if a regular hexagon is inscribed in a circle of radius 14 cm find the area of the region between the circle and the hexagon
Answers
Answer:
106.53 cm² (approx)
Step-by-step explanation:
Required area is
A = ( area of circle ) - ( area of hexagon)
The hexagon is made up of 6 equilateral triangles with side 14cm. So
A = ( area of circle ) - 6 × ( area of one of these equilateral triangles )
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Half an equilaterial triangle makes a right angled triangle with:
hypotenuse 14 cm, one side 7 cm
Pythagoras' Theorem says the other side is
√(14² - 7²) = √( (2×7)² - 7²) = √( 4 × 7² - 7² ) = √( 3 × 7²) = 7√3 cm
So
area of equilateral triangle = (1/2) × base × height
= (1/2) × 14 cm × 7√3 cm
= 49√3 cm²
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Area of circle = π r² = π × 14² cm² = 196π cm²
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So the required area is
A = ( area of circle ) - 6 × ( area of one of these equilateral triangles )
= 196π cm² - 6 × 49√3 cm²
= (196π - 294√3) cm²
≈ 106.53 cm²