Have you ever played with shapes and patterns? That's what geometry is all about! It's the study of lines, angles, shapes, and space. From drawing cool designs to building structures, geometry is an important part of our world.

The hypothesis of the experiment is that the ratio between the error in determining I using by inscribing polygons within and circumscribing polygons about a circle with (km) sides and that obtained using polygons with (kn) sides will approach (n/m)^2 as k increases.

How fast does your estimate of pi improve when you use polygons with more sides? Ancient math used shapes drawn inside and around circles to estimate pi. The more sides the polygon has, the closer the estimate gets.

You calculate pi by averaging the perimeters of polygons inscribed in and drawn around a circle. Then you compare the errors from polygons with different numbers of sides. You test six different pairs and track how the error ratio changes as the side count grows.

The error ratio follows a clear pattern. It approaches the square of the inverse of the side-count ratio. This means the error shrinks in a predictable way as you add more sides.

Title: The Pleasure of Pi

Objectives/Goals

The hypothesis of the experiment is that the ratio between the error in
determining I using by inscribing
polygons within and circumscribing polygons about a circle with (km) sides and
that obtained using
polygons with (kn) sides will approach (n/m)^2 as k increases.

Methods/Materials

To test my hypothesis, I needed to develop formulas to determine the perimeters
of the regular polygons
inscribed within and circumscribed about a circle. I discovered that the
perimeter of the regular polygon
with X sides inscribed in a circle with a diameter of 1 is X(sin(180/X)). The
perimeter of the regular
polygon of X sides circumscribed about a circle with a diameter of 1 is
X(tan(180/X). I estimated I by
using the expression:
(X(sin(180/X) + X(tan(180/X))) / 2, and I calculated the error in estimating pi
using polygons with the
formula:
error = ((X(sin(180/X) + X(tan(180/X))) / 2) - I.
I calculated the ratios of the errors of the estimates using polygons of m and n
sides employing six
different values for m and n [(m=8, n=10,) (m=6, n=8,) (m=4, n=6,) (m=4, n=8,)
(m=4, n=10,) and (m=4,
n=12)]. I then calculated the error ratios for polygons of km and kn sides using
those given m and n
values, and k = (1, 2, 3, 4, and 1000). Finally, I graphed the results.

Results

The graphs are consistent with the hypothesis. As k increases, the error ratio
approaches (n/m)^2, the
square of the inverse of the ratio of the number of sides.

Conclusions/Discussion

By completing this experiment, I discovered that the ratio between the error in
determining I using by
inscribing polygons within and circumscribing polygons about a circle with (km)
and (kn) sides
approaches (n/m)^2 as k increases.

Summary Statement

The summary is that I determined that the ratio between the error in estimating
pi using by inscribing
polygons within and circumscribing polygons about a circle with (km) and (kn)
sides approaches (n/m)^2
as k increases.

Help Received

Dad helped edit my report.

Polygon Sides and Pi Estimation Error

The pleasure of Pi

<h1><strong>1000 Science Fair Project Ideas by Subject</strong></h1><p>Looking for a science fair project idea? Each of our science fair project idea has complete instructions with video. Get ideas for your science projects, find extra resources for your class' scientific projects, or try a fun experiment at home! It doesn't matter what grade you're in, there's something for everyone!</p><p>You can find your perfect science fair project from the different science subjects below or from our <a href="https://www.all-science-fair-projects.com/category0.html#bottom">science fair project idea lists!</a></p>

<h2>Science Fair Project Idea Lists</h2><p>Looking for other ways to find good ideas for science fair projects? Check out these lists of science projects!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>Easy Science Fair Projects</h3><p><img src="/images2/flex/176_popular 2 (2).png" data-size="100" data-float="left" style="width:100px; float:left">Explore the exciting world of science with an easy science fair project that you can do at home using simple materials. Any one of these easy project ideas will teach you important scientific concepts and help you have fun while learning. An easy science fair project is a great way to introduce the wonders of science to kids and adults alike!</p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><ul><li><p>Changing the Color of Flowers</p></li><li><p>Baking Soda Volcano</p></li><li><p>Swimming Raisins</p></li><li><p>Egg in Vinegar</p></li><li><p>Color of Skittles Experiment</p></li><li><p>Jumping Rice Krispies</p></li><li><p>Mixing Colors of Light</p></li><li><p>Camouflage and Hide</p></li><li><p>Light and Dark on Seed Germination</p></li><li><p>Make a Parachute</p></li></ul><p>More science project details at <a href="https://www.all-science-fair-projects.com/science_fair_project_list/12">Easy Science Fair Projects <strong>→</strong></a></p><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><hr style="clear:both; border-style:dotted" data-marker="clearFloat"><h3>The Top 10 Best Science Fair Projects</h3><p><img src="/images2/flex/177_popular 2 (3).png" data-size="100" data-float="left" style="width:100px; float:left">To find the best science fair project for you, we've picked the best science projects from 10 different science categories, all with very interesting topics. 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Math is all about numbers, patterns, and solving problems. Computer science is about using math and technology to create computer programs and solve real-world problems. Together, they help us understand how things work, how to solve problems, and how to use technology to make our lives better!

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Polygon Sides and Pi Estimation Error

Hypothesis

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