Comprehensive Guide to 2D and 3D Geometrical Formulas

Circle Properties

• The circle is defined by its radius ($r$).
• Area: The total space enclosed within the boundary of the circle is calculated as:   $Area = \pi \times r^2$
• Circumference (Perimeter): The total length of the boundary around the circle is given by:   $Circumference = 2 \times \pi \times r$

Scalene Triangle Properties and Heron's Formula

• A scalene triangle is a triangle in which all three sides have different lengths, denoted here as $a$, $b$, and $c$.
• Semi-perimeter ($s$): To find the area, first calculate the semi-perimeter, which is half of the triangle's total perimeter:   $s = \frac{a+b+c}{2}$
• Area (Heron's Formula): The area can be determined using the lengths of the sides and the semi-perimeter:   $Area = \sqrt{s(s-a)(s-b)(s-c)}$

Semicircle Properties

• A semicircle is exactly one-half of a circle.
• Area: The area is half that of a full circle:   $Area = \frac{\pi \times r^2}{2}$
• Circumference: This refers specifically to the curved boundary of the semicircle:   $Circumference = \pi \times r$
• Total Perimeter [P]: The perimeter of a semicircle includes both the curved boundary and the flat diameter ($2 \times r$):   $Perimeter = \pi \times r + 2 \times r$

Rectangle Properties

• A rectangle is defined by its length ($l$) and breadth ($b$).
• Area: Calculated by multiplying the side lengths:   $Area = \text{side} \times \text{side} = l \times b$
• Perimeter: The total length of all four sides:   $Perimeter = 2 \times (l+b)$

Square Properties and Diagonal Calculation

• A square has four equal sides, each denoted as $a$.
• Area: The area is the square of the side length:   $Area = (\text{side})^2 = a^2$
• Perimeter: The sum of all four equal sides:   $Perimeter = 4 \times a$
• Diagonal of a Square ($d$): The diagonal can be calculated using the Pythagorean theorem:
  • Relationship: $d^2 = a^2 + a^2$
  • Simplified: $d = \sqrt{a^2 + a^2}$
  • Final form: $d = \sqrt{2} \times a$

Three-Dimensional Geometrical Figures: Sphere

• Total Surface Area (TSA): The surface area covering the entire exterior of the sphere:   $TSA = 4 \times \pi \times r^2$
• Volume: The total amount of 3D space occupied by the sphere (noted as highly important or "Imp"):   $Volume = \frac{4}{3} \times \pi \times r^3$

Three-Dimensional Geometrical Figures: Hemisphere

• A hemisphere is half of a sphere, possessing both a curved surface and a flat circular base.
• Total Surface Area (TSA): The TSA is the sum of the curved surface area (half of the sphere's TSA) and the area of the base circle:
  • Calculation: $TSA = \frac{4 \times \pi \times r^2}{2} + \pi \times r^2$
  • Intermediate step: $TSA = 2 \times \pi \times r^2 + \pi \times r^2$
  • Final result: $TSA = 3 \times \pi \times r^2$
• Volume: The volume is half that of a full sphere:   $Volume = \frac{2}{3} \times \pi \times r^3$

Mathematical Constants

• The following constant value is provided for geometric calculations:   $\sqrt{3} = 1.73$