Geometry Concepts Chapter 10 Polygons and Area Name polygons according to sides and angles Find measures of interior angles Find measures of exterior angles Estimate and find areas of polygons Estimate and find perimeters of polygons
Section 10.1 Naming Polygons Questions to think about: Definition Characteristics POLYGON Example Nonexample Definition Characteristics REGULAR POLYGON Example Nonexample Page 2 of 11
EXAMPLES 1.) Is this a polygon? If not explain why. 2.) Is this a polygon? If not explain why. 3.) Is this a polygon? If not explain why. 4.) Is this a polygon? If not explain why. 5.) Is this a polygon? If not explain why. 6.) Is this a polygon? If not explain why. 7.) Is this a polygon? If not explain why. 8.) Is this a polygon? If not explain why. 9.) Identify the polygons by the number of sides. 10.) Identify the polygons by the number of sides. 11.) Identify the polygons by the number of sides. 12.) Identify the polygons by the number of sides. 13.) Identify the polygons by the number of sides. 14.) Identify the polygons by the number of sides. Page 3 of 11
15.) Is the polygon regular? 16.) Is the polygon regular? 17.) Is the polygon regular? 18.) Is the polygon regular? Definition Characteristics CONVEX Example Nonexample Definition Characteristics CONCAVE Example Nonexample Page 4 of 11
EXAMPLES 19.) Is the polygon concave or convex? 20.) Is the polygon concave or convex? 21.) Is the polygon concave or convex? 22.) Is the polygon concave or convex? 23.) Is the polygon concave or convex? 24.) Is the polygon concave or convex? Section 10.2 Diagonals and Angle Measures Questions to think about: THEOREM 10.1 If a convex polygon has n sides, then the sum of the measures of its interior angles is (n 2)180. EXAMPLES 25.) Find the sum of measures of the interior angles. Find the measure of one interior angle. 26.) Find the sum of measures of the interior angles. Find the measure of one interior angle. Page 5 of 11
THEOREM 10.2 In any convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360. EXAMPLES 27.) Find the measure of one exterior angle of a regular heptagon. 28.) Find the measure of one exterior angle of a regular quadrilateral. Section 10.3 Areas of Polygons Questions to think about: POSTULATE 10.1 For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the polygon. POSTULATE 10.2 Congruent polygons have equal measures. POSTULATE 10.3 The area of a given polygon equals the sum of the areas of the nonoverlapping polygons that form the given polygon. Page 6 of 11
EXAMPLES 29.) Find the area of the polygon. Each square 30.) Find the area of the polygon. Each square 31.) Find the area of the polygon. Each square 32.) Estimate the area of the polygon. Each square 33.) Estimate the area of the polygon. Each square 34.) Estimate the area of the polygon. Each square Page 7 of 11
Section 10.4 Areas of Triangles and Trapezoids Questions to think about: THEOREM AREA of a TRIANGLE 10.3 If a triangle has an area of A square units, a base of b units, and a corresponding altitude of h units, then A = ½bh. EXAMPLES 35.) Find the area of each triangle. 36.) Find the area of each triangle. 37.) Find the area of each triangle. 38.) Find the area of each triangle. Page 8 of 11
39.) Find the area of each triangle. 40.) Find the area of each triangle. 41.) Find the area of each triangle. 42.) Find the area of each triangle. THEOREM AREA of a TRAPEZOID 10.4 If a trapezoid has an area of A square units, bases of b 1 and b 2 units, and an altitude of h units, then A = ½h(b 1 + b 2 ). EXAMPLES 43.) Find the area of each trapezoid. 44.) Find the area of each trapezoid. Page 9 of 11
45.) Find the area of each trapezoid. 46.) Find the area of each trapezoid. 47.) Find the area of each trapezoid. 48.) Find the area of each trapezoid. Section 10.5 Areas of Regular Polygons Questions to think about: Definition Characteristics CENTER and APOTHEM Example Nonexample Page 10 of 11
THEOREM AREA of a REGULAR POLYGON 10.5 If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then A = ½aP. A = ½ (6)(40) A = 3(40) A = 120 cm 2 EXAMPLES 49.) Find the area of the regular polygon. 50.) Find the area of the regular polygon. 51.) Find the area of the shaded region of the regular polygon. 52.) Find the area of the shaded region of the regular polygon. 53.) Find the area of the shaded region of the regular polygon. Page 11 of 11