Question

How do you find the largest possible area for a rectangle inscribed in a circle of radius 4?

Answer 1

Hint: As we know area of rectangle is \[A = l \times w\], i.e., length and width but here we need to find area for a rectangle inscribed in a circle of given radius as 4, hence for this we can use Pythagorean theorem to find out the area given by \[b = \sqrt {{{\left( {2r} \right)}^2} - {a^2}} \].

Formula used:
\[{A_0} = 2{r^2}\]
In which \[{A_0}\] is the area of a rectangle in a circle.
r is the radius.

Complete step by step solution:
To find the area for a rectangle inscribed in a circle of given radius, we can use Pythagorean theorem to get the area.
An inscribed rectangle has diagonals of length 2r and has sides (a, b) measuring \[0 < a < 2r\],\[b = \sqrt {{{\left( {2r} \right)}^2} - {a^2}} \],
So, the area of rectangle is length times width, hence we get
\[A = ab\]
\[A = a\sqrt {{{\left( {2r} \right)}^2} - {a^2}} \]
For, \[0 < a < 2r\]
Its maximum occurs at \[{a_0}\] such that
\[{\left( {\dfrac{{dA}}{{da}}} \right)_{{a_0}}} = 0\]
Differentiate with respect to a we get,
\[\dfrac{{2\left( {{a_0}^2 - 2{r^2}} \right)}}{{\sqrt {4{r^2} - {a_0}^2} }} = 0\]
Solving for a we get Area as
\[{a_0} = \sqrt {2r} \]
At this value \[{A_0}\] is
\[{A_0} = 2{r^2}\]
Substituting the given value of radius as 4, we get
\[{A_0} = 2{\left( 4 \right)^2}\]
\[{A_0} = 32\]

Therefore, area for a rectangle inscribed in a circle of radius is 32 units of area.

Additional information:
A circle closed plane geometric shape. In technical terms, a circle is a locus of a point moving around a fixed point at a fixed distance away from the point. Basically, a circle is a closed curve with its outer line equidistant from the center. The fixed distance from the point is the radius of the circle.
The radius of the circle is the line which joins the centre of the circle to the outer boundary and the diameter of the circle is the line which divides the circle into two equal parts.

Note: The key point to find is that the area of a rectangle depends on its sides. Hence, we can say that the region enclosed by the perimeter of the rectangle is its area. A circle is a closed curve with its outer line equidistant from center. The fixed distance from the point is the radius of the circle. But in the case of a square, since all the sides are equal, therefore, the area of the square will be equal to the square of side-length.