Mensuration 2D (Area & Perimeter)

Area: $A = \frac{1}{2} \times \text{Base} \times \text{Height}$

Heron's Formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (Semi-perimeter).

Equilateral Triangle:

• Area: $A = \frac{\sqrt{3}}{4} a^2$
• Height: $h = \frac{\sqrt{3}}{2} a$

Rectangle:

• Area: $L \times B$
• Perimeter: $2(L+B)$
• Diagonal: $\sqrt{L^2 + B^2}$

• Area: $a^2$ or $\frac{1}{2} \times (\text{Diagonal})^2$
• Perimeter: $4a$
• Diagonal: $a\sqrt{2}$

Area: $A = \pi r^2$
Circumference: $C = 2\pi r$
Semi-Circle: Area = $\frac{\pi r^2}{2}$, Perimeter = $\pi r + 2r$.

Sector:

• Area = $\frac{\theta}{360} \times \pi r^2$
• Arc Length ($l$) = $\frac{\theta}{360} \times 2\pi r$

Sum of Interior Angles: $(n-2) \times 180^\circ$
Each Interior Angle (Regular): $\frac{(n-2) \times 180}{n}$
Sum of Exterior Angles: $360^\circ$

Regular Hexagon:

• Area: $6 \times \frac{\sqrt{3}}{4} a^2 = \frac{3\sqrt{3}}{2} a^2$
• It consists of 6 Equilateral Triangles.

Path outside a Rectangle:
Area = $(L + B + 2w) \times 2w$

Path inside a Rectangle:
Area = $(L + B - 2w) \times 2w$

Cross Paths:
Area = $w(L + B - w)$
(where $w$ is the width of the path).