TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

Ancient Egyptian civilization flourished next to the Nile River. The ancient Egyptians

made great progress in geometry. They used geometry in their daily lives to measure

land and calculate area and volume. The word geometry comes from the ancient Greek

words geo and metron. It literally means earth measurement. This describes the

origins of the mathematical discipline. Geometry was born as a way to organise and

distribute land. People in ancient Egypt worked the land on the Nile riverbanks. Each

year the Nile flooded the land. After the flood, all of the plots of land had to be drawn

again. Geometry was really important for doing this! Ancient Egyptians not only used

geometry to calculate plots of land. Look at the pyramids. They’re perfect shapes,

carefully calculated and built. They’re physical proof of the great mathematical

knowledge that the ancient Egyptians had.

1. Polygons and other flat shapes

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

Look at the shapes above. 2 , 4 , 6 and 9 are polygons. Polygons are classified

depending on the number of sides (or angles) in triangles, quadrilaterals, pentagonals,

hexagonals, etc.

Shape 7 above is a closed, complex polygonal. It is called a star polygon. Despite their

name, star polygons are not exactly polygons.

Shapes 1 , 3 and 5 are formed by arcs and segments.

Shape 8 is an ellipse. Do you want to see one for yourself? Just take some salami, and

cut it with a knife at an angle. Or pour water into a glass and tilt it.

The next shape is a polygonal line, as it is formed by series of segments, where every

two segments have a common end. It is open because two of its segments have one

free end.

The two next polygonal lines are closed because there are no segments with free ends.

The blue line is simple because the segments don’t intersect each other. The green line

is complex because some segments intersect each other.

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

A polygon is a closed 2D shape made up of three or more line segments that only

meet at their endpoints.

Parts of a polygon:

* Side: each of the line segments that forms the polygon.

* Vertex : the point where two consecutive sides meet. (The plural of vertex is

vertices)

* Diagonal: the line segment that join two non-consecutive vertices.

* Interior angle: the angle formed by two consecutive sides (inside the

polygon).

We name polygons after the number of their sides:

Angles in polygons

The angles of any triangle have a sum of 180°. A polygon with n sides can be divided

into n – 2 triangles. So:

The sum of all interior angles in a polygon with n sides is equal to:

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

For example, a pentagon can be divided into 3 triangles. So the sum of its angles is

3 · 180° = 540°.

If the pentagon is regular, each angle measures:

540° : 5 = 108°

Diagonals in a polygon

A diagonal is a segment that connects two non-consecutive vertices in a polygon. The

number of diagonals in a polygon that can be drawn from any vertex in a polygon is

three less than the number of sides. To find the total number of diagonals in a polygon,

multiply the number of diagonals per vertex (n - 3) by the number of vertices, n, and

divide by 2 (otherwise each diagonal is counted twice).

The number of diagonals of an n-sided polygon is: d = n(n − 3) / 2

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

Types of polygons

* CONVEX POLYGON: If no line that contains a side of the polygon contains a

point in the interior of the polygon. (All their interior angles are less than 180º)

* CONCAVE POLYGON: When at least on line that contains a side of the

polygon contains points in the interior of the polygon. (At least one of their angles is

greater than 180º)

* REGULAR POLYGONS: All their sides are the same length and all their interior

angles are equal in measure.

* IRREGULAR POLYGONS: They have sides and angles with different measure.

TEMA 10 GEOMETRIA DEL PLANO / Geometric figures

Regular Polygons A polygon is regular when all angles are equal and all sides are equal (otherwise it is

"irregular").The names of the regular polygons come from the number of sides they

have.

Name Figure Sides Interior Angle

Equilateral triangle

3 equal sides Each angle is 60°

Square

4 equal sides Each angle is 90°

Regular Pentagon

5 equal sides Each angle is 108°

Regular Hexagon

6 equal sides Each angle is 120°

Regular Heptagon

7 equal sides Each angle is 128.57°

Regular Octagon

8 equal sides Each angle is 135°

In a regular polygon, we can find:

CENTRE: The point where all the diagonals intercept.

RADIUS: The segment that joins the centre with any vertex on a regular polygon.

APOTHEM: The line segment that joins the middlepoint of any side with the centre.

CENTRAL ANGLE: The angle which vertex is on the centre of the polygon and its legs

are two consecutive segments that pass by two consecutive vertices of the polygon.