1. Section 10.3 Polygons and Perimeter Math in Our World

2. Learning Objectives  Find the sum of angle measures of a polygon.  Find the angle measures of a regular polygon.  Find the perimeter of a polygon.

3. Polygons Closed geometric figures whose sides are line segments are classified according to the number of sides. These figures are called polygons.

4. Polygons

5. Sum of the Angle Measures The sum of the measures of the angles of a polygon with n sides is (n – 2)180°.

6. EXAMPLE 1 Finding the Sum of Angle Measures of a Polygon Find the sum of the measures of the angles of a heptagon. A heptagon has seven sides, so the sum of the measures of the angles of the heptagon is SOLUTION The sum of the measures of the angles of a heptagon is 900°.

7. Quadrilaterals Just as there are special names for certain types of triangles, there are names for certain types of quadrilaterals as well. A trapezoid is a quadrilateral that has exactly two parallel sides. A parallelogram is a quadrilateral in which opposite sides are parallel and equal in measure.

8. Quadrilaterals A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram in which all sides are equal in length. A square is a rhombus with four right angles.

9. Relationship of the Quadrilaterals Looking at the relationships, you can see that a square is also a rectangle and a rhombus. A rhombus and a rectangle are also parallelograms.

10. Regular Polygons In a regular polygon all of the sides have the same length, and all of the angles are equal in measure. The most common examples of regular polygons are squares and equilateral triangles.

11. EXAMPLE 2 Finding Angle Measure for a Regular Polygon Find the measure of each angle of a regular hexagon. SOLUTION First, find the sum of the measures of the angles for a hexagon. The formula is (n – 2)180°, where n is the number of sides. Since a hexagon has six sides, the sum of the measures of the angles is (6 – 2)180° = 720°. Next, divide the sum by 6 since a hexagon has six angles: 720° ÷ 6 = 120°. Each angle of a regular hexagon has a measure of 120°.

12. Perimeter The perimeter of a polygon is the sum of the lengths of its sides.

13. EXAMPLE 3 Finding the Perimeter of a Rectangle The Houser family finds their dream home perfect in every way except one: the backyard is not fenced in, and their dog Bunch needs room to roam. The rectangular portion they plan to enclose is 95 feet wide and 70 feet long. How much fence will they need to enclose the yard on all four sides?

14. EXAMPLE 3 Finding the Perimeter of a Rectangle SOLUTION The amount of fence needed is the perimeter of the rectangle. The Housers need 330 feet of fence.

15. EXAMPLE 4 Finding the Perimeter of a Polygon The length of each outside wall of the Pentagon is 921 feet. Suppose that a sentry must walk the outside wall six times during his 4-hour shift. How many miles does he walk in one shift? A pentagon has five sides, and each has length 921 feet, so the sum of the lengths of the sides is 5 x 921 = 4,605 ft. In walking the perimeter six times, the sentry covers 6 x 4,605 = 27,630 feet. Now we convert to miles: SOLUTION