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A circle is inscribed in a regular hexagon with side length 10 feet. What is the area of the shaded region?
A circle is inscribed in a regular hexagon with side length 10 feet. An apothem and 2 raddi are drawn to form 2 triangles with angles 30, 60, and 90 degrees. The area between the circle and the hexagon is shaded.
Recall that in a 30 – 60 – 90 triangle, if the shortest leg measures x units, then the longer leg measures xStartRoot 3 EndRoot units and the hypotenuse measures 2x units.
(150StartRoot 3 EndRoot – 75π) ft2
(300 – 75π) ft2
(150StartRoot 3 EndRoot – 25π) ft2
(300 – 25π) ft2
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GPT-5 mini
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The apothem (incircle radius) is r = 5√3 (from a 30-60-90 triangle with short leg 5).
Area(hexagon) = 1/2 · perimeter · apothem = 1/2 · (6·10) · 5√3 = 150√3.
Area(circle) = πr^2 = π(5√3)^2 = 75π.
Shaded area = 150√3 − 75π ft^2.
Answer: (150√3 − 75π) ft^2.
Area(hexagon) = 1/2 · perimeter · apothem = 1/2 · (6·10) · 5√3 = 150√3.
Area(circle) = πr^2 = π(5√3)^2 = 75π.
Shaded area = 150√3 − 75π ft^2.
Answer: (150√3 − 75π) ft^2.
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