Geometry and Mensuration

Geometry in placement tests means a small toolkit: angle facts, triangle and circle properties, areas and volumes, and a little coordinate geometry. The formulas are few, but accuracy depends on knowing exactly which one applies and keeping units straight. Every worked answer is checked by computation. Where a value of $\pi$ is needed, questions usually say whether to use $\frac{22}{7}$ or 3.14; this chapter states it each time.

1. Lines and angles

• Angles on a straight line sum to $180°$; angles around a point to $360°$.
• Vertically opposite angles are equal. With a transversal across parallel lines, corresponding and alternate angles are equal and co-interior angles sum to $180°$.
• Triangle: angles sum to $180°$; an exterior angle equals the sum of the two opposite interior angles.
• Polygon with $n$ sides: interior angles sum to $(n-2)\times 180°$; exterior angles sum to $360°$. Each interior angle of a regular polygon is $\frac{(n-2)\times 180°}{n}$.
• Number of diagonals of an $n$-gon: $\frac{n(n-3)}{2}$.

Example 1. In a triangle two angles are $48°$ and $67°$. The third angle is $180-115=65°$, and the exterior angle at the vertex of the $65°$ angle is $115°$.

Example 2. Each interior angle of a regular polygon is $150°$. How many sides? Exterior angle $=30°$, so $n=360/30=12$.

assert 180 - (48 + 67) == 65 and 48 + 67 == 180 - 65
assert 360 / (180 - 150) == 12
interior = lambda n: (n - 2) * 180 / n
assert interior(6) == 120 and interior(12) == 150 and (8 - 2) * 180 == 1080
assert 12 * (12 - 3) // 2 == 54

2. Triangles

Area and special triangles

• Area $=\frac{1}{2}\times \text{base}\times \text{height}$.
• Heron's formula: with semi-perimeter $s=\frac{a+b+c}{2}$, area $=\sqrt{s(s-a)(s-b)(s-c)}$.
• Equilateral with side $a$: area $=\frac{\sqrt{3}}{4}{a}^2$, height $=\frac{\sqrt{3}}{2}a$.
• Right-angled: $a^2+b^2=c^2$. Common triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and their multiples.
• Triangle inequality: the sum of any two sides exceeds the third.

Example 3. Area of a triangle with sides 13, 14, 15: $s=21$, area $=\sqrt{21\cdot 8\cdot 7\cdot 6}=\sqrt{7056}=84$.

Example 4. A ladder 13 m long leans against a wall with its foot 5 m from the wall. How high does it reach? $\sqrt{169-25}=12$ m.

Example 5. Area of an equilateral triangle of side 10: $\frac{\sqrt{3}}{4}\times 100=43.30$.

import math

def heron(a, b, c):
    s = (a + b + c) / 2
    return math.sqrt(s * (s - a) * (s - b) * (s - c))

assert heron(13, 14, 15) == 84.0
assert math.sqrt(13 ** 2 - 5 ** 2) == 12
assert round(math.sqrt(3) / 4 * 100, 2) == 43.30
assert all(a * a + b * b == c * c for a, b, c in [(3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (9, 40, 41)])

Similar triangles

Triangles with equal angles are similar: corresponding sides are in the same ratio $k$, perimeters are in ratio $k$, and areas are in ratio $k^2$. A line parallel to one side of a triangle divides the other two sides proportionally (the basic proportionality theorem).

Example 6. Two similar triangles have corresponding sides 6 cm and 9 cm. The ratio of their areas is $(6/9{)}^2=4:9$.

Example 7. A 6 m pole casts a 4 m shadow. At the same time a tower casts a 30 m shadow. Tower height? Ratio $6/4=h/30$, so $h=45$ m.

from fractions import Fraction
assert Fraction(6, 9) ** 2 == Fraction(4, 9)
assert Fraction(6, 4) * 30 == 45

Centres of a triangle (names and facts)

• Centroid: where medians meet; divides each median in the ratio $2:1$.
• Incentre: where angle bisectors meet; centre of the inscribed circle, radius $r=\frac{\text{area}}{s}$.
• Circumcentre: where perpendicular bisectors meet; for a right triangle it is the midpoint of the hypotenuse, so the circumradius is half the hypotenuse.
• Orthocentre: where altitudes meet.

area, s = 84, 21                       # the 13-14-15 triangle
assert area / s == 4                    # inradius
assert 13 * 14 * 15 / (4 * area) == 8.125   # circumradius R = abc / (4 * area)

3. Quadrilaterals

Example 8. The diagonal of a square is 10 cm. Its area is $10^2/2=50$ cm².

Example 9. A rhombus has diagonals 16 cm and 12 cm. Its side is $\sqrt{8^2+6^2}=10$ cm, its area $\frac{1}{2}\cdot 16\cdot 12=96$ cm², its perimeter 40 cm.

Example 10. A rectangular garden is 40 m by 30 m. A path 2 m wide runs around the outside. Area of the path? $(44\times 34)-(40\times 30)=1496-1200=296$ m².

assert 10 ** 2 / 2 == 50
assert math.hypot(8, 6) == 10 and 16 * 12 / 2 == 96
assert (40 + 4) * (30 + 4) - 40 * 30 == 296
assert (40 + 2 * 2) * (30 + 2 * 2) - 40 * 30 == 296       # a path of width w adds 2w to each side

4. Circles

• Circumference $=2\pi r$; area $=\pi r^2$.
• Arc length $=\frac{\theta }{360}\times 2\pi r$; sector area $=\frac{\theta }{360}\times \pi r^2$.
• Chord at distance $d$ from the centre has length $2\sqrt{r^2-d^2}$.
• The angle subtended by an arc at the centre is twice the angle at the circumference; an angle in a semicircle is $90°$; opposite angles of a cyclic quadrilateral sum to $180°$.
• A tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal in length.

Example 11. A circle has area 154 cm². Using $\pi =\frac{22}{7}$: $r^2=154\times \frac{7}{22}=49$, so $r=7$ and circumference $=44$ cm.

Example 12. A wheel has radius 35 cm. How many revolutions does it make over 1.1 km? Circumference $=2\times \frac{22}{7}\times 35=220$ cm $=2.2$ m; revolutions $=1100/2.2=500$.

Example 13. Find the length of an arc of $72°$ in a circle of radius 10 cm: $\frac{72}{360}\times 2\pi \times 10=4\pi =12.57$ cm.

Example 14. A circular field has area 1,386 m². Cost of fencing it at Rs 5 per metre? $r^2=1386\times \frac{7}{22}=441$, $r=21$; circumference $=132$ m; cost $=660$.

pi = Fraction(22, 7)
assert pi * 49 == 154 and 2 * pi * 7 == 44
assert 2 * pi * 35 == 220 and Fraction(110000, 220) == 500
assert round(72 / 360 * 2 * math.pi * 10, 2) == 12.57
r = math.sqrt(1386 * 7 / 22)
assert round(r) == 21 and 2 * Fraction(22, 7) * 21 * 5 == 660

5. Solids: volume and surface area

where for a cone the slant height is $l=\sqrt{r^2+h^2}$. Space diagonal of a cuboid: $\sqrt{l^2+b^2+h^2}$; of a cube: $a\sqrt{3}$.

Example 15. A cuboid is 10 m × 8 m × 6 m. Volume 480 m³, total surface area $2(80+48+60)=376$ m², space diagonal $\sqrt{200}=14.14$ m.

Example 16. A cylinder has radius 7 cm and height 10 cm ($\pi =\frac{22}{7}$). Volume $=1540$ cm³; curved surface area $=440$ cm².

Example 17. A cone has radius 3 cm and height 4 cm. Slant height 5 cm; volume $=\frac{1}{3}\pi \cdot 9\cdot 4=12\pi =37.70$ cm³; curved surface $=15\pi =47.12$ cm².

Example 18. A sphere has radius 6 cm: volume $288\pi =904.78$ cm³; surface area $144\pi =452.39$ cm².

assert 10 * 8 * 6 == 480 and 2 * (10 * 8 + 8 * 6 + 6 * 10) == 376
assert round(math.sqrt(10 ** 2 + 8 ** 2 + 6 ** 2), 2) == 14.14
assert pi * 49 * 10 == 1540 and 2 * pi * 7 * 10 == 440
assert math.hypot(3, 4) == 5 and round(math.pi * 9 * 4 / 3, 2) == 37.70 and round(math.pi * 3 * 5, 2) == 47.12
assert round(4 / 3 * math.pi * 6 ** 3, 2) == 904.78 and round(4 * math.pi * 36, 2) == 452.39

Melting and recasting: volume is conserved

Example 19. A solid metal sphere of radius 3 cm is melted and recast into a cylinder of radius 3 cm. Height of the cylinder? $\frac{4}{3}\pi \cdot 27=\pi \cdot 9\cdot h$, so $h=4$ cm.

Example 20. A cube of side 6 cm is cut into 27 equal small cubes. Side of each? $216/27=8$, so each has side 2 cm. The total surface area rises from $6\cdot 36=216$ to $27\cdot 24=648$ cm², exactly 3 times.

Example 21. A tank 5 m × 4 m × 3 m is filled by a pipe whose cross-section is 20 cm² and which delivers water at 2 m/s. Time to fill? Tank volume $=60$ m³; flow $=0.002\times 2=0.004$ m³/s; time $=15000$ s $=4$ h 10 min.

assert round(4 / 3 * math.pi * 27 / (math.pi * 9), 6) == 4
assert 6 ** 3 / 27 == 8 and round(8 ** (1 / 3)) == 2 and 27 * 6 * 2 ** 2 == 3 * 6 * 6 ** 2
assert 5 * 4 * 3 / (20 / 10000 * 2) == 15000 and 15000 / 3600 == 4 + 1 / 6

Scaling

If every linear dimension is multiplied by $k$: lengths by $k$, areas by $k^2$, volumes by $k^3$. Doubling a cube's side multiplies its surface area by 4 and its volume by 8.

assert (2 * 5) ** 3 / 5 ** 3 == 8 and 6 * (2 * 5) ** 2 / (6 * 5 ** 2) == 4

6. Coordinate geometry (the basics)

• Distance between $(x_1,y_1)$ and $(x_2,y_2)$: $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.
• Midpoint: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.
• Section formula: the point dividing the segment in the ratio $m:n$ internally is $\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$.
• Slope $m=\frac{y_2-y_1}{x_2-x_1}$; parallel lines have equal slopes, perpendicular lines have slopes whose product is $-1$.
• Area of a triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$: $\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$. Three points are collinear when this is 0.
• Line: $y=mx+c$; the line through $(x_1,y_1)$ with slope $m$ is $y-y_1=m(x-x_1)$.

Example 22. Distance between $(1,2)$ and $(4,6)$: $\sqrt{9+16}=5$.

Example 23. The point dividing $(1,2)$ and $(7,11)$ in the ratio $2:1$: $\left(\frac{14+1}{3},\frac{22+2}{3}\right)=(5,8)$.

Example 24. Area of the triangle with vertices $(0,0)$, $(4,0)$, $(0,3)$: $\frac{1}{2}\times 4\times 3=6$.

Example 25. Are $(1,1)$, $(3,5)$ and $(5,9)$ collinear? Slopes: $\frac{4}{2}=2$ and $\frac{4}{2}=2$, so yes.

assert math.dist((1, 2), (4, 6)) == 5
assert ((2 * 7 + 1 * 1) / 3, (2 * 11 + 1 * 2) / 3) == (5.0, 8.0)
tri = lambda p, q, r: abs(p[0] * (q[1] - r[1]) + q[0] * (r[1] - p[1]) + r[0] * (p[1] - q[1])) / 2
assert tri((0, 0), (4, 0), (0, 3)) == 6 and tri((1, 1), (3, 5), (5, 9)) == 0

7. Common traps

• Using the diameter as the radius, or the wrong $\pi$.
• Mixing units: cm with m, and square and cubic units ($1{\text{m}}^2=10,000{\text{cm}}^2$, $1{\text{m}}^3=10^6{\text{cm}}^3$, $1{\text{m}}^3=1000$ litres).
• Slant height versus vertical height in cones.
• Total versus curved surface area, and open versus closed containers.
• Path "inside" versus "outside" a rectangle (the outer rectangle grows by $2w$ on each dimension; inside it shrinks).
• Scaling: area changes by $k^2$, volume by $k^3$, not by $k$.
• Assuming a triangle is right-angled because the numbers look familiar; check $a^2+b^2=c^2$.

8. Practice set with answers

1. Find the area of a triangle with sides 13, 14 and 15 and its inradius.
2. A circular field has area 154 m². Cost of fencing it at Rs 12 per metre? (Use $\pi =\frac{22}{7}$.)
3. A cuboid measures 10 cm × 8 cm × 6 cm. Find its volume and total surface area.
4. A cylinder has radius 7 cm and height 10 cm. Find its volume and curved surface area ($\pi =\frac{22}{7}$).
5. A sphere of radius 3 cm is melted into a cylinder of radius 3 cm. Find the height of the cylinder.
6. Each interior angle of a regular polygon is $135°$. How many sides does it have, and how many diagonals?
7. A rectangle 40 m × 30 m has a path 2 m wide running around the outside. Area of the path?
8. Find the point that divides the segment from $(1,2)$ to $(7,11)$ in the ratio $2:1$ and its distance from the origin.

assert heron(13, 14, 15) == 84 and 84 / 21 == 4
assert 2 * Fraction(22, 7) * 7 * 12 == 528
assert (10 * 8 * 6, 2 * (80 + 48 + 60)) == (480, 376)
assert (Fraction(22, 7) * 49 * 10, 2 * Fraction(22, 7) * 7 * 10) == (1540, 440)
assert round(4 / 3 * 27 / 9, 6) == 4
n = round(360 / (180 - 135))
assert n == 8 and n * (n - 3) // 2 == 20
assert (40 + 4) * (30 + 4) - 40 * 30 == 296
point = (5, 8)
assert round(math.hypot(*point), 2) == 9.43

Answers: 1) area 84, inradius 4; 2) Rs 528 (radius 7 m, circumference 44 m); 3) 480 cm³ and 376 cm²; 4) 1,540 cm³ and 440 cm²; 5) 4 cm; 6) 8 sides and 20 diagonals; 7) 296 m²; 8) the point $(5,8)$, at distance $\sqrt{89}\approx 9.43$ from the origin.