Geometry Puzzle – Challenge 65

Challenge:

If a side of a square is increased by \(20\%\), what percent of the area of the square will be increased?

A- \(20\%\)

B- \(40\%\)

C- \(44\%\)

D- \(100\%\)

E- \(200\%\)

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Area of a square = side times side.
Let X be the side of a square. Then, the side of the square is \(X^2\).
If a side of a square is increased by \(20\%\), then the area of the square will be increased by 44 percent:
\(1.2X × 1.2X = 1.44X^2\)
\(1.44X^2\) is 0.44 or \(44\%\) bigger than \(X^2\)

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Frequently Asked Questions

Why does area scale by the SQUARE of the linear scale factor?

Because area depends on TWO dimensions (length and width). If you scale each by k, the product scales by k times k = k^2.

How do I compute the new area from a 20% side increase?

Let X be the original side. New side = 1.2X. New area = (1.2X)^2 = 1.44X^2. The increase factor is 1.44 – 1 = 0.44, or 44%.

Does this rule apply to any shape?

To any 2D shape scaled uniformly (same factor in all directions): yes. Triangle, circle, rectangle — all areas scale by k^2 when linear dimensions scale by k.

What happens to volume when sides scale?

Volume depends on THREE dimensions, so it scales by k^3. Side scaled by 1.2 means volume becomes 1.728 times original — a 72.8% increase.

Why is the answer not 40% (double the side change)?

Doubling the change-in-side and saying it equals the change-in-area is the most common wrong instinct. Area is a product of two scaled lengths, so the increase compounds rather than doubles.

What if the side DECREASES by 20%?

New side = 0.8X. New area = 0.64X^2, which is 36% less than the original. The decrease in area is more than double the decrease in side, for the same reason.

How does this connect to similar figures?

Similar figures share a scale factor k between all corresponding linear dimensions. Their areas are related by k^2 and volumes by k^3 — exactly the same rule.

What real-life examples use this principle?

Map scaling (a 1:1000 map shows 1,000,000 times less area than the real region), photo enlargements (doubling print width quadruples the ink), and pizza pricing (a 16-inch pizza has 1.78 times the area of a 12-inch).

How would I check the answer for a specific number?

Take a square with side 10 (area 100). Increase side to 12 (20% increase). New area = 144. The increase is 44 out of 100 = 44%.

What standards does this exercise?

Middle school proportional reasoning (Grade 7 and 8) plus area formulas. Foundational for high school geometry, where the area/volume scaling rules are formalized.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to solve angles and angle measure
• How to solve percent of change
• How to find volume and surface area of cubes

If your student enjoys puzzles like this, Geometry for Beginners works the same kinds of relationships inside a full curriculum. Pre-Algebra for Beginners covers the algebraic foundations.