Question

Is the area of a semicircle with a diameter of x greater than, less than, or equal to a circle with a diameter of 12x? Explain.

Answer 1

Area of circle with a diameter of 12x is greater than area of a semicircle with a diameter of x

Explanation:

Since semicircle is half of the full circle, then area of semicircle is also half of the full circle.

Area of circle is given by
`\pi r^2`

therefore, area of semicircle is
`\pi r^2/2`

Also radius for any circle is half of diameter

radius = Diameter/2

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For semicircle

Diameter = x

Radius = x/2

area of semicircle is given below

`area = \pi r^2 \\=>area = \pi (x/2)^2\\=>area = \pi (x^2/4)`

area of semicircle is

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For circle

Diameter = 12x

Radius = 12x/2 =6x

area of circle is given below

`area = \pi r^2 \\=>area = \pi (6x)^2\\=>area = \pi (36x^2)`

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we can see that

area of semicircle is
`\pi x^2/4`

and

area of circle is
`36\pi x^2`

Hence, area of circle with a diameter of 12x is greater than area of a semicircle with a diameter of x