What is the area of largest square that is inscribed in a semicircle of radius 10 cm?

Using this square in a circle calculator, you can find the biggest square in a circle. It also helps you find the largest circle inside a square. Be it geometry 📐, construction 🏗️, or daily life 🚶, we often come across composite shapes such as a square circumscribing a circle 🔵 or a square inscribed in a circle. This calculator helps you find the dimensions 📏 of such shapes when one of the measurements is known!

Have you ever wondered 'What is the largest circular pizza 🍕 I can fit into this square 🔲 box?' or 'What is the largest square piece of cake 🎂 I can fit into this circular plate 🍽️?' or 'What is the largest circular indoor pool 🏊 I can fit into this square room?' Well, wonder no more! Because our square in a circle calculator will help you find the answers to these questions and more!

Using the square in a circle calculator, you can find any of the following:

Dimensions of the biggest square in a circle:
- To find this, enter the value of the circle's radius or area.
- The calculator will display the side length and area of the largest square that can fit inside the circle!
Dimensions of the largest circle inside a square:
- To find this, enter the value of the square's side or area.
- The calculator will display the radius and area of the largest circle that can fit inside the square!
Dimensions of a square with the same area as a circle:
- To find this, enter the value of the circle's radius or area.
- The calculator will display the side length of the square with the same area as the circle!
Dimensions of a circle with the same area as a square:
- To find this, enter the value of the square's side or area.
- The calculator will display the radius of the circle with the same area as the square!

You can thus use this square in a circle calculator in several different ways, depending on your need!

To know how to find the largest square in a circle using the square inside a circle calculator, do the following:

Key in the value of the circle's radius or area.
The calculator will find what size square fits in the circle using the formula:
side length = √2 × radius
The side length and the area of the square inside the circle will be displayed!

In this manner, you can find the maximal square that you can draw within a given circle.

To know how to find the largest circle in a square using the square inside a circle calculator, do the following:

Key in the value of the side or area of the square circumscribed about a circle.
The calculator will find what size circle fits in the square using the formula:

$length2\large\text{radius} = \frac{\text{side length}}{2}$
The radius and the area of the circle inside the square will be displayed!

In this manner, when a square is circumscribing a circle, you can find the radius and area of the circle.

Squaring a circle refers to finding a square with the same area as that of the circle.

For a circle with radius r, a square with the same area will have a side length of r√π. So, for example, if a given circle has a radius of 10 cm, then a square with the same area as the circle will have a side length of 10√π cm.

Alternatively, we can also convert a given square to a round shape by doing the reverse operation.

It's interesting to note that we can approximate a square to a circle by incrementally increasing the number of sides to get regular polygons such as pentagon, hexagon, heptagon, octagon, etc. until we end up with a circle ⭕.

Converting a square to a circle refers to finding a circle with the same area as the square. So if we want to convert a square to a round figure, the radius of the resulting circle will be s/√π, where s is the side of the square.

If we have a circle of radius 10 cm, then we can do the following to find the largest square inscribed in the circle:

The largest square inscribed in a circle of radius r will have a side length of r√2.
So for a circle of radius 10 cm, the largest square in it will have a side length of 10√2 cm.
This value of the side length can be approximated to 14.1421 cm.
The area of the square will be 200 cm².

If we have a square circumscribed about a circle with side 10 cm, then we can find the largest circle inscribed in the square as follows:

The largest circle inscribed in a square of side s will have a radius of s/2.
So for a square of side 10 cm, the largest circle in it will have a radius of 5 cm.
The area of the circle will be 78.54 cm².

If we have a square of side 10 cm, its area will be 100 cm². A circle with the same area will therefore have a radius of 10/√π, or 5.64 cm.

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Given a semicircle with radius r, we have to find the largest square that can be inscribed in the semicircle, with base lying on the diameter.

Examples:

Input: r = 5 Output: 20 Input: r = 8 Output: 51.2

Approach: Let r be the radius of the semicircle & a be the side length of the square.
From the figure we can see that, centre of the circle is also the midpoint of the base of the square. So in the right angled triangle AOB, from Pythagoras Theorem:

a^2 + (a/2)^2 = r^25*(a^2/4) = r^2a^2 = 4*(r^2/5) i.e. area of the square

Below is the implementation of the above approach:

#include <bits/stdc++.h>
using namespace std;



float squarearea(float r)
{
    if (r < 0)
        return -1;
    float a = 4 * (pow(r, 2) / 5);
    return a;
}
int main()
{
    float r = 5;
    cout << squarearea(r) << endl;
    return 0;
}

import java.io.*;
class GFG {
static float squarearea(float r)
{
    if (r < 0)
        return -1;
    float a = 4 * (float)(Math.pow(r, 2) / 5);
    return a;
}
    public static void main (String[] args) {
         float r = 5;
    System.out.println( squarearea(r));
    }
}

def squarearea(r):
    if (r < 0):
        return -1
    a = 4 * (pow(r, 2) / 5)
    return a
if __name__ == "__main__":
    r = 5
    print(int(squarearea(r)))

using System;
class GFG
{
static float squarearea(float r)
{
    if (r < 0)
        return -1;
    float a = 4 * (float)(Math.Pow(r, 2) / 5);
    return a;
}
public static void Main ()
{
    float r = 5;
    Console.WriteLine(squarearea(r));
}
}

<?php
function squarearea($r)
{
    if ($r < 0)
        return -1;
    $a = 4 * (pow($r, 2) / 5);
    return $a;
}
$r = 5;
echo squarearea($r);
?>

<script>
function squarearea(r)
{
    if (r < 0)
        return -1;
    var a = 4 * (Math.pow(r, 2) / 5);
    return a;
}
var r = 5;
document.write( squarearea(r));
</script>

Time Complexity: O(1)

Auxiliary Space: O(1)