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Maths › Geometry › Area ›

edge triangle

Computes the area of a trapezium within a triangle with a fixed edge.

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Interface
Edge Triangle
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Interface

C++

Edge Triangle

doubleedge_triangle(	double	`a`
	double	`b`
	double	`c`
	double	`h`	)[inline]

This module computes the area of the trapezium formed between a triangle with a fixed edge on a reference line and a line found at a given distance distance from this reference line.

This situation is described by the following image. The area which we want to compute is that of the filled trapezium $\inline [BB_1C_1C]$ .

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1/edge_triangle-746.jpg cannot be found in /users/1/edge_triangle-746.jpg. Please contact the submission author.

Solution

Let $\inline \mathrm{xOy}$ be an orthogonal coordinate system and let $\inline \triangle ABC$ be an arbitrary triangle so that $\inline BC \subset \mathrm{Ox}$ and

$BC = a \qquad AC = b \qquad AB = c$

(2)

where $\inline a, b, c \in \mathbb{R}_+^*$ are fixed numbers. Also let $\inline d \parallel \mathrm{Ox}$ so that the distance between the line $\inline d$ and $\inline \mathrm{Ox}$ equals $\inline h \in \mathbb{R}_+$ and $\inline AB \cap d = \{B_1\}$ , $\inline AC \cap d = \{C_1\}$ .

Obviously $\inline \triangle AB_1C_1 \sim \triangle ABC$ which implies:

$\frac{B_1C_1}{BC} = \frac{H - h}{H} \qquad \Rightarrow \qquad B_1C_1 = a \left( 1 - \frac{h}{H} \right )$

(3)

where H is the height corresponding to vertex A.

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Example 1

#include <codecogs/maths/geometry/area/edge_triangle.h>
#include <stdio.h>
 
int main()
{
  // the lengths of the sides
  double a = 3.3, b = 4.5, c = 5.4;
 
  // display the lengths of the sides
  printf("a = %.1lf\nb = %.1lf\nc = %.1lf\n\n", a, b, c);
 
  // display the area for different values of h
  for (double h = 0.1; h < 1.09; h += 0.1)
    printf("h = %.1lf   Area = %.2lf\n", h, 
    Geometry::Area::edge_triangle(a, b, c, h));
 
  return 0;
}

Output:

a = 3.3
b = 4.5
c = 5.4
 
h = 0.1   Area = 0.33
h = 0.2   Area = 0.65
h = 0.3   Area = 0.96
h = 0.4   Area = 1.26
h = 0.5   Area = 1.56
h = 0.6   Area = 1.85
h = 0.7   Area = 2.13
h = 0.8   Area = 2.40
h = 0.9   Area = 2.67
h = 1.0   Area = 2.93

Note

The values of the sides must satisfy the triangle inequality.

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 $\inline d$ and $\inline \mathrm{Ox}$

Returns

The value of the desired area.

Authors

Eduard Bentea (September 2006)

Source Code

Source code is available when you agree to a GP Licence or buy a Commercial Licence.

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