A circle of radius r is inscribed in an equilateral triangle. Find the area of the shaded region.
Solution by Chaitanya Kanitkar , Novi High School
Label the vertices of the triangle as shown; then draw the angle bisector AE and the line segment OF perpendicular to AC . Then Now,
Thus, and Area of triangle Area of circle Area of shaded region |
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Also solved by: Alex Bergquist, Brooke Boer, Ashley Botham, Tessa Cantu, Sam Carlton, Garrett Cullen, Bill Darden, Max Dixon, Mischa Dylewski, Blaire Edgerton, Joan Gary, Rebecca Hammond, Melvin Joice, Ryan Klock, Kristin Kopec, Derek Mead, Hannah Novak, Drew Reyelts, Jake Rollenhagen, Brendan Ross, Aaron Sayfie, Gina Turner, Dani Tyner, Niki Tyner, John VanHouten, Marc Walker, Sheila Waslawski, Alex Zheutlin (Forest Hills Eastern H.S.) ; Carolyn Affholter, Emily Byl, Osama Chowdhury, Liam Considine, Alyssa Doster, Matt Emelander, Amanda Gade, Ryan Kamm, Alex Mead (Forest Hills Northern H.S.); Deanna Abrash, LeeAnne Carr, Armila Francis, Megan Greve, Ross Gerulis, Lauren Valentine (Plymouth Christian Academy) ; Manish Patel (Saginaw Heritage H.S.) ; Nonie Arora, Sandeep Palepin, Eshwar Sundararaju (Novi H.S.); Robert Harrison, Alex Robinson (Lansing Catholic H.S.); Shawna Wei (Forest Hills Central H.S.) ; Rob Brady, Sheila Saurman (St. Louis H.S.) ; James Frederick (Benzie Central H.S.) ; Matthew Nicholas (Detroit Catholic Central H.S.) ; Zach Baum (Lowell H.S.) ; Janki Patel (Saginaw Arts & Sciences Academy) ; Eliott Hoff (East Grand Rapids H.S.); and Sarah Avellar, Brady Morrison (Alamo Heights H.S., San Antonio, TX).
, 
, and 
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; so 
; so 
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