Okay, so that problem was a little hard.
If you were so inclined to look up “decagon, perpendicular, area” on yahoo you could have gone to a website that would have said the following:
If you were so inclined to look up “decagon, perpendicular, area” on yahoo you could have gone to a website that would have said the following:
3. Given the apothem (inradius)
If you know the apothem, or inradius, (the perpendicular distance from center to a side. See figure above)where
A is the length of the apothem (inradius)
N is the number of sides
TAN is the tangent function calculated in radians (see Trigonometry Overview).
And you would have computed the area like this:
Area = (14^2)(10)tan(3.1415/10) = (196*10)tan(0.314) = (1960)(.325) = 637 (approximately due to rounding)
The alternative is based on what steveegg was hinting at:
All of the angles in a regular decagon are equal to 144 degrees. If we draw a line from the center of the decagon to a vertex you will bisect the angle forming two 72 degree angles and a triangle. We can compute the area of the triangle using the following equation:
Area (triangle) = ½ base * height
We know the height and we can compute the base since we know the measurement of the interior angles using the following:
Tan (of interior angle) = opposite/adjacent
Or tan (72) = 14/?
? = 14/tan(72) = 14/3.1 = 4.5
Then using the area formula for a triangle:
Area (triangle) = ½ (4.5) (14) = 31.5
Since the decagon can be divided into 20 of these triangles the area of the triangle is multiplied by 20 and you get an approximate area of 630.
Now tell me that wasn’t fun! I know I’m a geek!
Area = (14^2)(10)tan(3.1415/10) = (196*10)tan(0.314) = (1960)(.325) = 637 (approximately due to rounding)
The alternative is based on what steveegg was hinting at:
All of the angles in a regular decagon are equal to 144 degrees. If we draw a line from the center of the decagon to a vertex you will bisect the angle forming two 72 degree angles and a triangle. We can compute the area of the triangle using the following equation:
Area (triangle) = ½ base * height
We know the height and we can compute the base since we know the measurement of the interior angles using the following:
Tan (of interior angle) = opposite/adjacent
Or tan (72) = 14/?
? = 14/tan(72) = 14/3.1 = 4.5
Then using the area formula for a triangle:
Area (triangle) = ½ (4.5) (14) = 31.5
Since the decagon can be divided into 20 of these triangles the area of the triangle is multiplied by 20 and you get an approximate area of 630.
Now tell me that wasn’t fun! I know I’m a geek!
This week we will take a small step backward.
Solve for x, y, and z:
3y + 9 = -3x/2
2x – y – 3z = -17
6z – 8x + 4 = 32
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