8.3/8.4 Areas of Circles, Sectors and Regular Polygons Name Geometry 3313
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1 8.3/8.4 Areas of Circles, Sectors and Regular Polygons Name Geometry 3313 Key Learning Targets: a. I can derive the formula for the area of a circle. b. I can apply the area of a sector formula to solve problems. c. I can apply the area formulas of a circle and a regular polygon to solve problems. Investigation #1: Deriving the Area of a Circle 1. After the circle s circumference is unfolded and the pieces are fit together, what shape does the figure start to resemble? Rectangle Parallelogram 2. How would you find the area of this shape? A=bh 3. What are the dimensions of the shape in terms of r (the radius of the circle)? b=g=25i=ar h=r 4. Based on this information, make a conjecture about the area formula for a circle. A=bh A.ir.r As,Yr2 Area and Circumference Formulas for Circles: Tip Area: A = Circumference: C = air
2 Examples: 1. What is a better deal, one 16-inch diameter pizza for $20 or two 8-inch diameter pizzas for $10 each? Why? 2. If the area of the circle is 196π ft 2, 3. If the circumference of the circle is 196π ft, find the circumference. find the area. anetiiiin.saaieiaiiixaoaiifnoeeris8eidgire@ei.eee.eiitoeeai.e :# Sector: The part of a circle enclosed by two radii of a circle and their intercepted area. aaaifef.tt#r2s196r=fa6r: Investigation #2 Step 1: For each shaded region, find what fraction of the circle each sector is. Step 2: Find the area of each full circle. Step 3: Combine the results in Steps 1 and 2 to find the area of each sector. Circle Fraction of Circle Area of Circle (in terms of ) Area of Sector (in terms of ) o= I, a. 62 tt.36t36tcm@9tcmgee3aydyiemg3yiffideo-ts5tt9igeeetigcent3rgloani.t Area Formula for a Sector: Area: A =, r2
3 TGP Example: 1. Suppose the slices of pizza below have the same price. Which piece of pizza is the better deal? Why? TC ins 26.2 ins FeeaYs nee%htoe+re9yagiiteee*rf Regular Polygon: a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Suppose the following two pizzas cost the same price. Which pizza is the better deal, a circular pizza with a 16 inch diameter or a regular hexagonal pizza where both the radius and side length 8 inches long? Why? AsTCSY Ab=tsC4X6.9) Anex. Dazs ) A = ins As Anex= 165.6in '?fhfnayaargpeizaajetekekrdedn=f
4 Investigation #3: Deriving the Area of a Regular Polygon Step 1: Consider a regular pentagon with side length s, divided into congruent isosceles triangles. Each triangle has base length s and height a. What is the area of one isosceles triangle in terms of a and s? As = tasa Step 2: What is the area of this pentagon in terms of a and s? = = A (E) 5Sa 2.5 Sa pent Step 3: Repeat Steps 1 and 2 with other regular polygons and complete the table below. 2. Ssa 3s a 3.55A 4sa 4.5 Sa Esa tansa The distance a appears in the area formula for a regular polygon, and it has a special name apothem. An apothem of a regular polygon is a perpendicular segment from the center of the polygon s circumscribed circle to a side of the polygon. You may also refer to the length of the segment as the apothem. Step 4: What is the perimeter of a regular polygon in terms of n (number of sides) and s? Use your answer to this question and your last entry in the table to fill in this conjecture. Area Formula for Regular Polygons Conjecture: The area of a regular polygon where A is the area, P is the perimeter, a is the apothem, is given by: A = Pens +2Pa
5 Examples: 1. Use the area formula for a regular polygon to find the area of the pizza below. 2. Find the area of the regular pentagon Pi8CcDA-tzC48k6.DP-49As1G5.6i@P-7.5GTA-tzC37.5X5.D below. P=37.5 A=95.G25u@ 3. Suppose a regular hexagon has an area of 522 m 2 and a side length of 12 m. Find the length of the apothem of the hexagon. P=l2C6) A=tzPa P = EYE.3Ea 4. Suppose a regular pentagon has an area of 33 ft 2 and an apothem length of 3 ft. Find the length of each side of the pentagon. A=ZPa 22 33=tap( 3) 33=1.5P 22 =P 5- =4.4f
6 5. Suppose we have the regular hexagon below. 60 a. Use the interior angle of a regular polygon formula, Each Interior Angle = (n 2)180 :, n to find the measure of each interior angle of the regular hexagon =120 b. Draw in the apothem of the regular hexagon to form a triangle. What are the angle measurements of the triangle? triangle c. Use SOHCAHTOA or special right triangles to find the length of the apothem. go. L 2h10 OR =nb n=5 10 d. Use SOHCAHTOA or special right triangles to find the side length = fool Osin6O=a 2h10 a=5b 7=a zda ORzn.to#n=55=ya=5F3 Y sin3o= 6dL Yot y=n 60L 2h10 y=5 a µ losin3e=y e. Find the area of the regular hexagon above =10 P=1oCa)=6o A=ts(Go)C5B) A=259.8iTf
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