Ch 06 - Measuring Space: Perimeter and Area9th Ganita Manjari
Constant Circumference to Diameter Ratio
What fundamental geometric relationship defines the constant for all circles?
What fundamental geometric relationship defines the constant for all circles?
Invariant Ratio
The ratio of a circle's Circumference () to its Diameter () is identical for all circles, regardless of their scale.
The Constant
This fixed, universal ratio is defined as the mathematical constant (Pi).
Formula Link
This gives the foundational relation .
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Archimedes' Bounds for Pi
What were the mathematical bounds for established by Archimedes?
What were the mathematical bounds for established by Archimedes?
Method
Archimedes trapped the circle between inscribed and circumscribed 96-sided polygons.
Bounds
- Lower Bound:
- Upper Bound:
Significance
He proved that the true value of strictly lies between these two fractions.
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Zu Chongzhi's Rational Approximations of Pi
Which two famous rational fractions did the Chinese mathematician Zu Chongzhi discover to approximate ?
Which two famous rational fractions did the Chinese mathematician Zu Chongzhi discover to approximate ?
- Yuelü (Approximate Ratio):
- Miü (Close Ratio):
Significance
The fraction is the most accurate rational fraction with a denominator under 15,000, and it remained the best approximation in the world for over 800 years.
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Also in this chapter
41 more cards to study in Prism
- 4Irrationality of Pi vs 22/7
- 5Madhava's Exact Infinite Series for Pi
- 6Perimeters of Basic Polygons
- 7Distance Traveled by a Rotating Wheel
- 8Calculating Number of Wheel Revolutions
- 9Area of a Rectangle and Square
- 10Area of a Triangle (Standard)
- 11Median-Area Theorem of Triangles
- 12Area of a Parallelogram
- 13Parallelogram Area Constraint
- 14Area of a Trapezium
- 15Area of a Rhombus
- 16Heron's Formula
- 17Area of an Equilateral Triangle
- 18Area of an Isosceles Triangle
- 19Brahmagupta's Formula
- 20Brahmagupta's Formula Application Limits
- 21Heron's Formula as a Special Case
- 22Area of Triangle via Circumcircle Radius
- 23Area of Triangle via Incircle Radius
- 24Area of a Regular Polygon via Incircle
- 25Length of a Circular Arc
- 26Athletics Track Stagger Calculation
- 27Area of a Circular Sector
- 28Area of a Circular Segment (Definition)
- 29Area of a Circular Ring (Annulus)
- 30Finding Side Lengths from Algebraic Ratios
- 31Identical Area and Perimeter Constraints
- 32Area of Midpoint Parallelogram in a 4-gon
- 33The Subtraction Strategy for Complex Regions
- 34Geometric Setup for Centers-Intersecting Circles
- 35Area Fraction of Identical Circles Packed in a Rectangle
- 36Scale Invariance of Perimeter-Squared to Area Ratio
- 37Area Ratio: Inscribed Equilateral Triangle to Circle
- 38Area Ratio: Inscribed Regular Hexagon to Circle
- 39Area Ratio: Inscribed Square to Circle
- 40Geometric Area Model for (a+b)^2
- 41Geometric Area Model for Difference of Squares
- 42Algebraic Basis of Squaring a Rectangle (Baudhayana)
- 43Semicircle Generalisation of Pythagoras
- 44Semicircular Path Length Invariance
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9th Ganita Manjari
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