ABCD is a square with length of each side 1cm. An octagon is formed by lines joining the vertices of the square to the mid points of opposite sides. Find the area of the octagon?

Let the corners of the square be at (0,0),(1,0),(0,1),(1,1)

y=1-2x intersects y=(1-x)/2 at (1/3,1/3)

So, in each of 4 quadrants, the exterior of the octagon is a trapezoid and a triangle

trapezoid has area 3/16
triangle has area 1/48
sum has area 5/24
The 4 have area 5/6
The octagon thus has area 1/6

Or, consider that the octagon has
side = √5/12
apothem = 1/2√5

area = 1/2 * 8√5/12 * 1/2√5 = 1/6

To find the area of the octagon formed by lines joining the vertices of the square to the midpoints of opposite sides, we can follow these steps:

1. Start by drawing the square ABCD with side length 1cm.

2. Identify the midpoints of the opposite sides of the square. Let's label them E, F, G, and H. These midpoints divide the sides of the square into halves.

3. Draw lines connecting the vertices of the square to the midpoints of the opposite sides. This will form an octagon.

4. Now, we can divide this octagon into 8 congruent isosceles triangles. Four of these triangles have a base of length 1cm (the sides of the square), and the remaining four triangles have a base of length 0.5cm (the halves of the sides of the square).

5. The area of each of these triangles can be calculated using the formula A = 0.5 * base * height. In this case, the height of the triangle will be the distance from the midpoint of the side of the square to the vertex (which is also the side length of the square).

6. Calculate the area of one of the triangles with base 1cm: A = 0.5 * 1cm * 1cm = 0.5cm^2.

7. Calculate the area of one of the triangles with base 0.5cm: A = 0.5 * 0.5cm * 0.5cm = 0.125cm^2.

8. Since there are 4 triangles with base 1cm and 4 triangles with base 0.5cm, sum up the areas of these triangles: 4 * 0.5cm^2 + 4 * 0.125cm^2 = 2cm^2 + 0.5cm^2 = 2.5cm^2.

Therefore, the area of the octagon formed by lines joining the vertices of the square to the midpoints of opposite sides is 2.5 square centimeters (cm^2).

To find the area of the octagon formed by joining the vertices of the square to the midpoints of opposite sides, we can use the concept of subtracting areas.

First, let's find the area of the square. Since each side of the square has a length of 1cm, the area of the square is given by the formula: A = s^2, where s is the length of a side.
So, the area of the square is A = 1^2 = 1 square cm.

Next, let's find the total area of the triangles formed by joining the vertices of the square to the midpoints of opposite sides. Since there are four sides of the square, there will be four such triangles.

To find the area of each triangle, we need the length of the base and height. Since the base of each triangle is half the length of a side of the square (0.5cm), and the height is the length of the side of the square (1cm), the area of each triangle is given by the formula: A = 0.5 * base * height = 0.5 * 0.5 * 1 = 0.25 square cm.

Therefore, the total area of the four triangles is 4 * 0.25 = 1 square cm.

Finally, to find the area of the octagon, we subtract the total area of the four triangles from the area of the square:
Octagon area = Square area - Triangle area
Octagon area = 1 - 1 = 0 square cm.

Hence, the area of the octagon formed by joining the vertices of the square to the midpoints of opposite sides is 0 square cm.