7.1 Angle Measures in Polygons POLYGON INTERIOR ANGLES THEOREM. n = and. n-gon. Name Period. Sum of the measures of the interior angles of any polygon

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1 7.1 Angle Measures in Polygons Name Period POLYGON INTERIOR ANGLES THEOREM Sum of the measures of the interior angles of any polygon n = and Classify polygons by # of sides 3 Triangle 6 Hexagon 9 Nonagon 4 Quadrilateral 7 Heptagon 10 Decagon 5 Pentagon 8 Octagon 12 Dodecagon Example 1: Use the Interior Angles Theorem 1. Find the sum of the measures of the interior angles of a convex 22-gon. n n-gon 2. The sum of the measures of the interior angles of a convex polygon is Classify the polygon by the number of sides. Example 2: Apply the Interior Angles Theorem 1. a. n = b. Classify the polygon c. Solve for x. 2. a. n = b. Classify the polygon c. Find the 1

2 3. The measures of the interior angles of a pentagon are, and What is the measure of the largest angle? POLYGON EXTERIOR ANGLES THEOREM The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is. Example 3: Find missing angles. 1. Find the value of x. 2. What is the sum of the exterior angles of a 14-gon? 3. Two interior angles of a triangle have measures 42 and 56. Which of the following could not be a measure of an exterior angle of the triangle? A. 98 B. 82 C. 124 D. 138 REGULAR POLYGON: A polygon with all sides and angles. Example: Regular Pentagon 1) What is the measure of each exterior angle? 2) What is the measure of each interior angle? Example 4: With Regular Polygons 1. What is the measure of each exterior angle of a regular nonagon? 2. Each interior angle of the regular n-gon measures of 165. Classify the polygon by the number of sides. 2

3 3. What is the measure of an interior angle of a regular hexagon? 4. If an interior angle of a regular polygon is, what is the measure of an exterior angle? 5. Given a regular decagon, find the following information: (Don t forget your diagram). A. Sum of the exterior angles B. Each exterior angle C. Each interior angle D. Sum of the interior angles RECAP (note: n = and ) ANY convex polygon REGULAR convex polygon S E = E E = S I = E I = At a vertex, an interior and exterior angle are 1. If the interior angle sum of a regular polygon is 2160, then find the measure of one exterior angle. 2. If an interior angle of a regular polygon is 165, then find (a) the measure of an exterior angle (b) the interior angle sum of the polygon 3. If each exterior angle of a regular polygon is, then classify the polygon by the number of sides. 4. Given a regular nonagon, what is the sum of the measures of the exterior angles? 3

4 7.2 Circumference and Area of Circles CIRCUMFERENCE C = in terms of - leave π in your answer - NO DECIMAL answer! AREA OF A CIRCLE A = **NOTE** Circumference is measured in single units (not square units like area) Example 1: Find the circumference or area. 1. Find the circumference and area of a circle with a diameter of 26 meters. Leave your answers in terms of π. 2. Find the diameter of a circle with an area of 361π m Find the area of the shaded region. Round your answer the nearest tenth. 4. Find the area of the shaded region. Leave your answer in terms of. 2m 6m 5. If the area of a circle is 121 cm 2, what is the circumference? 6. If the circumference of a circle is 48 inches, then what is its area? 4

5 7. The circle below is inscribed in the square. If the circumference of the circle is, then find the area of the square. 8. The square is circumscribed about the circle. If the area of the square is 64 square meters, then find the circumference of the circle. If a tire has a circumference of 13 inches, how far would it travel if it rotated twice? Distance traveled = x Example 2: Finding distance traveled 1. How far does a tire with a diameter of 32 inches travel in 30 revolutions. Round your answer to the nearest tenth. 2. The dimensions of the skateboard wheel shown at the right are in millimeters. To the nearest millimeter, how far does the wheel travel when it makes 35 revolutions? The tires of an automobile have a diameter of 24 inches. Find the number of revolutions the tire will make if it travelled 3120 inches. (Round answer to the tenths place). 4. The tire of a tricycle travels 216 feet in 12 revolutions. What is the circumference of the tricycle tire? What is its radius? MINI REVIEW 1. Four of the interiors angles of a pentagon are 90, 143, 77, and 103. Find the measure of the missing angle. 2. Solve for x. 2x x 60 5

6 7.3 Area of Basic Polygons AREA OF A PARALLELOGRAM (includes rectangles, rhombuses, and squares) A = b = h = Example 1: Name the figure then use a formula to find area of the shaded region 1. ft ft m AREA OF A TRAPEZOID A = h b b b **NOTE** The height of a trapezoid is a perpendicular segment between the bases. (BASES parallel sides) b h Example 2: Find the area of the shaded region in in 12 in m 28 m 18 m 25 m m 15 m 35 m 5. is an isosceles trapezoid with legs and. If, find and. 6

7 AREA OF A RHOMBUS AREA OF A KITE A = [You can still use A = bh (parallelogram)] A = Example 3: Name the figure then use a formula to find its area cm 5 cm Example 4: Find the area of the equilateral triangle * Remember! Each angle in an equilateral is So when the altitude cuts the triangle in half, you get two triangles Find the area of the figure below. 2. Find the area of an equilateral triangle with a side length of 18. Notice a Pattern? EQUILATERAL Area Formula 1. Use the formula to find the area of an equilateral triangle with a side length of 16 cm. 2. Find the area of an equilateral triangle with a side length of 30 in. 7

8 Example 5: Solve for unknown measures 1. A triangle has an area of 126 square feet and a height of 14 feet. What is the length of the base? 2. The parallelogram shown at the right has an area of 70 square feet. Find the value of x. SHADED AREA: find the area of the shaded region Find the area of the figure comprised of a kite and a trapezoid. 4. Find the area of the shaded region. 8

9 7.4 Perimeter/Area of Similar Figures + Review of Angles in Polygons PERIMETER OF SIMILAR POLYGONS If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their perimeters is : AREA OF SIMILAR POLYGONS If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is : ( ) ( ) SIDE : SIDE = PERIMETER : PERIMETER SIDE 2 : SIDE 2 = AREA : AREA Example 1: Use the similar polygons to find the unknown perimeters and areas I I II II P I = 15 in P II = P I = P II = A I = A II = 27 in 2 A I = 72 m 2 A II = 3. JANE is similar to BURT where the ratio of similarity is 5:12. If BURT is the smaller quadrilateral with a perimeter of 32.5, then what is the perimeter of JANE? A. 78 B C. 93 D Two similar trapezoids have a similarity ratio of 5:3. If the area of the smaller trapezoid is 54 square feet, then find the area of the larger trapezoid. A. 270 B. 90 C D

10 5. If the ratios of the areas of two similar hexagons is 225:64. What is the ratio of the lengths of corresponding sides? 6. The hexagons below are similar. The perimeter of the larger hexagon is 64 inches and the perimeter of the smaller hexagon is 24 inches. (a) Find the ratio of the corresponding side lengths. (b) Find the ratio of the areas of the larger hexagon to the smaller hexagon. 7. Your mother has taken her favorite baby picture of you and wants to enlarge the size of the photo to place over the fireplace. She wants to place a nice ribbon along the edges of the enlarged photo. What total length of ribbon would she need to cover the border? 8. Jill built a smaller version of a door that she liked for her dollhouse. She would like to paint the door on the doll house red. If the orginial door needed enough paint to cover 80 square inches, then how many square inches would need to covered by paint in the doll house? cm cm i cm i i i cm cm i Review So Far 1. What is the measure of one exterior angle of a regular dodecagon? A. 36 B. 360 C. 30 D Find the area of the shaded region. A. 36 B. C. D If an interior angle of a regular polygon is, then classify the polygon. 4. The diagram below shows the size of the tire that Billy uses on his unicycle. The area of the tire is ft 2. If Billy rode his unicycle feet, then how many revolutions did the tire make? 10

11 5. Find the area of the shaded region. 6. If an interior angle of a regular polygon is, then what is the measure of an exterior angle? 7. Find the area of the figure below. A B. 980 C D The ratio of similarity between to pentagons is 3:7. If the area of the larger pentagon is 147 square units, then what is the area of the smaller pentagon? A. 63 B. 27 C. 142 D If the circumference of a circle is meteres, then find the area of the circle. A. 225π B. 225 C. 15 D. 30π 10. Find the area of the shaded region. Leave your answer in terms of. A. 144 B. C. D. i 11. Find the area of the parallelogram. A. 30 B. 28 C. 24 D If the ratio of the perimeters of two similar figures is 3:5, then what is the ratio of the areas? A. 3:5 B. 9:25 C. 25:9 D. 5:3 13. Daisy made a reduced model of her backyard for a school project. She wants to build a white picket fence enclosing the model of the yard. What length of fencing would Daisy need in order to enclose the model of the yard? 14ft 5ft 6ft Original 8ft yard 13ft 11

12 7.5 Spiral Questions I. SOH-CAH TOA 1) Use ΔJAM to solve for x below. A. B. J 58 x A 22 2) Use the triangle below to find the length of. P A B. i C. D. i M T W C. D. 3) Find the area of the given shape. 4) Find the area of the kite. 19 m m II. CONGRUENT TRIANGLES 1) If you are given two triangles, ΔMNZ and ΔUTH where and What additional information would be sufficient to show that ΔMNZ ΔUTH? A. B. C. D. and is a right angle 2) Given,, and. Why are the triangles congruent? K A. ASA B. AAS C. SAS D. HL I M E Y 3) Given the diagram below, determine whether the triangles are congruent are not. If so, why? A. ASA B. SAS C. AAS D.NONE P R T T I 4) If you are given two triangles, ΔPLH and ΔRTV where and What additional information would not be sufficient to show that ΔPLH ΔRTV? A. B. C. and are right angles D. 12

13 III. Quadrilaterals 1) A quadrilateral has interior angles,, and. What is the measure of the missing interior angle? 2) Given JANE is an isosceles trapezoid with legs and. If, then find. Explain. 3) Given is an isosceles trapezoid with bases and. If, then find. Explain. A. B. C. D. 4) Three exterior angles of a quadrilateral are,, and. Which of the following could not be the measure of an interior angle? A. B. C. D. 13

14 7.6 Review Day 1. Find the area of the equilateral triangle below. 2. Find the area of the shaded region. 10 cm 6 cm 13 cm 3. If, find the perimeter of each polygon. 4. What is the measure of an interior angle of a regular nonagon? 5. Find the area of the shaded region. 5. A tire travels 324 feet in 9 revolutions. What is the length of the radius of the tire? 6. What is the sum of the exterior angles of the regular octagon? 7. The measures of the interior angles of a pentagon are,,,, and. What is the value of 14

15 8. Two similar shapes have a similarity ratio of If the smaller shape has an area of 32 inches, what is the area of the larger shape? 9. Find the area of the kite below. 10. Solve for x. 11. Find the area of the figure below. ( x ) 15 m x 39 m 12. Find the area of the figure below. 13. Each interior angle of a regular polygon has a measure of 156. Classify the polygon by the number of sides. 14. Four interior angles of a pentagon are 90, 143, 77, and 103. Find the measure of the missing angle. 15. If the radius of a tire measures 14 inches, how many inches will the tire roll in 7 revolutions. Leave your answer in terms of. 16. Find the area of the shaded region. 17. If the area of a circle is 144 in 2, find the circumference of the circle. 15

16 7.7 Area of a Regular Polygon Vocabulary REGULAR POLYGON Center Radius Apothem Central angle MEASURE OF A CENTRAL ANGLE Central Angle = ** notice that the # of central angles is the same as the # of sides. Example 1: Find the measure of the indicated angle 1. m ONH 2. m EGH 3. m ARS and m ERA T Q U P A N R E G H O S E A Finding the APOTHEM or SIDE LENGTH of a regular polygon * In a regular polygon, the apothem, radius, and ½ side length make a right triangle Use the right triangle to find the missing lengths GIVEN: 2 side lengths GIVEN: 1 side length a r 1) Use 1) Find ½ the central angle 2) Use ½ s 16

17 Example 2: Leave your answer as a simp. radical or round to the nearest tenth if necessary. 1. Find the length of the apothem of the regular polygon 2. Find the length of the apothem of the regular polygon. with a side length of 14 m and a radius of 10 m. 19 What do we need to find the Apothem and Side length for? To find the area of the and eventually the area of the entire! Find the area of the shaded region. a * notice that the base of the triangle is the same as the * notice that the height of the triangle is that same as the SO: s Example 3 1. Find the area of. 2. Find the area of. A M 7 C Finding the area of a regular polygon Ex: Find the area of the regular hexagon. 1) Find the area of one triangle A = = = = 2) Multiply the area of the triangle by the # of triangles **NOTICE** # of triangles = # of sides (n) A polygon = A n = = 17

18 AREA OF A REGULAR POLYGON A = Area of # of sides ( ) OR Height (h) of the Apothem (a) of the polygon Base (b) of the Side length (s) of the polygon Number of Number of sides (n) A = a = n = s = Example 4: Find the area of the regular polygon 1. a = n = s = A polygon = 2. a = n = s = A polygon = 3. a = n = s = A polygon = 18

19 7.8 Review ANGLE MEASURES IN POLYGONS SUM I = EACH I = SUM E = EACH E = 1. Given a regular decagon, what is the measure of each exterior angle. 2. The measures of the interior angles of a hexagon are,,,,, and. What is the value of 3. What is the measure of one interior angle of a regular 14-gon? 4. If an exterior angle of a regular polygon is 20, what is the measure of one interior angle? CIRCLES AREA of a circle CIRCUMFERENCE Revolution Problems 1. If the area of a circle is 121π meters 2, what is the circumference of the circle. 2. If the radius of a bicycle wheel measures 12 inches, how many inches will the wheel roll in 20 revolutions? AREA OF POLYGONS Parallelogram Triangle Trapezoid Equilateral Triangle Rhombus (2 nd formula) Kite 1. Find the area of the parallelogram. 2. Find the area of the rhombus. 3. Find the area of the shaded region created by the parallelogram and trapezoid. 4. Find the area of the shaded region created by the rectangle and 2 circles. 19

20 SIMILAR POLYGONS PERIMETER of similar polygons AREA of similar polygons 1. Two similar shapes have a similarity ratio of. If the larger shape has an area of 256 mm, find the area of the smaller shape. 2. The ratio of the areas of two similar figures is 196:64. What is the ratio of the length of their corresponding sides? AREA OF REGULAR POLYGONS To find the area of a regular polygon you will need 3 things: 1) 2) 3) To find the apothem (a) or side length (s), use a right created by a radius and apothem. a s r a 1 s r (Notice that the right has only half the side length of the polygon) If two sides are given, use the Pyth. TH (a 2 + b 2 = c 2 ) If only one side is given, use a *Must find 1 central first! Central = To find the AREA of the polygon Use the formula * SOH-CAH-TOA or special right triangle 1. Find the area of the regular polygon. 2. Find the area of a regular dodecagon with a side length of 30 inches. a = n = s = A polygon = a = n = s = A polygon = 20

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