Exploring Some Geometric Themes – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part II, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the Exploring Some Geometric Themes chapter of Ganita Prakash Part II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
R_n = 8^n
R_n represents the number of remaining squares at the nth step in the Sierpinski Carpet sequence. Each square remaining results in 8 smaller squares in the next step, demonstrating exponential growth.
H_(n + 1) = H_n + R_n
H_n denotes the number of holes at the nth step. This formula connects the holes with the squares and shows how holes accumulate as squares are removed.
Perimeter (P) = 3^n
For the Koch Snowflake, P is the perimeter at the nth step, where each side is further divided into 3 segments, showing a fractal-like increase.
Area remaining after nth step (Sierpinski Triangle) = (1/2)^n
This formula gives the area of the remaining shape after n iterations, showing how the area decreases with each iteration.
Total Faces (F) = E - V + 2
This is Euler's formula for polyhedra where E is the number of edges and V is the number of vertices, linking basic properties of solids.
Net area of a square pyramid = B + (1/2)Pl
Where B is the base area and P is the perimeter of the base, and l is the slant height. This is used to find the surface area of pyramid shapes.
Volume (V) of a cuboid = l × w × h
Here, l, w, and h represent the length, width, and height respectively. It is foundational for calculating the volume of three-dimensional shapes.
Volume (V) of a cylinder = πr²h
Where r is the radius and h is the height, this formula helps in calculating the volume of cylindrical shapes in practical applications.
Volume (V) of a cone = (1/3)πr²h
Like the cylinder, this formula accounts for the radius and height but includes the factor of 1/3 due to tapering.
Volume of a triangular prism = (1/2) × base × height × length
This calculates the volume of a prism using its triangular base, height of the triangle, and length extending to the back.
Equations
R_0 = 1
Base case for the number of squares in the Sierpinski Carpet at the zeroth step, being the initial square.
H_0 = 0
This indicates that there are no holes at the zeroth step of the Sierpinski Carpet sequence; it starts with a complete square.
R_1 = 8
At the first step of the Sierpinski Carpet, there are 8 remaining squares after removing the center square.
H_1 = 1
At the first step, one hole is created after the center square is removed from the Sierpinski Carpet.
Area of triangle remaining at nth step = (B×H)/2 - (1/2)×(B×H×Sum(1/2)^(n-1))
This calculates the remaining area after each step in constructing Sierpinski's Triangle.
Perimeter of Koch Snowflake = 3 × (4/3)^n
The initial perimeter of an equilateral triangle is multiplied by a fraction due to the addition of segments in subsequent steps.
V = (4/3)πr³
Formula for the volume of a sphere, showing how the radius directly affects the overall space occupied.
V = (n × (n-1))/2 for n-sided polygon
This describes the number of diagonals in a polygon based on the number of sides it has.
Surface Area of a cylinder = 2πrh + 2πr²
This encompasses both the curved surface area and the areas of the circular bases.
Surface Area of a cone = πr(l + r)
This combines both the curved surface and the base area, revealing how varying the radius and slant height change the total area.
