Quadratic Functions: Your Garden Fence Has Opinions

Sasha Sweder

Oct 9

2 min read

A square garden has five-meter sides, twenty meters of perimeter, and twenty-five square meters of area.

You have 20 meters of imaginary fencing and a plan to become a gardening person. The fence does not care about your aesthetic. It cares about how much rectangle it can surround. That turns out to be a quadratic problem wearing muddy shoes.

Start here

Perimeter measures the distance around a shape. Area measures the space inside it. For a rectangle, perimeter is 2 × length + 2 × width. With 20 meters of fence, length + width = 10. This example encloses all four sides, with no gate gap or existing wall.

Try some shapes

A 1-by-9-meter rectangle uses 20 meters of fence and holds 9 square meters. A 3-by-7 rectangle holds 21. A 5-by-5 square holds 25. Same fence, different planting space. Measuring around is not the same as measuring inside.

Where the quadratic appears

Let width be x. Length must be 10 − x. Multiply them: A = x(10 − x) = 10x − x². The squared term makes this a quadratic function. In the actual garden, x must be between 0 and 10; negative widths are not a bold landscaping choice.

Level up: find the best size

Rewrite the formula as A = 25 − (x − 5)². A square is never negative, so subtracting that square cannot give more than 25. The largest area is 25 square meters, when x = 5. The graph’s highest point is its vertex.

Try it

With the same 20 meters of fencing, what area does a 4-by-6 rectangle enclose? How much less is that than the square?

Check your answer

4 × 6 = 24 square meters. That is 1 square meter less than 25. Its perimeter is still 2 × 4 + 2 × 6 = 20 meters.

Save this

Perimeter goes around. Area fills the inside. A fixed perimeter can enclose different areas. Quadratics can help find the largest or smallest result—not just decorate an algebra worksheet.