I have a triangle with equal sides. Then a s cons triangle is put in the middle so I have four triangles. Triangles are continued to be put instead so I have more and more triangles. As more triangles are added in the same space, the size of them are getting smaller.
Could you explain why the area is getting smaller but the perimeter is getting larger?
I think you are missing something.
Each added triangle has a smaller area and a smaller perimeter.
I think I am missing something, too. If all the triangles are inscribed into the same area, the total area is unchanged, right? By "the perimeter" do you mean the sum of all the perimeters?
Sure! First, let's break down the situation step by step:
1. You start with a triangle that has equal sides. Let's call this triangle T1.
2. You then place a smaller triangle inside T1, making the new shape look like a larger triangle with a smaller triangle removed from its center. Let's call this new shape T2.
3. If you continue this pattern, placing smaller triangles inside the new shapes each time, we can call the subsequent shapes T3, T4, T5, and so on.
Now, let's look at why the area is getting smaller and the perimeter is getting larger:
1. Area: When you add smaller triangles inside the original triangle, the total area of the shape decreases because the additional triangles take up some of the space that was originally part of the larger triangle. As you continue to add more triangles, the total area keeps decreasing because the new triangles occupy smaller and smaller spaces within the overall shape.
2. Perimeter: The perimeter, on the other hand, increases because each new triangle adds more sides to the overall shape. When you add smaller triangles inside the original triangle, each additional triangle creates new sides along the inner edges of the larger shape, thus increasing the total perimeter.
To visually understand this concept, imagine drawing a large equilateral triangle and dividing it into smaller equilateral triangles. Each time you add more triangles inside, you will notice that the individual areas of the smaller triangles decrease, but the number of sides and therefore the perimeter of the overall shape keeps increasing.
In summary, the area gets smaller because the added triangles take up space within the original shape, while the perimeter gets larger because each new triangle adds more sides to the overall shape.