![]() ![]() This can be easily done through some algebra: From here, we can use the formula for the area of the square: ĭoubling this gives us the diagonal of both the square and the octagon. Using the area of a triangle as, we can determine the length of apothem fromįrom the apothem, we can once again use the Pythagorean Theorem, giving us the length of the circumradius. ![]() Further dividing by gives us the area of a smaller segment consisting of the right triangle with legs of the apothem and. ĭividing this by gives us the area of each triangular segment which makes up the octagon. We know from the Pythagorean Theorem that, meaning that. Letting the side length be, we can create a square of length around it (see figure).Ĭreating a small square of side length from the corners of this figure gives us an area of. In order to do this, we first determine the area of the octagon. To determine the area of this square, we can determine the length of its diagonals. This is because, as, they all have the same base, meaning that. The first thing to notice is that is a square. We can easily compute AF to be from splitting one of the sides into two triangles. There are many ways to find the area of the octagon, but one way is to split the octagon into two trapezoids and one rectangle. Since each angle measures in an octagon, then Assume each of the sides of the octagon has length. Since quadrilateral ACEG is a square, the area of the square would just be, which we can find by applying Law of Cosines on one of the four triangles. Call one of the side lengths of the square. ![]() WLOG then using Law of Cosines, The area of the octagon is just plus the area of the four congruent (by symmetry) isosceles triangles, all an angle of in between two sides of length 1. 8 Video Solution (Using Law of Cosines).Therefore, the measures of the exterior angles of the pentagon are 90°, 60°, 70°, 75°, and 65°. Therefore, we can subtract the interior angle from 180° to find the measure of the exterior angle.įor example, if we have interior angles 90°, 120°, 110°, 105°, and 115° in a pentagon, we have to subtract each 180° angle to find the corresponding exterior angles: For this, we consider that the sum of an interior angle and its corresponding exterior angle is equal to 180°. For this, we have to add all the known angles and subtract from 360°.įor example, if we have the exterior angles 60°, 70°, 80°, and 85° in a pentagon, we have to start by determining their sum and then subtract it from 360°:Īdditionally, we can also calculate the exterior angle measures if we know the interior angle measures. We can determine the measure of a missing exterior angle if we know the measures of the other exterior angles. How to calculate exterior angle measures of irregular polygons? ![]()
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