A Square and A Cube – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part I, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the A Square and A Cube chapter of Ganita Prakash Part I. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Area of a square: A = s²
A is the area (in square units), and s is the length of one side. This formula calculates the area of a square, useful in geometry.
Square of a number: n²
n is any number. Squaring a number multiplies it by itself; for example, 4² = 16.
Perfect square criterion
A number is a perfect square if it has an odd number of factors. Identifying square numbers is crucial in various mathematical applications.
Difference of squares: a² - b² = (a - b)(a + b)
This shows that the difference between two square numbers can be factored into the product of the sum and difference; useful in algebra.
Relationship of consecutive square numbers: n² - (n-1)² = 2n - 1
This tells us how the difference between consecutive squares increases; it’s always an odd number.
Volume of a cube: V = s³
V is the volume (in cubic units), and s is the length of one side. This formula is applied in three-dimensional geometry.
Cube of a number: n³
n is any number. Cubing a number means multiplying it by itself three times; for example, 3³ = 27.
Sum of the first n odd numbers: S = n²
This states that the sum of the first n odd numbers equals the square of n. This relation is fundamental in number theory.
Square root definition: √y = x, if x² = y
This defines the square root. For instance, √36 = 6 because 6² = 36. Useful in simplifying expressions.
If y = x², then x = √y
This shows the inverse relation between squaring a number and taking its square root, essential for solving quadratic equations.
Equations
1² = 1
Square of 1, establishing the base property of perfect squares.
2² = 4
This demonstrates the square of the first natural number.
3² = 9
Another example of squaring, confirming sequential square growth.
4² = 16
Continues the pattern of perfect squares; helps in recognizing composite numbers.
5² = 25
Shows that squares can represent products; useful in problem-solving.
1³ = 1
Square of a cube number; establishes a base for understanding volume.
2³ = 8
A critical example of cube growth, integral for geometry.
3³ = 27
Demonstrates increasing values within cubic progression.
4³ = 64
Continues the pattern of cubic values and their applications.
5³ = 125
Final example in the sequence, encapsulating properties of cube numbers.
