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Hi Robi!
I came across this question when looking through this book. How do you find the area of a do decagon if you know the side length is 4? That was all the information it gave me...
Thanks for your help!
-Danielle
Hi Danielle,
For this question, it is best to start with one center point in the middle. Next, if you connect it to each point of the decagon, you get 12 congruent triangles. Now, the vertex angle of each triangle is 360/12 = 30 since there is each triangle is congruent and twelve of them make the complete round. So, now draw a height down to the midpoint of each side from the center. You will see two right triangles in each with one side being 2. Now, the height will then be 2/tan(15) using some trigonometry. (If you do not know trigonometry, then you should learn a bit since these problems require it). So then, the area is 1/2 * 4 * 2/tan(15) * 12 since there are 12 triangles and the height of each triangle is 2/tan(15). So this turns out to equal 48/tan(15).
So the answer is 48/tan(15).
I hope this helped,
Robi
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| Comment | Thank you for taking the time to answer my question. I now feel like I understand the problem better. I have another question, so expect one from me soon. I think it is going to be on the area of a decagon. Thank you very much! -Danielle | ||
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Answers by Expert:
Robi Bhattacharjee
Expertise
I can answer a variety of questions on mathematics. Questions on trigonometry, calculus(preferably single variable), algebra, geometry, and number theory will be answered. I cannot answer questions on abstract branches of mathematics such as group theory. I also cannot answer questions on statistics. In number theory, I can answer questions on congruences, prime numbers, units, functions, and the riemann-zeta function.
Experience
I have studied advanced math my entire life. I started calculus in sixth grade. I have attended numerous math competitions and I am attending math organizations such as the San-Diego math circle. Also, this year I have been invited to the USAMO which is a prestigious math competition (Every year the USAMO invites 500 students from across the USA to participate in this competition. The top 6 go to represent the USA in the International Math Olympiad).
Organizations
I am in the San Diego Math Circle
Education/Credentials
I am entering high school and have received a perfect score and the STAR test 5 times in a row. I also have gotten recognitions in the AMC 10, AIME, Math Counts, and ARML. Additionally, I have won the San Diego Math Olimpiad twice in a row.
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