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Area of a Road Around a Square Garden: S=at Proof
Published on May 26, 2026
Geometry
Learn how to prove that the area of a road surrounding a square garden is equal to the product of the road's width and the length of its centerline. This problem involves algebraic manipulation and geometric area calculations.
Question
Answer
Based on the image provided, here is the step-by-step solution to the math problem.
Problem Statement (Translated)
A road of width
Step-by-Step Solution
1. Finding the Area
The area of the road
- The side length of the inner garden is
. - The width of the road is
on all sides. Therefore, the side length of the outer square is .
The area
Expanding the square:
Factoring out
Looking at the worksheet’s structure, it seems to guide you to a specific form. Let’s re-examine the “middle line” approach.
2. Finding the length
The problem states that
The “middle line” forms a square that sits exactly halfway between the inner and outer boundaries.
The “middle line” forms a square that sits exactly halfway between the inner and outer boundaries.
- The distance from the inner garden to the middle of the road is
. - Therefore, the side length of this “middle square” is
. - The length of the line
passing through the middle (the perimeter of this middle square) is:
3. Proving
Now we substitute
From step 1, we found
From step 2, we found
From step 1, we found
From step 2, we found
Therefore:
Filling in the Worksheet Blanks
Based on the logic above, here is how the blanks in the image should be filled:
Area
The length
Therefore,
From ① and ②,