Question

its to find area of a composite fiqure with 6 sides and I on/ly have 4 mesuarmen/ts to the sides

Answer 1

Answer on Question #84873 – Math – Geometry

Question

Find an area of the composite figure with 6 sides, if I only have 4 measurements to the sides.

Solution

Consider a composite figure with 6 sides $A_1A_2A_3A_4A_5A_6$ . This figure can be divided into four triangles as shown in figure 1:

If we only have 4 sides, for example, $a_1$ , $a_2$ , $a_3$ and $a_4$ , then an area $S$ of $A_1A_2A_3A_4A_5A_6$ will depend on angles between the sides of the figure. If we have these angles, then we can find unknown sides and the area of each triangle:

1) From the cosine theorem in triangles $A_{1}A_{2}A_{3}$ and $A_{3}A_{4}A_{5}$ :

$$x = \sqrt {\left(a 1 ^ {2} + a 2 ^ {2} - 2 a 1 a 2 \cos \varphi 2\right)};$$ $$y = \sqrt {\left(a 3 ^ {2} + a 4 ^ {2} - 2 a 3 a 4 \cos \varphi 4\right)}.$$

2) From the sine theorem in triangles $A_1A_2A_3$ and $A_3A_4A_5$ :

$$x / \sin \varphi_ {2} = a _ {1} / \sin \theta_ {2} = a _ {2} / \sin \theta_ {1};$$ $$\theta_ {2} = \arcsin \left(a _ {1} / x * \sin \varphi_ {2}\right);$$ $$\theta_1 = \arcsin(a_2 / x * \sin \varphi_2);$$ $$y / \sin \varphi_4 = a_3 / \sin \theta_4 = a_4 / \sin \theta_3;$$ $$\theta_3 = \arcsin(a_4 / y * \sin \varphi_4);$$ $$\theta_4 = \arcsin(a_3 / y * \sin \varphi_4);$$ $$\alpha_3 = \varphi_3 - (\theta_2 + \theta_3).$$

3) From the cosine theorem in the triangle $A_1A_3A_5$:

$$z = \sqrt{(x^2 + y^2 - 2 \cdot \arccos \alpha^3)}.$$

4) From the sine theorem in the triangle $A_1A_3A_5$:

$$x / \sin \alpha_5 = y / \sin \alpha_1 = z / \sin \alpha_3;$$ $$\alpha_1 = \arcsin(y / z * \sin \alpha_3);$$ $$\alpha_5 = \arcsin(x / z * \sin \alpha_3);$$ $$\theta_5 = \varphi_5 - (\theta_4 + \alpha_5);$$ $$\theta_6 = \varphi_1 - (\theta_1 + \alpha_1).$$

5) From the sine theorem in the triangle $A_1A_5A_6$:

$$a_6 / \sin \theta_5 = a_5 / \sin \theta_6 = z / \sin \varphi_6;$$ $$a_5 = z * \sin \theta_6 / \sin \varphi_6;$$ $$a_6 = z * \sin \theta_5 / \sin \varphi_6.$$

6) Find the area of the composite figure:

$$S_1 = \frac{1}{2} * a_1a_2\sin \varphi_2;$$ $$S_2 = \frac{1}{2} * a_3a_4\sin \varphi_4;$$ $$S_3 = \frac{1}{2} * \chi_3\sin \alpha_3;$$ $$S_4 = \frac{1}{2} * a_5a_6\sin \varphi_6;$$ $$S = S_1 + S_2 + S_3 + S_4.$$

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