In a 10th-grade geometry course, students learn about the properties of objects and the relationships among them. Students study types of angles, shapes and lines. They investigate the concepts of similarity, congruence, parallelism and transformations. They construct proofs and perform calculations, such as finding distances and circumference. Although students may find much of this material dry, they may be surprised to learn some interesting facts about geometry throughout their course of study.

### Historical Trivia

Geometry is an ancient subject -- Egyptians used certain principles of geometry as far back as 3,000 BC. The word “geometry” is of Greek origin and translates literally to “earth measure,” as it was originally developed as a means of quantifying elements of the natural world. The type of geometry that 10th grade students study is formally known as Euclidean geometry. It takes its name from a Greek mathematician named Euclid who lived in the third century B.C. Until the 1800s, the world knew no other type of geometry aside from Euclidean geometry.

### A Standard of Beauty

A geometric ratio known as the Golden Mean has been a hallmark of beauty in artwork, architecture and the human form for millennia. Also known as the Golden Ratio or Divine Proportion, this phenomenon is encountered when calculating ratios of side lengths in basic figures like the pentagon, and approximates a number close to 1.62. Because it is considered the most attractive geometric ratio to the human eye, it has been used as the basis for creating well-known aesthetically pleasing works, such as the Great Pyramid at Giza, the Parthenon and the Mona Lisa.

### Pi

In any circle -- no matter how large or how tiny – its circumference divided by its diameter always produces the same number, known as pi. Pi is commonly approximated to 3.14; so, in other words, the circumference of any given circle is always slightly more than three times its diameter. However, 3.14 is merely an estimate; in fact, the digits of pi continue endlessly with no discernible pattern. Mathematicians have used computers to calculate pi to more than one trillion digits.

### The Banach-Tarksi Paradox

Suppose a solid sphere is cut into a number of pieces. Those pieces can be reassembled into two spheres -- each exactly the same size and shape as the original. During reassembling, the pieces need only be moved and rotated, not shrunken or enlarged. It is quite hard for students to believe, but the results are real and make for an interesting experiment.

### Euler's Formula

Consider a polyhedron -- a three-dimensional solid with flat faces -- such as a cube, pyramid or soccer ball. In any given polyhedron, adding the number of faces to the number of vertices and then subtracting the number of edges always produces the same answer: two. This is known as Euler’s formula. For example, a cube possesses six faces, eight vertices, and 12 edges: 6 + 8 – 12 = 2.

#### References

- Wolfram MathWorld: Geometry
- Regents Exam Prep Center: Geometry
- Regents Exam Prep Center: Euclidean and Non-Euclidean Geometry
- Encyclopedia Britannica: Euclidean Geometry
- Kids Math Games: Geometry Facts
- The Geometry Center: The Golden Ratio
- The Golden Number: The Golden Section
- Math.com: A Brief History of Pi
- Pi Day: Learn About Pi
- Harvey Mudd College: Banach-Tarski Paradox

#### Resources

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