Interface

#include <codecogs/geometry/area/free_triangle.h>

using namespace Geometry::Area;

	double	free_triangle (double a, double b, double c, double h, double alpha)[inline] Computes the area of a triangle or a trapezium within a triangle with a fixed vertex.
	Real	cc_free_triangle (Real a, Real b, Real c, Real h, Real alpha) This function is available as a Microsoft Excel add-in.

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Function Documentation

This module computes the area of the triangle or trapezium formed between a triangle with a fixed vertex on a reference line and a line found at a given distance 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

Let be an orthogonal coordinate system and let be an arbitrary triangle so that and

(1)

where are fixed numbers. Consider and so that the distance between the line d and equals . Also let and .

We will first find the lengths of the projections of B and C on because we can separate different cases based on these values. Let and . Since these values are independent of the relations set between them and we can consider, without loss of generality, the following case: and .

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Let be the area which we must determine. Based on the values of , and , we find the following solutions:

1) ,

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2) ,

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3) ,

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4) ,

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Output:

a = 2.613
b = 1.878
c = 1.209
alpha = 0.61086
 
h = 0.0   Area = 0.00
h = 0.3   Area = 0.14
h = 0.6   Area = 0.56
h = 0.9   Area = 0.97
h = 1.2   Area = 1.04
h = 1.5   Area = 1.04

Example:

#include <codecogs/geometry/area/free_triangle.h>
#include <stdio.h>
 
int main()
{
  // the lengths of the sides
  double a = 2.613, b = 1.878, c = 1.209;
 
  // the value of the alpha angle in radians
  double alpha = 0.61086;
 
  // display the lenghts of the sides and the alpha angle
  printf("a = %.3lf\nb = %.3lf\nc = %.3lf\nalpha = %.5lf \n\n", a, b, c, alpha);
 
  // display the area for different values of h
  for (double h = 0.0; h < 1.6; h += 0.3)
  printf("h = %.1lf   Area = %.2lf\n", h, 
  Geometry::Area::free_triangle(a, b, c, h, alpha));
  
  return 0;
}

Parameters:

a	first side of the triangle (BC)
b	second side of the triangle (AC)
c	third side of the triangle (AB)
h	the distance between line and
alpha	the input angle in radians

Returns:

The value of the desired area.

Note:

The vertices must be denoted such that the abscissa of is less than that of , i.e. as in the given examples.

Authors:

Eduard Bentea (September 2006)

Source Code:

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