Comprehensive Geometry and Mensuration Study Guide

Sum of all interior angles: The total sum of all interior angles in any convex polygon with $n$ sides is calculated using the formula: $(n - 2) \times 180^{\circ}$ .
Each exterior angles: According to the provided material, the individual exterior angles are given by the formula: $\frac{(n - 2) \times 180^{\circ}}{n}$ .
Sum of each exterior angle: The cumulative sum of all exterior angles for any regular polygon is always constant at $360^{\circ}$ .
Each exterior angle: To calculate the measure of a single exterior angle in a regular polygon with $n$ sides, use: $\frac{360^{\circ}}{n}$ .
Number of Diagonals: The total number of diagonals that can be drawn in a polygon with $n$ sides is: $\frac{n \times (n - 3)}{2}$ .
Sum of each interior and exterior angle: At any vertex of a polygon, the sum of the adjacent interior and exterior angle remains: $180^{\circ}$ .

Perimeter (Circumference): The perimeter of a circle is expressed as $P = \pi \times d$ or $2 \times \pi \times r$ .
Area: The total space enclosed by a circle is calculated as: $\pi \times r^2$ .
Perimeter of rubberband enclosing n no. of circles: When a rubber band is stretched around $n$ identical circles, the formula for its perimeter is: $n \times d + 2 \times \pi \times r$ .
Area of shaded region: The area of the shaded region in specific configurations is given by: $\frac{3}{14} \times a^2$ .

Volume: The total capacity or space occupied by a prism is: $\text{base area} \times \text{height}$ .
LSA/CSA (Lateral Surface Area / Curved Surface Area): The total area of the side faces of a prism is: $\text{base perimeter} \times \text{height}$ .
TSA (Total Surface Area): The entire surface area, including the top and bottom bases, is: $\text{LSA} + 2 \times \text{base area}$ .

Volume: The cubic space occupied by a cube with side length $a$ is: $a^3$ .
LSA (Lateral Surface Area): The area of the four side faces is: $4 \times a^2$ .
TSA (Total Surface Area): The area of all six faces of the cube is: $6 \times a^2$ .

Volume: The capacity of a cuboid defined by length ( $l$ ), breadth ( $b$ ), and height ( $h$ ) is: $l \times b \times h$ .
LSA (Lateral Surface Area): The surface area of the four vertical sides is: $2 \times (l + b) \times h$ .
TSA (Total Surface Area): The total surface area including the top and bottom faces is: $2 \times (l + b) \times h + 2 \times l \times b$ .

Volume: The volume of a pyramid is one-third of a prism with the same base: $\frac{1}{3} \times \text{base area} \times \text{height}$ .
LSA/CSA: The Lateral Surface Area is calculated using the slang height: $\frac{1}{2} \times \text{base perimeter} \times \text{slang height}$ .
TSA (Total Surface Area): The sum of the lateral area and the base: $\text{LSA} + \text{base area}$ .

Volume: The total space inside a sphere is: $\frac{4}{3} \times \pi \times r^3$ .
LSA/TSA: For a complete sphere, the lateral surface area and total surface area are identical: $4 \times \pi \times r^2$ .
Solid sphere cut into two pieces: When a solid sphere is divided into two parts, the combined total surface area of both halves becomes: $6 \times \pi \times r^2$ .

Volume: The volume of a half-sphere is: $\frac{2}{3} \times \pi \times r^3$ .
LSA (Lateral Surface Area): The area of the curved surface only: $2 \times \pi \times r^2$ .
TSA (Total Surface Area): The sum of the curved surface and the flat circular base: $3 \times \pi \times r^2$ .

Perimeter: The total distance around the triangle, defined as the sum of all sides: $a + b + c$ .
Area (Heron's Formula): Calculates area using the semiperimeter ( $s$ ): $\sqrt{s \times (s - a) \times (s - b) \times (s - c)}$ .
Area (Trigonometric): Based on two sides and the included angle: $\frac{1}{2} \times a \times b \times \sin(\theta)$ .
Radius of incircle: This is found by dividing the area by the semiperimeter: $\frac{\text{area of } \Delta}{\text{semiperimeter}}$ .
Radius of circumcircle: The distance from the center to the vertices is: $\frac{a \times b \times c}{4 \times \text{area of } \Delta}$ .

Area: Specific area calculation for right triangles: $\frac{1}{2} \times \text{base} \times \text{altitude}$ .
Radius of incircle: Calculated from the perimeter (P), base (B), and hypotenuse (H): $\frac{P + B - H}{2}$ .
Radius of circumcircle: This is exactly half of the hypotenuse: $\frac{H}{2}$ .

Perimeter (P): The sum of the three equal sides: $3 \times a$ .
Area: The area specific to equilateral triangles: $\frac{\sqrt{3}}{4} \times a^2$ .
Radius of incircle: Defined as: $\frac{a}{2 \times \sqrt{3}}$ .
Radius of circumcircle: Defined as: $\frac{a}{\sqrt{3}}$ .
Altitude: The perpendicular height of the triangle: $\frac{\sqrt{3}}{2} \times a$ .

Squares:
- Area: $a^2$
- Diagonal: $a \times \sqrt{2}$
Rhombus:
- Area: Calculated using the two diagonals ( $d_1$ and $D_2$ ): $\frac{1}{2} \times d_1 \times D_2$
Rectangles:
- Perimeter (P): $2 \times (l + b)$
- Area: $l \times b$
- Diagonal: The distance between opposite corners: $\sqrt{l^2 + b^2}$
Parallelograms:
- Area: $\text{base} \times \text{height}$
Trapezium:
- Area: Based on the sum of parallel sides ( $a$ and $b$ ): $\frac{1}{2} \times (a + b) \times h$

Area: Comprised of six equilateral triangles: $6 \times \frac{\sqrt{3}}{4} \times a^2$ .
Radius if incircle: Calculated as: $\frac{\sqrt{3}}{2} \times a$ .
Radius if circumcircle: Equal to the side of the hexagon: $a$ .

Area: The total area for a regular octagon is: $2 \times \text{side}^2 \times (\sqrt{2} + 1)$ .