📐 Geometry, Area, Shapes & Perimeter

Day 1-3 | Supplementary Topic | TCS Interview Prep

📌 Subtopics

• Triangles: Area, perimeter, types, Heron’s formula, Pythagoras theorem
• Circles: Area, circumference, sector, arc
• Rectangles & Squares: Area, perimeter, diagonal
• Quadrilaterals: Parallelogram, trapezium, rhombus
• 3D Shapes: Cube, cuboid, cylinder, cone, sphere — surface area & volume

📖 Comprehensive Notes

1. Triangles

A triangle is a 3-sided polygon.

• Basic Area: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$
• Heron’s Formula: For a triangle with sides $a, b, c$: $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ where semi-perimeter $s = \frac{a+b+c}{2}$.
• Pythagoras Theorem: For a right-angled triangle with base $b$, height $a$, and hypotenuse $c$: $a^2 + b^2 = c^2$
  • Common Triplets: $(3,4,5)$, $(5,12,13)$, $(8,15,17)$, $(7,24,25)$.
• Equilateral Triangle: Sides are equal ($a$): $\text{Area} = \frac{\sqrt{3}}{4} a^2$ $\text{Height} = \frac{\sqrt{3}}{2} a$

Example: Find the area of a triangle with sides $13\text{ cm}$, $14\text{ cm}$, and $15\text{ cm}$. $s = \frac{13 + 14 + 15}{2} = 21$. $\text{Area} = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84\text{ cm}^2$.

2. Circles

• $\text{Area} = \pi r^2$
• $\text{Circumference (Perimeter)} = 2\pi r$
• Sector of a Circle: A pie-slice part of a circle with angle $\theta$: $\text{Length of Arc} = \frac{\theta}{360^\circ} \times 2\pi r$ $\text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$

Example: Find the area of a sector of a circle of radius $7\text{ cm}$ with an angle of $60^\circ$. $\text{Area} = \frac{60}{360} \times \frac{22}{7} \times 7 \times 7 = \frac{1}{6} \times 154 = 25.67\text{ cm}^2$.

3. Rectangles & Squares

• Rectangle: Length $l$, width $b$.
  • $\text{Area} = l \times b$
  • $\text{Perimeter} = 2(l + b)$
  • $\text{Diagonal} = \sqrt{l^2 + b^2}$
• Square: Side $a$.
  • $\text{Area} = a^2 = \frac{1}{2} \times d^2$ (where $d$ is diagonal)
  • $\text{Perimeter} = 4a$
  • $\text{Diagonal} = a\sqrt{2}$

Example: A rectangular park has length $20\text{ m}$ and width $15\text{ m}$. Find its diagonal length. $\text{Diagonal} = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25\text{ m}$.

4. Quadrilaterals

• Parallelogram: $\text{Area} = \text{Base} \times \text{Height}$
• Rhombus: A parallelogram with all sides equal.
  • $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are diagonals)
  • $\text{Side} = \frac{1}{2} \sqrt{d_1^2 + d_2^2}$
• Trapezium: One pair of parallel sides ($a$ and $b$) and distance between them ($h$): $\text{Area} = \frac{1}{2} \times (a + b) \times h$

Example: Find the area of a trapezium whose parallel sides are $12\text{ cm}$ and $18\text{ cm}$, and the distance between them is $10\text{ cm}$. $\text{Area} = \frac{1}{2} \times (12 + 18) \times 10 = \frac{1}{2} \times 30 \times 10 = 150\text{ cm}^2$.

5. 3D Shapes (Mensuration)

• Cube: Side $a$.
  • $\text{Volume} = a^3$
  • $\text{Total Surface Area (TSA)} = 6a^2$
  • $\text{Diagonal} = a\sqrt{3}$
• Cuboid: Length $l$, width $w$, height $h$.
  • $\text{Volume} = l \times w \times h$
  • $\text{TSA} = 2(lw + wh + hl)$
• Cylinder: Radius $r$, height $h$.
  • $\text{Volume} = \pi r^2 h$
  • $\text{Curved Surface Area (CSA)} = 2\pi rh$
  • $\text{TSA} = 2\pi r(r + h)$
• Cone: Radius $r$, height $h$, slant height $l = \sqrt{r^2 + h^2}$.
  • $\text{Volume} = \frac{1}{3} \pi r^2 h$
  • $\text{CSA} = \pi rl$
  • $\text{TSA} = \pi r(r + l)$
• Sphere: Radius $r$.
  • $\text{Volume} = \frac{4}{3} \pi r^3$
  • $\text{Surface Area} = 4\pi r^2$

Example: Find the volume and surface area of a sphere of radius $3\text{ cm}$ (take $\pi \approx 3.14$). $\text{Volume} = \frac{4}{3} \times 3.14 \times 3^3 = 4 \times 3.14 \times 9 = 113.04\text{ cm}^3$. $\text{Surface Area} = 4 \times 3.14 \times 3^2 = 113.04\text{ cm}^2$.

🌐 Learning Resources

✅ Practice Links

📝 Key Formulas

2D SHAPES:
  Rectangle:  Area = l × b,         Perimeter = 2(l+b)
  Square:     Area = a²,             Perimeter = 4a
  Triangle:   Area = ½ × b × h,     Heron's: √[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2
  Circle:     Area = πr²,            Circumference = 2πr
  Trapezium:  Area = ½ × (a+b) × h

3D SHAPES:
  Cube:     Volume = a³,             SA = 6a²
  Cuboid:   Volume = l×b×h,          SA = 2(lb+bh+hl)
  Cylinder: Volume = πr²h,           SA = 2πr(r+h)
  Cone:     Volume = ⅓πr²h,          SA = πr(r+l)  [l = slant height]
  Sphere:   Volume = 4/3 πr³,        SA = 4πr²

🔗 Related Topics

• Ratios and Proportion — Scaling and ratio of areas or volumes
• Equations — Solving for dimensions or variables in geometric word problems
• Data Interpretation — Interpreting spatial or geometric dimensions from tables or charts
• Time Speed and Distance — Tracking pathways or tracks (circular/linear)
• Permutations Combinations and Probability — Geometric probability and permutations of points