Semicircle In a Square!

Geometry Level 2

What is the area of the largest semicircle that can be inscribed in a unit square?

(5+2\sqrt{2})\pi

(3+2\sqrt{2})\pi

(5-2\sqrt{2})\pi

(3-2\sqrt{2})\pi

4 solutions

Andrew Ellinor
Jan 18, 2016

Since the figure is symmetrical over the diagonal, the angle formed by the diameter of the semicircle and the side of the square is $45^\circ.$

Consider the point A on the left side of the square where the semicircle is tangent. A line perpendicular to the side of the square containing A passes through the midpoint of the diameter of the semicircle (point O), thus giving the measurements as seen above.

The length of the side of the square is 1, so this means

$r + \dfrac{r}{\sqrt{2}} = 1 \longrightarrow r = 2 - \sqrt{2}.$

From here, we can easily find the area is $\pi (2 - \sqrt{2})^2 = (3 - 2\sqrt{2})\pi$

Area if semicircle = Π×r^2/2 = (Π(2-√2)^2)/2 = (3-2√2).Π

Bhupendra Jangir - 5 years, 4 months ago

Shourya Pandey
Jan 21, 2016

Please change the options. Only one of the options is less than $1$ sq. unit, so it must be the answer.

Largest possible semicircle possible would have the straight part of the semicircle collinear with a side of the square - area of the semi circle would therefore be 1/2 x 3.14 x (1/2)^2

Ron Fay - 5 years, 4 months ago

A Former Brilliant Member
Apr 13, 2018

Without computing anything, the only possible answer is $(3-2\sqrt{2})\pi$ since it is the only one that is less than 1.

Kieran Kaempen
Jan 21, 2016

None of the answers are correct. You have to divide by 2, because it's a semicircle and not a full circle.

Its already divided by 2, but not mentioned it.

Bhupendra Jangir - 5 years, 4 months ago

Semicircle In a Square!

4 solutions

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