CodeCogs

Computes the area of the circular segment within a circle tangent to a reference line.

Interface

#include <codecogs/maths/geometry/area/circle.h>

using namespace Maths::Geometry::Area;

inline double circle(double r, double h)

This module computes the area of the circular segment formed between a circle tangent to a reference line and a line found at a given distance from this reference line.

The various cases are described by the images given in the following documentation, where the area which we want to compute is that of the filled shape.

Solution

Consider an orthogonal coordinate system and $\mathcal{C}(O, r)$ a circle so that there exists $P \in \mathcal{C}(O, r)$ with $P$ on the x-axis. Also let $d$ be parallel to the x-axis so that the distance from it to the x-axis is $h \in \mathbb{R}_+$ and $d \cap \mathcal{C} = \{A, B\}$ .

Let $S$ be the area which we must determine. Based on the relation between $h$ and $r$ we get the following cases:

1) $h<r \Rightarrow S = \mathcal{A}_{APB}$

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2) $h = r \Rightarrow S = \mathcal{A}_{\mathcal C} /2$

Because $\mathcal{A}_{\mathcal C} = \pi r^2$ , we find the solution:

$S = \frac{\pi r^2}{2}$

(1)

3) $r<h<2r \Rightarrow S = \mathcal{A}_{APB}$

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4) $h \geq 2r \Rightarrow S = \mathcal{A}_{\mathcal{C}}$

Because $\mathcal{A}_{\mathcal{C}} = \pi r^2$ , the solution is:

$S = \pi r^2$

(2)

Output:

r = 2.5

h = 0.0   Area = 0.000
h = 0.5   Area = 1.022
h = 1.0   Area = 2.796
h = 1.5   Area = 4.954
h = 2.0   Area = 7.334
h = 2.5   Area = 9.817
h = 3.0   Area = 12.301
h = 3.5   Area = 14.681
h = 4.0   Area = 16.839
h = 4.5   Area = 18.613
h = 5.0   Area = 19.635
h = 5.5   Area = 19.635

Example 1

#include <codecogs/maths/geometry/area/circle.h>
#include <stdio.h>

int main()
{
  // the length of the radius
  double r = 2.5;

  // display the lenghts of the radius
  printf("r = %.1lf\n\n", r);

  // display the area for different values of h
  for (double h = 0; h < 5.6; h += 0.5)
  printf("h = %.1lf   Area = %.3lf\n", h, 
  Geometry::Area::circle(r, h));

  return 0;
}

Parameters

the radius of the circle

the distance between line $d$ and the reference line

Returns

The value of the desired area.

Interactive Calculator

Result –