2D Mensuration: Master Formula Sheet
Handwritten Notes for Teaching Exams (CTET, KVS, DSSSB, State TET)
1. Triangles (त्रिभुज)
General Triangle
- Perimeter ($P$): $a + b + c$
- Semi-perimeter ($s$): $\frac{a+b+c}{2}$
- Area (Base & Height): $\frac{1}{2} \times b \times h$
- Heron's Formula:
$Area = \sqrt{s(s-a)(s-b)(s-c)}$
Right-Angled Triangle
- Pythagoras: $H^2 = P^2 + B^2$
- Area: $\frac{1}{2} \times P \times B$
- 🌟 TET Super Tricks:
- Inradius ($r$): $\frac{P + B - H}{2}$
- Circumradius ($R$): $\frac{H}{2}$
Equilateral Triangle
- Area: $\frac{\sqrt{3}}{4} a^2$
- Height/Altitude ($h$): $\frac{\sqrt{3}}{2} a$
- Inradius ($r$): $\frac{a}{2\sqrt{3}}$
- Circumradius ($R$): $\frac{a}{\sqrt{3}}$
- Ratio of $r : R = 1 : 2$
- Ratio of Incircle Area : Circumcircle Area $= 1 : 4$
Isosceles Triangle
(Two sides equal '$a$', base '$b$')
- Height ($h$): $\frac{1}{2} \sqrt{4a^2 - b^2}$
- Area: $\frac{b}{4} \sqrt{4a^2 - b^2}$
- Tip: Usually faster to split it into two right-angled triangles instead of memorizing this!
2. Quadrilaterals (चतुर्भुज)
Rectangle (आयत)
- Area: $L \times B$
- Perimeter: $2(L + B)$
- Diagonal ($d$): $\sqrt{L^2 + B^2}$
- Room Walls Area: $2h(L+B)$
Square (वर्ग)
- Area: $a^2$ OR $\frac{d^2}{2}$ (Important!)
- Perimeter: $4a$
- Diagonal ($d$): $a\sqrt{2}$
- Incircle radius $r = \frac{a}{2}$
- Circumcircle radius $R = \frac{a}{\sqrt{2}}$
Rhombus (समचतुर्भुज)
- Area: $\frac{1}{2} \times d_1 \times d_2$
- Perimeter: $4a$
- Relation: $4a^2 = d_1^2 + d_2^2$
- Note: Diagonals bisect at $90^\circ$
Trapezium (समलम्ब)
- Area: $\frac{1}{2} \times (a + b) \times h$
- Median length: $\frac{a + b}{2}$
Parallelogram (समान्तर चतुर्भुज)
- Area: $Base \times Height$ ($b \times h$)
- Perimeter: $2(a + b)$
- Diagonals bisect each other (but NOT at $90^\circ$ unless it's a rhombus).
- Sum of adjacent angles $= 180^\circ$.
3. Circles & Curves (वृत्त)
Circle & Semi-Circle
- Circle Area: $\pi r^2$
- Circumference: $2\pi r$
-
Semi-Circle Perimeter:
$\pi r + 2r$ (Don't forget the base!) - Semi-Circle Area: $\frac{\pi r^2}{2}$
Sector & Circular Ring (Annulus)
Sector (त्रिज्यखंड)
- Arc Length ($l$): $\frac{\theta}{360} \times 2\pi r$
- Area: $\frac{\theta}{360} \times \pi r^2$
- Shortcut Area: $\frac{1}{2} \times l \times r$
Ring (Outer $R$, Inner $r$)
- Area of Ring: $\pi (R^2 - r^2)$
- Calculation shortcut: $\pi (R + r)(R - r)$
- Width of path: $R - r$
4. Regular Polygons (सम बहुभुज)
Angles & Diagonals
For a polygon with '$n$' sides:
- Sum of Int. Angles: $(n - 2) \times 180^\circ$
- Each Int. Angle: $\frac{(n - 2) \times 180^\circ}{n}$
- Sum of Ext. Angles: $360^\circ$ (Always!)
- Each Ext. Angle: $\frac{360^\circ}{n}$
- Number of Diagonals = $\frac{n(n - 3)}{2}$
Regular Hexagon (समषट्भुज)
A regular hexagon is made of 6 equilateral triangles.
- Area: $6 \times \left( \frac{\sqrt{3}}{4} a^2 \right) = \frac{3\sqrt{3}}{2} a^2$
- Perimeter: $6a$
- Circumradius ($R$): $a$ (Radius = side)
- Inradius ($r$): $\frac{\sqrt{3}}{2} a$
🔥 EXAM MASTER TRICKS (TET/KVS Favorites) 🔥
🛣️ Pathways in Rectangles
(Field dimensions $L, B$, path width $w$)
-
1. Path OUTSIDE the field:
Area = $2w(L + B + 2w)$ -
2. Path INSIDE the field:
Area = $2w(L + B - 2w)$ -
3. Two CROSSED paths in middle:
Area = $w(L + B - w)$
📈 Percentage Changes
-
If $L$ increases by $x\%$ and $B$ by $y\%$:
Area Change = $\left(x + y + \frac{xy}{100}\right)\%$ -
If radius/side increases by $x\%$:
Area increases by $\left(2x + \frac{x^2}{100}\right)\%$
🎡 Wheel Revolutions
- Distance in 1 Rev: Circumference ($2\pi r$)
-
No. of Revolutions ($n$):
$n = \frac{\text{Total Distance}}{\text{Circumference}}$
📌 Golden Rule of Wire Reshaping:
If a wire is bent from one shape to another (e.g., Circle to Square), their Perimeters are EQUAL ($2\pi r = 4a$).
