This unit will focus on measurement for two dimensional figures. You have discovered that the sum of the angle measures in a triangle is \(180^{\circ}\). In this lesson you will investigate the classifying polygons and determining angle measures.
All of the figures that you grouped in the Discuss This can be found in part of the MC Escher woodcut Metamorphisis III. Can you see them?
All of the figures that you grouped in the Discuss This can be found in part of the MC Escher woodcut Metamorphisis III. Can you see them?
Classifying Polygons
Let's look at some vocabulary.
Polygon - a closed figure created by line segments which intersect only at their endpoints
Convex - a polygon with each interior angle measures less than \(180^{\circ}\).
Let's look at some vocabulary.
Polygon - a closed figure created by line segments which intersect only at their endpoints
Convex - a polygon with each interior angle measures less than \(180^{\circ}\).
Non-convex or Concave - a polygon with at least one interior angle that measures greater than \(180^{\circ}\).
Regular Polygon - a polygon with all sides congruent and all angles are congruent.
Polygons are named according to their number of sides, whether they are regular or not. Some have specific names, as listed below. Any number of sides not on the list can be called either a #-gon or simply a polygon. For example, a polygon with \(11\) sides can either be called an \(11\)-gon or a polygon.
Example 1:
Is each figure a polygon? Explain your reasoning.
Is each figure a polygon? Explain your reasoning.
Solution:
Figure 1: No. A polygon is defined as a closed figure created by line segments which intersect only at their endpoints.
Since Figure 1 has a curved side, that is not a line segment.
Figure 2: Yes! This figure is created by line segments which intersect only at their endpoints.
Figure 3: No. This figure’s segments do not only intersect at their endpoints.
Figure 4: No. This is not a closed figure (the top is open).
Example 2:
Classify each polygon. Use “concave” or “convex”, name the polygon according to its number of sides, and mention other special features (such as “regular”).
Figure 1: No. A polygon is defined as a closed figure created by line segments which intersect only at their endpoints.
Since Figure 1 has a curved side, that is not a line segment.
Figure 2: Yes! This figure is created by line segments which intersect only at their endpoints.
Figure 3: No. This figure’s segments do not only intersect at their endpoints.
Figure 4: No. This is not a closed figure (the top is open).
Example 2:
Classify each polygon. Use “concave” or “convex”, name the polygon according to its number of sides, and mention other special features (such as “regular”).
Solution
a) Convex Heptagon
b) Concave Octagon
c) Regular Pentagon
d) Concave Decagon
a) Convex Heptagon
b) Concave Octagon
c) Regular Pentagon
d) Concave Decagon
From the Investigation for this target we discovered that polygons can be divided into triangles using the diagonals from a vertex. This led to creating an expression for the sum of the interior angles in a polygon.
If \(n\) represents the number of sides in the polygon, you can use the following equations to find the measure of the following angles:
- Sum of the measures of the interior angles of a polygon = \(180^{\circ}(n - 2)\)
- Measure of each interior angle of a regular polygon = \(\dfrac{(180^{\circ}(n - 2))}{n}\)
- Sum of the measures of the exterior angles of a polygon = \(360^{\circ}\)
- Measure of each exterior angle of a regular polygon = \(\dfrac{360^{\circ}}{n}\)
Example 3:
Determine the sum of the interior angles for each polygon.
Determine the sum of the interior angles for each polygon.
Solution:
a) The polygon is a heptagon. The sum of the interior angles is \(180^{\circ}(7-2)=180^{\circ}(5)=900^{\circ}\)
b) The polygon is an octagon. The sum of the interior angles is \(180^{\circ}(8-2)=180^{\circ}(6)=1080^{\circ}\)
Example 4:
Classify the polygon with the following angle measures and determine the missing angle measure:
\(63^{\circ}\), \(194^{\circ}\), \(120^{\circ}\),\(90^{\circ}\), \(150^{\circ}\), \(x\)
Solution:
Since there are six angles, the figure has six sides and is a hexagon
Since one angle is greater than \(180^{\circ}\), then it is a concave hexagon.
The sum of the interior angles of a hexagon is \(180^{\circ}(6-2)=180^{\circ}(4)=720^{\circ}\).
To calculate the value of \(x\):
\(63^{\circ} + 194^{\circ} + 120^{\circ} + 90^{\circ} + 150^{\circ} + x = 720^{\circ}\)
\(617^{\circ} + x= 720^{\circ}\)
So \(x=103^{\circ}\)
a) The polygon is a heptagon. The sum of the interior angles is \(180^{\circ}(7-2)=180^{\circ}(5)=900^{\circ}\)
b) The polygon is an octagon. The sum of the interior angles is \(180^{\circ}(8-2)=180^{\circ}(6)=1080^{\circ}\)
Example 4:
Classify the polygon with the following angle measures and determine the missing angle measure:
\(63^{\circ}\), \(194^{\circ}\), \(120^{\circ}\),\(90^{\circ}\), \(150^{\circ}\), \(x\)
Solution:
Since there are six angles, the figure has six sides and is a hexagon
Since one angle is greater than \(180^{\circ}\), then it is a concave hexagon.
The sum of the interior angles of a hexagon is \(180^{\circ}(6-2)=180^{\circ}(4)=720^{\circ}\).
To calculate the value of \(x\):
\(63^{\circ} + 194^{\circ} + 120^{\circ} + 90^{\circ} + 150^{\circ} + x = 720^{\circ}\)
\(617^{\circ} + x= 720^{\circ}\)
So \(x=103^{\circ}\)