Mensuration 2D: SSC CGL Quantitative Aptitude

Formulas (4)

• Area and perimeter of a rectangle and a square · Rectangle and square
  $$A = lb \qquad P = 2(l + b) \qquad A = a^2 \qquad P = 4a$$
• Diagonal of a rectangle and of a square · Diagonals
  $$d = \sqrt{l^2 + b^2} \qquad d_{\text{square}} = a\sqrt{2} \qquad A_{\text{square}} = \frac{d^2}{2}$$
• Tiles, paths, cut-outs and the cost of covering · Tiles and paths
  $$N = \frac{\text{area to cover}}{\text{area of one tile}} \qquad A_{\text{path}} = (l + 2w)(b + 2w) - lb = 2w(l + b) + 4w^2$$
• Area of a parallelogram, a trapezium and a rhombus · Parallelogram, trapezium and rhombus
  $$A = bh \qquad A = \frac{1}{2}(a + b)h \qquad A = \frac{1}{2}d_1 d_2$$

Watch out for (9)

• Dividing a rectangle's perimeter by 4
  → Area and perimeter of a rectangle and a square
• Multiplying lengths in different units
  → Area and perimeter of a rectangle and a square
• Squaring the diagonal for a square's area
  → Diagonal of a rectangle and of a square
• Adding the sides to get the diagonal
  → Diagonal of a rectangle and of a square
• Adding the path width only once
  → Tiles, paths, cut-outs and the cost of covering
• Rounding the tile count down
  → Tiles, paths, cut-outs and the cost of covering
• Charging fencing by area
  → Tiles, paths, cut-outs and the cost of covering
• Using the slanting side as the height
  → Area of a parallelogram, a trapezium and a rhombus
• Forgetting the half in a trapezium
  → Area of a parallelogram, a trapezium and a rhombus

Formulas (4)

• Half base times height, and two sides with the angle between them · Area of a triangle
  $$A = \frac{1}{2}bh \qquad A = \frac{1}{2}ab\sin C$$
• Three sides: semi-perimeter, Heron's formula and the inradius · Heron's formula and the inradius
  $$s = \frac{a + b + c}{2} \qquad A = \sqrt{s(s - a)(s - b)(s - c)} \qquad A = rs \qquad A_{\text{isosceles}} = \frac{b}{4}\sqrt{4a^2 - b^2}$$
• Area of an equilateral triangle and a regular hexagon · Equilateral triangle and regular hexagon
  $$A = \frac{\sqrt{3}}{4}a^2 \qquad h = \frac{\sqrt{3}}{2}a \qquad A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}a^2$$
• Similar figures: scale the sides, square the area · Scaling of area
  $$\frac{A_2}{A_1} = \left(\frac{s_2}{s_1}\right)^2 \qquad \text{area change} = 2x + \frac{x^2}{100}\ \%$$

Watch out for (8)

• Pairing a base with the wrong height
  → Half base times height, and two sides with the angle between them
• Reading an area ratio as a base ratio
  → Half base times height, and two sides with the angle between them
• Putting the perimeter into Heron's formula
  → Three sides: semi-perimeter, Heron's formula and the inradius
• Running Heron on a right triangle
  → Three sides: semi-perimeter, Heron's formula and the inradius
• Mixing up the height and the area
  → Area of an equilateral triangle and a regular hexagon
• Taking a hexagon as three triangles
  → Area of an equilateral triangle and a regular hexagon
• Doubling the side doubles the area
  → Similar figures: scale the sides, square the area
• A percentage change in area is not the side's change
  → Similar figures: scale the sides, square the area

Formulas (3)

• Circumference, wheel revolutions and fencing a circle · Circumference and revolutions
  $$C = 2\pi r = \pi d \qquad N = \frac{\text{distance}}{C}$$
• Area of a circle and how it scales with the radius · Area of a circle
  $$A = \pi r^2 \qquad \frac{A_2}{A_1} = \left(\frac{r_2}{r_1}\right)^2 \qquad R^2 = r_1^2 + r_2^2$$
• Area of a ring, a circular track or a path round a circle · Area of a ring
  $$A = \pi(R^2 - r^2) = \pi(R - r)(R + r) \qquad A = \pi\left(\frac{c}{2}\right)^2$$

Watch out for (6)

• A whole-number ratio for circumference to diameter
  → Circumference, wheel revolutions and fencing a circle
• Dividing kilometres by centimetres
  → Circumference, wheel revolutions and fencing a circle
• Using a circumference ratio as an area ratio
  → Area of a circle and how it scales with the radius
• Adding radii to add areas
  → Area of a circle and how it scales with the radius
• Squaring the width
  → Area of a ring, a circular track or a path round a circle
• Swapping inner and outer radii
  → Area of a ring, a circular track or a path round a circle

Formulas (3)

• Arc length as a fraction of the circumference · Arc length
  $$l = \frac{\theta}{360^\circ} \times 2\pi r \qquad C = \frac{360^\circ}{\theta} \times l$$
• Perimeter of a sector · Perimeter of a sector
  $$P = 2r + \frac{\theta}{360^\circ} \times 2\pi r \qquad P_{\text{semicircle}} = \pi r + 2r$$
• Radians: arc length equals radius times angle · Radian measure
  $$l = r\theta \qquad \pi \text{ radians} = 180^\circ \qquad \theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180^\circ}$$

Watch out for (6)

• Doubling a diameter that is already a diameter
  → Arc length as a fraction of the circumference
• Taking the minor arc for the major one
  → Arc length as a fraction of the circumference
• Leaving out the radii
  → Perimeter of a sector
• Counting the radius once
  → Perimeter of a sector
• Putting degrees into l = rθ
  → Radians: arc length equals radius times angle
• Taking the chord for the arc
  → Radians: arc length equals radius times angle

Formulas (4)

• Sector area as a fraction of the circle's area · Area of a sector from its angle
  $$A = \frac{\theta}{360^\circ} \times \pi r^2$$
• Sector area from its arc: half the arc times the radius · Area of a sector from its arc
  $$A = \frac{1}{2}lr \qquad A = \frac{1}{2}r^2\theta$$
• Comparing two sectors given in degrees and in radians · Ratio of two sector areas
  $$\frac{A_1}{A_2} = \frac{\theta_1 r_1^2}{\theta_2 r_2^2}$$
• Finding a sector from its perimeter · Sector perimeter and area together
  $$P = 2r + l \qquad r(P - 2r) = 2A$$

Watch out for (7)

• Using 2πr for an area
  → Sector area as a fraction of the circle's area
• Stopping at r squared
  → Sector area as a fraction of the circle's area
• Putting degrees into half r squared theta
  → Sector area from its arc: half the arc times the radius
• Dropping the half
  → Sector area from its arc: half the arc times the radius
• Comparing a degree number with a radian number
  → Comparing two sectors given in degrees and in radians
• Comparing by angle when the radii are unknown
  → Comparing two sectors given in degrees and in radians
• Keeping the wrong root
  → Finding a sector from its perimeter

Formulas (3)

• Area of a segment: sector minus triangle · Area of a segment
  $$A_{\text{segment}} = \frac{\theta}{360^\circ} \pi r^2 - \frac{1}{2}r^2\sin\theta$$
• Shaded areas: the whole minus the holes, or the sum of the pieces · Corner sectors of equal radius
  $$\text{three corners of a triangle: } \frac{1}{2}\pi r^2 \qquad \text{four corners of a rectangle: } \pi r^2$$
• Inscribed figures: a circle in a square or triangle, a square or hexagon in a circle · Inscribed figures
  $$r_{\text{in square}} = \frac{a}{2} \qquad A_{\text{square in circle}} = 2r^2 \qquad r_{\text{in equilateral}} = \frac{a}{2\sqrt{3}} \qquad R_{\text{equilateral}} = \frac{a}{\sqrt{3}}$$

Watch out for (6)

• Giving the sector as the segment
  → Area of a segment: sector minus triangle
• Using the equilateral triangle at every angle
  → Area of a segment: sector minus triangle
• Counting each corner sector as a full circle
  → Shaded areas: the whole minus the holes, or the sum of the pieces
• Squaring the difference for a hole of the same shape
  → Shaded areas: the whole minus the holes, or the sum of the pieces
• Taking a square's side as the diameter of the circle round it
  → Inscribed figures: a circle in a square or triangle, a square or hexagon in a circle
• Mixing up a triangle's two circles
  → Inscribed figures: a circle in a square or triangle, a square or hexagon in a circle