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Constructions and Tilings – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash II, tailored for Class 7 in Mathematics.
This one-pager compiles key formulas and equations from the Constructions and Tilings chapter of Ganita Prakash II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
AB = 2r
AB represents the length of the perpendicular bisector, and r is the radius used to create arcs above and below the line segment XY. This equals the diameter of a circle with radius r, forming the basis for constructing perpendicular bisectors.
∠AOX = ∠AOY = 90°
This states that the angles formed between line segments and the perpendicular bisector at the midpoint of XY are each 90 degrees, establishing the perpendicular property of line AB to XY.
m + n = total length in tiling
Where m is the number of 2 × 1 tiles and n is the number of 1 × 1 tiles used in a grid. This equation helps determine how to efficiently cover a rectangular grid.
Area of a regular hexagon = (3√3/2)(s²)
s is the side length of the hexagon. Knowing the area is essential for constructing and visualizing hexagonal designs.
θ = 360°/n (for regular polygons)
Where θ is the individual angle at each vertex and n is the number of sides. This formula helps in constructing regular shapes like pentagons and hexagons.
∠COA = ∠AOB = 60°
This states that the angles in an equilateral triangle (which can be a basis for hexagon construction) are 60 degrees, essential for angle bisection and polygon construction.
Perimeter of a regular polygon = n × s
Where n is the number of sides and s is the length of each side. This is crucial for ensuring constructed shapes maintain their geometric integrity.
Diagonal of a rectangle = √(l² + w²)
Where l is the length and w is the width. Useful for checking the dimensions of tiles during tiling problems.
Length of arc = (θ/360°) × 2πr
Where θ is the angle in degrees and r is the radius. This formula is essential when constructing circular arcs in geometric designs.
Area of a trapezium = 1/2 × (a + b) × h
Where a and b are the lengths of the parallel sides and h is the height. It helps in tiling calculations with irregular shapes.
Equations
If XY is divided into 2 equal parts, then AX = AY = BX = BY
This equation defines the property of a perpendicular bisector in geometric constructs. It is essential for constructing symmetrical shapes.
A + B = 180° (Angles on a straight line)
Where A and B are angles formed by intersecting lines. Understanding this relationship aids in ensuring the integrity of constructed angles.
Total area = Area covered by tiles + Area left uncovered
This equation is foundational in tiling problems to understand coverage and efficiency.
For 2 × 1 tiles on a grid: Total tiles = total area/2
This indicates how many tiles can cover a defined area. Utilized in practical tiling scenarios.
A single unit covers 2 squares (if working with 2 × 1 tiles)
Defines the capacity of each tile type in covering areas within grids, which is beneficial in tiling questions.
∠XAO + ∠YAO = 180° (Straight angles)
Reflects properties of angles formed through constructions, aiding in angle division and bisection.
A1 + A2 + ... + An = total area
Ensures the cumulative area of multiple shapes is calculated, finding applications in complex tiling and construction problems.
x + y = total length of arc (in concentric circles)
Where x and y are individual arcs in a circular arrangement. This is relevant in advanced constructions involving arcs.
Total degrees in a polygon = (n-2) × 180°
Where n is the number of sides in any polygon, guiding students in understanding the relationship between sides and angles.
Number of congruently spaced points = n × k (where k is a constant multiplier)
This helps in finding equally spaced points necessary in various geometric constructions and designs.
