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What shapes have the same perimeter but different areas?
Most shapes can have the same area and different perimeters. For example the right size square and circle will have the same are but they will have different perimeters. You can draw an infinite number of triangles with the same area but different perimeters. This is before we think about all the other shapes out there.
How can two parallelograms have the same areas but different perimeters?
Start with a rectangle, which is a special type of a parallelogram. Suppose its height is A units and length is B units. Then its area is A*B sq units and its perimeter is 2*(A+B) units. Now slide the top side horizontally, keeping the vertical height between the top and the bottom the same. The area will not change but the perimeter will increase because the sides have now become slanted and longer. The rather griim figures below may help illustrate (the . are for spacing): +------+.....+------+ |........|........\........\ +------+..........+------+
How many rectangles have the same area but different perimeters?
Infinitely many. Suppose the area of the rectangle is 100. We could create rectangles of different areas: 100x1 50x2 25x4 20x5 10x10 However, the side lengths need not be integers, which is why we can create infinitely many rectangles. Generally, if A is the area of the rectangle, and L, L/A are its dimensions, then the amount 2(L + (L/A)) can range from a given amount (min. occurs at L = sqrt(A), perimeter = 4sqrt(A)) to infinity.
Is it true that when equilateral triangles have equal perimeters they must also have equal areas?
Yes.
Who invented area and perimeter?
The first recorded use of areas and perimeters in the west was in ancient Babylon, where they used it to measure the amount of land that was owned by different people for taxation puposes.
