Same Perimeter, Different Area

Key idea

Figures can have the same perimeter but different areas. Perimeter measures the total length around a shape, while area measures the space inside it. So two shapes with equal boundary lengths do not necessarily enclose the same amount of space.

How to solve these exercises

1. Find the perimeter condition. If side lengths are given, write an equation for the perimeter and use it to relate the unknowns.
2. Express the dimensions. Rewrite one side or set of sides in terms of the others so that the perimeter stays the same.
3. Compute each area separately. Use the correct area formula for each shape. Keep expressions exact if the problem uses fractions, radicals, or algebraic expressions.
4. Simplify the final answer. Combine like terms, reduce fractions, and write results in simplest form.

Important check

Before finishing, verify that the perimeters match exactly. Then compare the areas to see that they are not equal. If the question asks for a difference in area, subtract carefully and simplify.

Common strategy

When two shapes share a perimeter, changing the side lengths can make one shape more “spread out” and another more compact. A more compact shape often has a larger area when the perimeter is fixed, but always confirm this by calculation.