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Warm-Up

Warm-Up . 1. Find the perimeter of the polygon:. 160 m. 2 . Find the area of the green shaded region:. 7 cm. 24 cm². 4 cm. 2 cm. Math Jokes of the Day!!!. Which triangles are the coldest? . Ice- sosceles triangles . .

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Warm-Up

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  1. Warm-Up 1. Find the perimeter of the polygon: 160 m 2. Find the area of the green shaded region: 7cm 24 cm² 4 cm 2 cm

  2. Math Jokes of the Day!!! Which triangles are the coldest? Ice-sosceles triangles. What did the student say when the witch doctor removed his curse? Hex-a-gon!

  3. Section 11.3 (Part 1)Areas of Circles and Regular Polygons 5/1/14

  4. Learning Goals for Day 1 of 11.3 • You will all be able to: • Define a polygon and use prefixes that describe polygons according to their number of sides • Compute the area of a circle • Define our new vocabulary word of the day! • Use what you know about perimeters and common area formulas to compute the area of a Regular Polygon • Compute areas of shaded regions within another polygon or circle

  5. What is a polygon? • Polygon – The word "polygon" derives from the Greek words poly (many) and gonu (knee). So a polygon is a thing with many knees! It must have straight edges that connect at vertices. • Polygons can be concave or convex and irregular or regular. • “Regular” – all sides congruent • Today we will just be focusing on REGULAR Polygons (Specifically equilateral triangles, squares, & hexagons)

  6. Polygon Prefixes…

  7. Fun Fact: The more sides a polygon has, the more “circular” it becomes!

  8. Area of a Circle Area of ⊙ = π times the (radius)²

  9. Example #1: Find the exact Area of ⊙ C , which has a diameter of 14 cm. C A ⊙ = πr² A ⊙ = π(7cm)² Don’t forget your units and that they are squared, because it’s an area problem!!! A ⊙ = 49πcm²

  10. Let’s Break Down a Regular Polygon… • A regular polygon with “n” sides is comprised of “n” congruent, isosceles triangles. • The base of each triangle is one side of the polygon. • The two congruent legs of each isosceles triangle are considered the radii of the polygon. The radii start from the center point and end at each vertex. • The height of each triangle is called an apothem.

  11. Central Angles of Regular Polygons • The central angles of a regular polygon are all congruent and add up to 360° (Central ∡ Sum Theorem ⊙) • To find the measure of one central angle, use the following formula: • One Central ∡ = 360° ÷ n “n” being the number of sides the polygon has

  12. Example #2: • Find the central angle for each regular polygon: A. B. • n = 6 • n = 10 • Central ∡ = 360° ÷ 6 • Central ∡ = 360° ÷ 10 • Central ∡ = 60° • Central ∡ = 36°

  13. New Vocabulary Word of the Day! • Apothem – a line segment drawn from the center of a polygon perpendicular to one of its sides, bisecting that sideand the central angle. • *** It is a PERPENDICULAR BISECTOR! *** Sounds like …. A Opossum

  14. Why is the Apothem Important? • The apothem is needed when finding the area of a regular polygon. (It’s the height of each isos. △ in the polygon) • The formula for the area of a regular polygon is: A = ½ the apothem times the perimeter or You can think of area of the entire polygon as _____________________ up the area of all of the ___________________ ________________ that make up the polygon; hence using the PERIMETER! The perimeter is just the base of each triangle added together!!! ADDING ISOSCELES TRIANGLES

  15. When finding the Area of a Regular Polygon…. • You might be given just the perimeter, or one side length, or the apothem, or the radius… Or you may be given a combination of these facts. • (Best case scenario is where you are given the perimeter/or side length, and the apothem because you are given everything for the Area formula!) • Depending on what you are given, just reminder that you always have the central angles to rely on and the fact that the apothem bisects this angle and the side length.

  16. Quick Refresher of Special △’s 30°-60°-90° 45°-45°-90° Hypot = twice (short) or 2(short) Long Leg = (short) Hypot = leg

  17. Example #3: • Find the area of the following regular polygon with a radius of 4 cm. ** What do we still need in order to find the area? A = ½ a·P Central ∡= 360°÷3 = 120° One side length = 4 cm P = 3(4 cm) = 12 cm A = ½ a·P A = ½ (2cm)·(12 cm) A = 12 cm² 4 cm 2 cm 2 cm 60° 4cm 30° a Apothem bisects each central angle, so 120° ÷ 2 = 60° each

  18. Example #4: • Find the area of the following regular polygon: A = ½ a·P A = ½ a·60cm 60° A = 150 cm² A = ½ 5 cm·60cm 60° 60° 60° 60° 30° 5 cm 30° 60° 5cm 60°

  19. Example #5: • A square is circumscribed about a circle. Find the area of the shaded region if the circle has a radius of 8m: Area of Shaded Region = Area  - Area  Area  = (16m)(16m) = 256 m² Area  = π·(8m)² = 64π m² Area = (256 – 64π) m² 8m 16m

  20. Section 11.3 (Part 2)Areas of Circles and Regular Polygons 5/2/14

  21. Right Triangle Trigonometry! SOH – CAH - TOA

  22. Ex 1: Find the area of the regular nonagon with a perimeter of 180 cm .

  23. Ex 2: Find the area of the regular heptagon with an apothem of 6 in.

  24. Ex 3: Find the area of the non-shaded region if the circle has a radius of 17 ft.

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