Question

In the regular octagon, the numeric value of the area of the shaded region is twice more than the numeric value of the length of AB. Find the area of the unshaded region.

Answer 1

The area of the shaded region in a regular octagon is given to be three times the length of AB. Since there are no specific measurements for AB, a numerical answer cannot be provided. The relationship between the area of shapes and their dimensions is key to solving problems like this, and the area of the larger square is four times that of the smaller square when the sides are doubled.

Step-by-step explanation:

To solve the problem related to the regular octagon and shaded region, we need to determine the relationship between the area of the shaded region and the length of segment AB. Although the question does not provide specific numbers, the concept that the area of the larger square is four times larger than the smaller square must be understood. This is represented by the rule stating that the ratio of areas of similar figures is the square of the scale factor.

If in Marta's problem, the side length of the larger square is 8 inches (since 4 inches x 2 = 8 inches), then the area would be 64 square inches (8 inches x 8 inches), which is indeed four times the area of the smaller square with sides of 4 inches (which is 16 square inches). Consequently, if the area of the shaded region is 'twice more' (which can be interpreted as three times) than the length of AB, and considering the properties of a regular octagon where all sides are equal, we can find the area of the unshaded region by calculating the total area and subtracting the supposedly known area of the shaded region. However, without the specific length of AB, we cannot provide a numerical answer.

Answer 2

A = (2+√8)L² - 2L, where L is the length of AB

Step-by-step explanation:

Lets call L the length of one side. The octagon can be divided in 8 triangles composed by the vertices of the octagon and the center. The base of each triangle has length L and the height is L/2 + M, with M such that 2M² = L². This is because the center of the octagon is located at middle height of the center sides. You can reach that height by 'travelling' from the bottom side throught one slanted side (which is the hypotenuse of a isosceles rectangle triangle of height M) and then you travel throught a center side half its length. Thus, h = L/2 + L/√2.

The area of each triangle of the octagon therefore is, as a result, h*L/2 = L²/4 + L²/2√2

Hence, the area of the octagon is 8*(L²/4 + L²/2√2) = (2+√8)L². The unshaded region area can be computed by taking this number and substract to it the value of 2L, which is the area of the shaded region (The double of the length of a side)

As a result,

A = (2+√8)L² - 2L