Regular Polygons Finding Area

Isosceles Triangles 2 sides congruent 2 base angles congruent Non base angle will be referred to as the vertex.

Isosceles Height or Altitude What is special about the Height of an isosceles Triangle? Perpendicular to the base Bisects base Bisects vertex Therefore creates a right triangle

Find the Area Given the following Isosceles Triangle find its area using previously discussed material. Draw Altitude to make a right triangle Use trig ratios to calculate the height Find area of isosceles triangle

Regular Polygons What does Regular mean? All angles congruent All sides congruent How would you find the area of a polygon? Divide into triangles and rectangles and add the areas together Draw a rectangle around the figure, find its area, then subtract the parts that are not part of the orignal figure

What about a Regular Polygon If only given a side length? Is there enough information given to find different shapes? Can we use the idea previously discusses with Isosceles triangles?

Area of Regular Polygons You can divide a regular polygon into congruent isosceles triangles. From the center of the polygon draw segments to all vertices, these lengths are all congruent because they are actually the radii of the circumscribed circle around the polygon, making all the triangles congruent because of SSS

Example of Idea

Area of Regular Polygon Could you find the area knowing you have these isosceles triangles? We would know a side because of perimeter, but do we know an angle? Central Angle would be the vertex angle, how do we find this? All the triangles come together and add to 360 could then calculate the angle measure

Steps for Area Draw in isosceles Triangles – I usually do not draw the polygon just the isosceles triangle Find side length perim/ number of sides Find vertex angle 360/number of sides Draw in Altitude – this is where I draw the right triangle that is formed and cut my base and vertex angle in half Use trig to find the height, this should always be the tan ratio Find Area of the Isosceles triangle bh/2 Find area of regular polygon Triangle area times number of sides

Example Find area of a Regular Pentagon with perimeter of 300 m.

Short Cut Area = 1/2asn a=apothem ( which is the height of the isosceles triangle) s=side length n= number of sides

Short Cut sn=the perimeter, number of sides times side length 1/2as=area of the isosceles triangle You can use this if you are given apothem or short cut to find other parts

Examples Given a regular pentagon with a=3 and s=4 find the area Given the area of a regular hexagon as 234, with apothem of 8 find the perimeter

Homework Pg 443 1,2,3,7,8 Find Area of a regular Octagon with Perimeter 1600 in Find Area of a regular decagon with perimeter of 1200 cm