Polygon Properties and Tiling Concepts

Study Notes

A polygon is a closed shape composed of at least three sides and three angles.
A convex polygon has all interior angles measuring less than 180 degrees.
The sum of the interior angles of any n-sided convex polygon can be expressed as 180(n-2) degrees. This is derived by dividing the polygon into n triangles and subtracting the central angles.

A hexagon is a six-sided polygon.
A regular hexagon has all sides and angles equal.
To find the measure of each interior angle of a regular hexagon:
- Calculate the sum of interior angles using the formula 180(n-2), where n = 6. The sum is 720 degrees.
- Divide the sum (720 degrees) by the number of angles (6) to find the measure of each angle, which is 120 degrees.

To tile a floor with congruent regular octagons and squares with the same side length, the angles around each vertex must add up to 360 degrees.
For each vertex:
- The square contributes 90 degrees.
- The octagon contributes 135 degrees (calculated by dividing the sum of interior angles (180(8-2) = 1080 degrees) by 8 angles).
The angles add up to 360 degrees (90 + 135 + 135), confirming the tiles will fit together without gaps or overlap.

A regular pentagon has five equal sides and five equal interior angles.
To find the measure of each interior angle of a regular pentagon:
- Calculate the sum of interior angles using the formula 180(n-2), where n = 5. The sum is 540 degrees.
- Divide the sum (540 degrees) by the number of angles (5) to find the measure of each angle, which is 108 degrees.
In an isosceles triangle, two sides are equal, and the angles opposite those sides are also equal.
Triangle ACE:
- The base angles are 36 degrees each.
- The apex angle is 108 degrees.
- The triangle is isosceles because two angles are equal (36 degrees each).