![Page 1: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/1.jpg)
Formula-Free Geometry
![Page 2: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/2.jpg)
Area and Volume
![Page 3: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/3.jpg)
Exact GeometryEasy! Pretty easy!
11
1 1
12 12
= ½ = ¼
![Page 4: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/4.jpg)
Inexact Geometry
1
𝑨=?
No formula for green area
![Page 5: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/5.jpg)
Random Procedure
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
𝑁 𝑖𝑛
𝑁𝑡𝑜𝑡
≈𝐴𝑖𝑛
𝐴𝑡𝑜𝑡
Uniform RandomOccurring with equal probability
![Page 6: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/6.jpg)
Random Points
24 points total 11 points inside
green area ≈
𝑁 𝑖𝑛
𝑁𝑡𝑜𝑡
≈𝐴𝑖𝑛
𝐴𝑡𝑜𝑡
1
![Page 7: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/7.jpg)
Real GeometryCan we use this for real problems?
25000 km
YES!
![Page 8: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/8.jpg)
Real Geometry Use a computer to generate MANY random points
25000 km
![Page 9: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/9.jpg)
Real Geometry
![Page 10: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/10.jpg)
Real Geometry
10 cm
![Page 11: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/11.jpg)
Today’s 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
![Page 12: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/12.jpg)
Circle in a SquareKnown formula: if you know
2
![Page 13: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/13.jpg)
Simulated Random Points
Simulate many random 2-D points
Use Use to estimate area
Use the NXT Brick to…
![Page 14: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/14.jpg)
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
![Page 15: Formula-Free Geometry. Area and Volume Exact Geometry Easy! Pretty easy! 11 1 1](https://reader035.vdocuments.us/reader035/viewer/2022062722/56649f345503460f94c5236d/html5/thumbnails/15.jpg)
How Close? Standard Error Samples: {3.1635, 3.1393, 3.1453} Average: = 3.1494 Sum of deviations:
S = S = 0.0003176
Standard Error: SE = SE = .007276