Explore Geometry Concepts and Challenges with Visual Representations
Delve into the world of geometry through visually engaging images covering various topics such as area, volume, exact geometry, and random procedures. From simulated random points to real-world applications, witness the beauty and practicality of geometric principles in action.
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Presentation Transcript
Formula-Free Geometry
Exact Geometry Easy! Pretty easy! 1 1 1 2 1 2 1 1 ? = ?? = ? = ?? =
Inexact Geometry No formula for green area 1 ? = ?
Random Procedure Uniform Random Occurring with equal probability Drop a point at any location within the square The likelihood of a point falling inside the region is determined by the proportion of area that the shaded region fills ??? ???? ??? ????
Random Points 24 points total 11 points inside ??? ???? 1 ??? ???? green area 11 24
Real Geometry Can we use this for real problems? YES! 25000 km
Real Geometry Use a computer to generate MANY random points 25000 km
Real Geometry 10 cm
Todays Challenge The constant ? describes the geometry of any circle of any size. Ever wonder why ? =3.14159? Today, we ll use approximated geometry to investigate ?
Circle in a Square Known formula: if you know ? 2
Simulated Random Points Use the NXT Brick to Simulate many random 2-D points Use ?2+ ?2= ?2 Use ??? ???? to estimate area
How Close to ?? Find average of five estimates Find the percent error relative to 3.14159 Find the standard error of samples Tighter error better confidence
How Close? Standard Error Samples: {3.1635, 3.1393, 3.1453} Average: ? = 3.1494 Sum of deviations: S = (3.1635 ?)2+ (3.1393 ?)2+(3.1453 ?)2 S = 0.0003176 Standard Error: ? SE = ? = 3 3 1 SE = .007276
