Entry Task Find the unknown side lengths in each special right triangle. 1. a 30°-60°-90° triangle with hypotenuse 2 ft 2. a 45°-45°-90° triangle with leg length 4 in. 3. a 30°-60°-90° triangle with longer leg length 3m

I can…... Learning Target Areas of Regular Polygons 10.3 … work with the relationship between the dimensions of and the area of any Regular Polygon. (Section 10.3)

Today’s Group Task Find the area of a regular “n-gon” with a side length of x. If you need other side lengths, define a variable and use it. In other words, create a formula that will work for ALL regular polygons. (a triangle, a hexagon, a square)

The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Vocabulary Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

apothem radius

10.3 Area of Regular Polygons Example: Find the measure of each numbered angle m ∠ 1= m ∠ 2= m ∠ 3= 360∕6 = 60° 60∕2 = 30° =60°

10.3 Area of Regular Polygons Example: Find the measure of each numbered angle m ∠ 4= m ∠ 5= m ∠ 6= 360∕8 = 45° 45∕2 = 22.5° =67.5°

Let’s see how this works… A = 1/2ap A = ½(6.88)(50) A = 172 sq.units

10.3 Area of Regular Polygons Example: Find the area of the regular polygon 8 in 12.3 in

Example: Find the area of a regular pentagon with 7.2ft sides and a 6.1ft radius. 6.1ft a A = ½ ap = ½ (4.92ft)(36ft) = 88.6ft ft p = (5)(7.2ft) = 36ft a 2 + b 2 = c 2 a = a = a 2 = a = 4.92

Area of Regular Polygons Example: Find the area of the regular polygon 18 ft 23.5 ft 9 ft 21.7 ft Use Pythagorean Theorem!

Things to Remember… s s 2s s Special Triangle shortcuts for and triangles. Remember!

Example 2: Find the apothem of a reg. hexagon with sides of 10mm. 10mm m  1 = 360/6 = 60 m  2 = ½ (60) = 30 m  3 = 60 a mm Δ shortcut a = √ 3 short side a = √ 3 (5mm) a = 5 √ 3 = 8.66 Could you find the area of it now??

Find the area of this one! 12 ….. A circle has 360°… ….. How many degrees would the top angle of each Δ have? ….. Since the Δs are isosceles, what are the measures of the base angles? ….. If the apothem is an angle bisector, then what is the measure of the small top angle? 60° 30° 60° The short side = 6 The apothem = 6√3 A = 1/2ap A = ½(6√3)(72) = 216√3 (exact) A = (approx.)