The octagon can be decomposed into four triangles that have a base of 10.3 in. and a height of 10 in., and four rectangles that have a length of 20.6 in. and a width of 10 in.
An octagon is a polygon with eight sides and eight angles. The sum of the interior angles of an octagon is 1080°.
To find the length of each side of the octagon, you can use the Pythagorean theorem. The octagon can be divided into eight right triangles, each with a hypotenuse of 24 in. and a leg of 10 in. The other leg is the side of the octagon. Using the formula a^2 + b^2 = c^2, you can find that the side of the octagon is about 20.6 in.
To find the area of the octagon, you can use the formula A = 2(1 + √2)s^2, where s is the side length. Plugging in s = 20.6, you get A = 2(1 + √2)(20.6)^2 ≈ 1184.1 in^2.
Alternatively, you can use the method given in the question, which is to decompose the octagon into four triangles and four rectangles. The base of each triangle is half of the side of the octagon, which is 10.3 in.
The height of each triangle is the same as the leg of the right triangle, which is 10 in. The area of each triangle is A = (1/2)bh = (1/2)(10.3)(10) = 51.5 in^2. The length of each rectangle is the same as the side of the octagon, which is 20.6 in.
The width of each rectangle is the same as the leg of the right triangle, which is 10 in. The area of each rectangle is A = lw = (20.6)(10) = 206 in^2. The total area of the octagon is the sum of the areas of the four triangles and the four rectangles, which is 4(51.5) + 4(206) = 1184 in^2.