Given a semicircle of radius r, we have to find the largest rectangle that can be inscribed in the semicircle, with base lying on the diameter.
Input : r = 4 Output : 16 Input : r = 5 Output :25
Let r be the radius of the semicircle, x one half of the base of the rectangle, and y the height of the rectangle. We want to maximize the area, A = 2xy.
So from the diagram we have,
y = √(r^2 – x^2)
So, A = 2*x*(√(r^2 – x^2)), or dA/dx = 2*√(r^2 – x^2) -2*x^2/√(r^2 – x^2)
Setting this derivative equal to 0 and solving for x,
dA/dx = 0
or, 2*√(r^2 – x^2) – 2*x^2/√(r^2 – x^2) = 0
2r^2 – 4x^2 = 0
x = r/√2
This is the maximum of the area as,
dA/dx > 0 when x > r/√2
and, dA/dx < 0 when x > r/√2
Since y =√(r^2 – x^2) we then have
y = r/√2
Thus, the base of the rectangle has length = r/√2 and its height has length √2*r/2.
So, Area, A=r^2
- Largest trapezoid that can be inscribed in a semicircle
- Largest triangle that can be inscribed in a semicircle
- Largest square that can be inscribed in a semicircle
- Area of Largest rectangle that can be inscribed in an Ellipse
- Area of largest triangle that can be inscribed within a rectangle
- Radius of the biggest possible circle inscribed in rhombus which in turn is inscribed in a rectangle
- Area of a triangle inscribed in a rectangle which is inscribed in an ellipse
- Biggest Reuleaux Triangle inscirbed within a square inscribed in a semicircle
- Largest sphere that can be inscribed within a cube which is in turn inscribed within a right circular cone
- Largest right circular cylinder that can be inscribed within a cone which is in turn inscribed within a cube
- Largest square that can be inscribed within a hexagon which is inscribed within an equilateral triangle
- Largest right circular cone that can be inscribed within a sphere which is inscribed within a cube
- Largest sphere that can be inscribed in a right circular cylinder inscribed in a frustum
- Largest subset of rectangles such that no rectangle fit in any other rectangle
- The biggest possible circle that can be inscribed in a rectangle
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.