Geometry Puzzle – Challenge 63

Challenge:

A rectangle is made of 7 equal squares. If the area of the rectangle is 567 feet, what is the perimeter of the rectangle?

A- 120 feet

B- 132 feet

C- 144 feet

D- 169 feet

E- 180 feet

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The rectangle is made of 7 equal squares. Therefore, the ratio of the width and length of the rectangle is 1 to 7. Why?
Let X be the width of the rectangle. So,
\(X × 7X = 567 → 7X^2 = 567 → X^2 = 81 → X = 9\)
The width of the rectangle is 9 feet and the length is 63 feet.
(7 × 9 = 63)
The perimeter of the rectangle is:
2 × (9 + 63) = 144 feet

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Frequently Asked Questions

Can a square be a rectangle?

Yes. A rectangle is any four-sided shape with four right angles, and a square has four right angles plus four equal sides, so every square is a rectangle. The reverse is not true: the 9 ft by 63 ft rectangle in this puzzle is not a square, but each of the 7 equal pieces inside it is a 9 ft by 9 ft square.

Is a rectangle made of two squares?

A rectangle can be made of two equal squares placed side by side, giving a length twice its width. In the same way, this puzzle’s rectangle is made of 7 equal squares in a row, so its length is 7 times its width. With area 567 square feet, each square has area 567 ÷ 7 = 81, so each side is 9 ft.

How do you solve area and perimeter word problems?

Draw a quick sketch, label the unknown, and write an equation from the facts. Here, let the width be x, so the length is 7x and the area is 7x² = 567. Then x² = 81 and x = 9, making the rectangle 9 ft by 63 ft. Finish with P = 2(L + W) = 2(63 + 9) = 144 ft, and check that 9 × 63 = 567.

What area and perimeter word problems do 6th and 7th graders solve?

Middle school word problems often give an area or perimeter plus a relationship, such as a ratio, and ask for the missing measure. The 7-equal-squares puzzle is a typical grade 6 or 7 challenge because it needs a ratio, an equation and a square root. A similar one: 5 equal squares with total area 405 sq ft give sides of 9 ft and a perimeter of 108 ft.

Is there an area and perimeter word problems worksheet with answers?

You can build a quick worksheet from this puzzle by changing the numbers and keeping the answers handy. Try 4 equal squares with area 196 (sides 7, perimeter 70), 3 equal squares with area 75 (sides 5, perimeter 40), and 7 equal squares with area 567 (sides 9, perimeter 144). Students check each one by multiplying length × width back to the area.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to find complementary, supplementary, vertical, and adjacent angles
• How to solve angles and angle measure
• How to solve multi-step word problems

If your student enjoys puzzles like this, Geometry for Beginners dives into the same kinds of relationships inside a full curriculum. For the algebra you will lean on, Pre-Algebra for Beginners fills in the foundations gently.