Trigonometry - Practice

For a start, we need to be clear in our mind that, in order to find out the area, we have to know the base and the height of the triangle.

To find the height, we could divide the triangle in two smaller ones. Now, knowing its opposite cathetus (4 cm), we have to work out h, which is the cathetus next to te angle. Which Trigonometric function connects the opposite cathetus to the adjacent one? The tangent. Then, with the calculator,

tan 25º= 2cm/h => h = 4 cm/tan 25º =
= 4 cm/0.47 = 8.51 cm

And now, we can use the formula A = (b*h)/2:

A = (8 cm * 8.51 cm) / 2 = 34.31 cm²

First of all, we have to split the main problem into smaller ones. An advantage of Trigonometry is that its rules and fuctions are really easy to put into practice. For that reason, we will split the octogon into eight triangles. As we already know, an entire turn is 360º.

Therefore, the inner angle of the triangles will be 360º/8 = 45º. They are isosceles triangles, because both remainder angles are equal. Using the rule "the three angles in the triangle must add up to 180 degrees", we will be able to calculate the angles: 67,5º.

At this point, if we split again the triangle into two rectangle ones, we can put in practice the Trigonometric functions.

We have come up against a situation similar to the one of the previous problem.

tan 22.5º = 6cm / ap => ap = 6 cm / tag 22.5º =
= 6 cm / 0,414 = 14.5 cm

At this moment, we can use the regular polygon area formula:

A = (P * ap) / 2

Its perimeter, P, is 8 * 12 cm = 96 cm. In conclusion,

A = (96 cm * 14.5 cm) / 2 = 695.3 cm²