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Area of Regular Polygons. Break the figure down into triangles. 4. 5. 3. 6. 2. 1. Notice there are 6 triangles and 6 sides. 4. 5. 3. 6. 2. 1. Let’s look at 1 of the triangles closer. The area of a triangle is:. h. b. Area = ½bh.
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4 5 3 6 2 1 Notice there are 6 triangles and 6 sides
4 5 3 6 2 1 Let’s look at 1 of the triangles closer
The area of a triangle is: h b Area = ½bh
When the triangle is part of a polygon, the height is called the Apothem apothem h b
There are 6 triangles, so we have to multiply our area by 6! apothem base Of course, using the equation you’ll only find the area for one of the triangles.
Another way to look at it is that we have 6 bases and one apothem apothem base Remember, 6 times the base is the perimeter of the object!
Therefore we can use the formula: apothem base 1 __ ap Area = 2 Perimeter
Ex: Find the area of the hexagon 8 5 1 __ ap Area = 2 First we must find the perimeter: p = 5(6) = 30
Find the area of the hexagon 8 5 1 1 __ __ ap 8(30) Area = Area = 2 2 Plug everything in p = 5(6) = 30
Find the area of the hexagon 8 5 1 __ 8(30) Area = 2 Solve = 120
Practice 1 • Find the area of the following regular polygons: • Octagon with base of 4 and apothem 7 • Hexagon with base of 8 and apothem 3 • The figure below 12 10
But what if we’re not given the apothem?? 1 __ ap Area = 2
Ex: Find the area of the polygon 10 1 __ ap Area = We need the perimeter and apothem 2
4 5 3 6 2 1 Let’s look at 1 of the triangles closer again
a 10 Inside we have two right triangles
We can find the angle at the top and use it to find the apothem a 10
There’s 3600 around the center of the figure Since there’s 6 sides we can find the angle at the top of each triangle by: 360 = 60 6
600 The top angle is 60 and the apothem cuts it in half a 300
300 Half the triangle means half the base a 5 This is now a 30-60-90 right triangle! The apothem is:
Ex: Find the area of the polygon 10 1 __ ap Area = 2 We need the perimeter and apothem
Ex: Find the area of the polygon 10 1 __ ap Area = 2
6 But what if it’s not a special right triangle?? 1 __ ap Area = 2 360 = 72 5
360 Break it down to the small triangle a 3 The angle gets cut in half The base is cut in half
360 Use the trig function to find apothem hyp a adj 3 opp