Quick Revision

Triangles: Area

• Area $= \frac{1}{2} \times \text{base} \times \text{height}$; any side can be the base.
• Heron: $s = \frac{a + b + c}{2}$, area $= \sqrt{s(s - a)(s - b)(s - c)}$.
• Isosceles (equal sides $a$, base $b$): $\frac{b}{4}\sqrt{4a^2 - b^2}$.
• Two sides and the angle between them: $\frac{1}{2}ab\sin C$.
• Area $= rs = \frac{abc}{4R}$.
• 13-14-15 has area 84; sides times $k$ give area times $k^2$.
• A median halves the area; joining the midpoints leaves one quarter.
• Trap: altitude $= \frac{2 \times \text{area}}{\text{base}}$. Keep the 2.

Equilateral and Right Triangles

• Equilateral, side $a$: height $\frac{\sqrt{3}}{2}a$, area $\frac{\sqrt{3}}{4}a^2$.
• From the height: area $= \frac{h^2}{\sqrt{3}}$. From the perimeter: $\frac{\sqrt{3}}{36}P^2$.
• $r = \frac{a}{2\sqrt{3}}$, $R = \frac{a}{\sqrt{3}}$, height $= 3r$.
• A point inside: the three perpendiculars add up to the height.
• Right triangle: area $\frac{1}{2}ab$, altitude to the hypotenuse $\frac{ab}{c}$, $r = \frac{a + b - c}{2}$, $R = \frac{c}{2}$.
• $(a + b)^2 = c^2 + 4 \times \text{area}$.
• Triplets: 5-12-13, 8-15-17, 7-24-25, 20-21-29, 9-40-41, 12-35-37.
• Trap: $\frac{h^2}{\sqrt{3}}$, not $h^2\sqrt{3}$.

Rectangles and Squares

• Rectangle: area $lb$, perimeter $2(l + b)$, diagonal $\sqrt{l^2 + b^2}$.
• Two clues: $(l + b)^2 = d^2 + 2lb$ and $(l - b)^2 = d^2 - 2lb$.
• Square: area $a^2 = \frac{d^2}{2}$, perimeter $4a$, diagonal $a\sqrt{2}$.
• Same perimeter: the square has the largest area, ahead by $\left(\frac{l - b}{2}\right)^2$.
• Wire bent into a new shape: the perimeter stays the same, the area changes.
• Trap: halve the perimeter to get $l + b$ before squaring.
• Trap: an area question with a perimeter-sized option is a decoy.

Paths, Tiles and Cost

• Path outside: $2w(l + b + 2w)$. Inside: $2w(l + b - 2w)$.
• Square field: outside $4w(a + w)$, inside $4w(a - w)$.
• Cross roads: $w(l + b) - w^2$; the yellow square is counted once.

• Largest square tile: HCF of the sides, in one unit.
• Carpet length = floor area ÷ carpet width.
• Cost = area × rate (flooring) or perimeter × rate (fencing).
• Trap: change m to cm first, and remember the $-w^2$.

Parallelogram, Rhombus and Trapezium

• Parallelogram: base × height, each side with its own height; also $ab\sin\theta$.
• Rhombus: area $\frac{1}{2}d_1d_2$, side $\frac{1}{2}\sqrt{d_1^2 + d_2^2}$, height = area ÷ side.
• Trapezium: $\frac{1}{2}(a + b)h$. Isosceles: each overhang is $\frac{a - b}{2}$.
• Unequal slant sides: the triangle with sides (slant, slant, $a - b$) has the same height.
• Kite: $\frac{1}{2}d_1d_2$. Any quadrilateral: $\frac{1}{2} \times d \times (h_1 + h_2)$.
• Trap: $\frac{1}{2}d_1d_2$ does not work for a parallelogram or a rectangle.

Regular Polygons

• Hexagon, side $a$: six equilateral triangles, area $\frac{3\sqrt{3}}{2}a^2$, perimeter $6a$.
• Longest diagonal $2a$, shorter one $\sqrt{3}a$.
• Circle around a hexagon has radius $a$; circle inside has radius $\frac{\sqrt{3}}{2}a$.
• Joining alternate corners gives a triangle of half the hexagon's area.
• Any regular polygon: area $= \frac{1}{2} \times \text{perimeter} \times r$.
• Hexagon and triangle with equal perimeters: areas 3 : 2.
• [Tier 2] Octagon: $2(1 + \sqrt{2})a^2$.
• Trap: the shorter diagonal is not $2a$.

Inscribed and Circumscribed Figures

• Square, side $a$: $r = \frac{a}{2}$, $R = \frac{a}{\sqrt{2}}$; circle areas 1 : 2.
• Equilateral triangle: $r = \frac{a}{2\sqrt{3}}$, $R = \frac{a}{\sqrt{3}}$; 1 : 4.
• Hexagon: $r = \frac{\sqrt{3}}{2}a$, $R = a$; 3 : 4.
• Square inside a circle of radius $r$: area $2r^2$.
• Midpoints of a square joined: half the area.
• Largest circle in a rectangle: diameter = breadth.
• Same wire: the circle encloses most, then the square, then the triangle.
• Trap: $R = \sqrt{2}r$ in a square but $R = 2r$ in a triangle.

Circle Area and Circumference

• $C = 2\pi r = \pi d$ and $A = \pi r^2 = \frac{1}{2}Cr$.
• Options in multiples of 22: use $\frac{22}{7}$. Multiples of 3.14: use 3.14. $\pi$ shown: keep it.
• $r = 7$: 44 and 154. $r = 14$: 88 and 616. $r = 21$: 132 and 1386.
• $C - d = \frac{15}{7}d$.
• Adding areas: $R^2 = r_1^2 + r_2^2$. Adding circumferences: $R = r_1 + r_2$.
• Discs from a sheet: each needs a $2r \times 2r$ square.
• Trap: diameter used as radius.

Semicircles, Quadrants and Rings

• Semicircle: area $\frac{1}{2}\pi r^2$, perimeter $\pi r + 2r = \frac{36}{7}r$.
• Quadrant: area $\frac{1}{4}\pi r^2$, perimeter $\frac{\pi r}{2} + 2r = \frac{25}{7}r$.
• Ring: $\pi(R^2 - r^2) = \pi(R + r)(R - r)$.

• Path of width $w$ round a circle of radius $r$: $\pi w(2r + w)$.
• Width from two circumferences: $\frac{C_1 - C_2}{2\pi}$.
• Trap: $\pi(R - r)^2$ is wrong.
• Trap: a semicircle's perimeter includes the diameter.

Wheels and Revolutions

• One revolution covers one circumference, $2\pi r$.
• Revolutions $= \frac{\text{distance}}{2\pi r}$. Use one unit: 1 km = 100000 cm.
• Distance per minute = revolutions per minute × circumference. Metres per minute × $\frac{60}{1000}$ gives km/h.
• Two wheels on the same road: revolutions are in the inverse ratio of the radii.
• Front wheel $n$ more turns than the back: $\frac{D}{C_1} - \frac{D}{C_2} = n$.
• Trap: taking the diameter as the radius halves the count.
• Trap: forgetting to change km into cm.

Arcs and Sectors

• Take the fraction $\frac{\theta}{360}$ of the circle: arc $= \frac{\theta}{360} \times 2\pi r$, area $= \frac{\theta}{360} \times \pi r^2$.
• Sector area $= \frac{1}{2} \times \text{arc} \times r$.
• Perimeter of a sector = arc + $2r$.
• Clock: minute hand $6^\circ$ a minute, hour hand $30^\circ$ an hour.
• Grazing: corner of a rectangle gives a quarter circle; the three corners of a triangle give one semicircle; four corners of a quadrilateral give one circle.
• Trap: a major sector uses $360^\circ - \theta$.
• Trap: the perimeter needs both radii.

Segments and Shaded Regions

• Segment = sector minus triangle. At $90^\circ$ the triangle is $\frac{r^2}{2}$; at $60^\circ$ and $120^\circ$ it is $\frac{\sqrt{3}}{4}r^2$.
• Major segment = circle minus minor segment.
• Four quarter circles at the corners of a square make one circle: shaded middle $\frac{3}{14}a^2$.
• Three touching circles: gap = equilateral triangle on the centres minus one semicircle.
• Leaf from two quadrants of a square: $\frac{4}{7}a^2$.
• [Tier 2] Crescents on the legs of a right triangle equal the triangle.
• Trap: find each piece, then add or subtract.

Change in Area

• Every length times $k$: perimeter times $k$, area times $k^2$.
• One length up $x\%$, the other up $y\%$: area changes by $x + y + \frac{xy}{100}$ percent (a fall is negative).
• Every length up $x\%$: area up $2x + \frac{x^2}{100}$ percent. 10 gives 21, 20 gives 44, 30 gives 69.
• To keep the area fixed, the other length falls by $\frac{x}{100 + x} \times 100\%$.
• Backwards: area times 2.25 means lengths times $\sqrt{2.25} = 1.5$.
• Trap: the perimeter changes by $x\%$, not $2x\%$.

Before the Exam

• Read the units first: cm or m, area or perimeter, rate per m² or per metre.
• Choose $\pi$ from the options, and draw a quick figure with every length on it.
• Size check: a part is smaller than the whole, and an altitude is shorter than the sides beside it.
• Usual decoys: perimeter for area, diameter for radius, $k$ for $k^2$, no $\frac{\theta}{360}$, no $-w^2$, a semicircle without its diameter.
• Practise with the PYQ Practice Sets of this chapter, at about 45 seconds a question.