doesn't fit the figure
  → Join the centres
• The sectors add to half a circle, not a whole one
  → The gap between touching circles

Formulas (4)

• The whole minus the holes · Shaded area
  $$\text{shaded} = \text{whole} - \sum \text{removed pieces}$$
• Semicircles on a divided diameter · Arcs on a split diameter
  $$\pi r_1 + \pi r_2 + \cdots = \pi R \quad \text{when } r_1 + r_2 + \cdots = R$$
• Overlaps — lenses and crescents · Lens from two quarter-circles
  $$\text{lens} = a^2\left(\frac{\pi}{2} - 1\right)$$
• Figures built from pieces · Composite figure
  $$\text{Area} = \sum \text{pieces}, \qquad \text{Perimeter} = \text{outer edges only}$$

Watch out for (2)

• Read the hatching, not the words
  → The whole minus the holes
• Half the lens is a single segment
  → Overlaps — lenses and crescents