Geometry Puzzle – Challenge 68

Challenge:

The sum of three sides of a rectangular is 63. The sum of three sides of the same rectangle is 51. What is the area of the rectangle?

A- 168

B- 169

C- 240

D- 300

E- 325

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A rectangle has two widths and two lengths. Therefore, the perimeter equals: 2 × (width + length)
Since, the sum of three sides of a rectangular is 63 and the sum of other three sides of the rectangle is 51. Let’s assume the width is smaller the length. Let W be the width and L be the length. Therefore:
2W + L = 51 and W + 2 L = 63
Solve the system of equations:
2W + L = 51 → L = 51 to 2W
W + 2 L = 63 → W + 2 (51 to 2W) = 63→ W + 102 to 4W = 63 →
W = 13 and W + 2 L = 63 → 13 + 2L = 63 → 2L = 50 → L = 25
The area of the rectangle is:
W × L = 13 × 25 = 325

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Frequently Asked Questions

How do you find the length and width of a rectangle?

You need two facts about the rectangle, then write one equation for each and solve. In this puzzle, two widths and a length make 51 and a width and two lengths make 63, so 2W + L = 51 and W + 2L = 63. Adding gives 3W + 3L = 114, so W + L = 38, and subtracting gives L − W = 12. That makes W = 13 and L = 25.

How do you find length and width of a rectangle with area and perimeter?

Half the perimeter gives L + W, and the area gives L × W. Find two numbers with that sum and product. For the rectangle in this puzzle, the perimeter is 2(13 + 25) = 76, so L + W = 38, and the area is 325. Which two numbers add to 38 and multiply to 325? Testing factor pairs of 325 gives 13 and 25.

How do you find the length and width of a rectangle given the perimeter?

The perimeter alone only tells you L + W, so you need one more fact, such as how much longer the length is. If a rectangle has perimeter 76 and its length is 12 more than its width, then W + L = 38 and L = W + 12. So 2W + 12 = 38, W = 13 and L = 25, the same rectangle as in this puzzle.

How do you find length and width of a rectangle with only area?

With only the area you cannot get one exact answer, because many rectangles share the same area. An area of 325 could be 13 by 25, 5 by 65 or 1 by 325. You need a second clue, like the perimeter, a ratio, or sums of sides as in this puzzle, to pin down a single length and width.

How do you find length and width of a rectangle with diagonal?

Use the Pythagorean theorem, W² + L² = d², together with one more fact. For the 13 by 25 rectangle here, the diagonal is √(169 + 625) = √794 ≈ 28.18. If you knew only the diagonal and W + L = 38, you could square the sum: 38² = 1444, so 2WL = 1444 − 794 = 650 and WL = 325.

What is the rectangle length and width formula?

The key formulas are perimeter P = 2(L + W) and area A = L × W. Rearranged, L = P/2 − W and L = A ÷ W. In this puzzle, once W = 13 is known, L follows from 13 + 2L = 63, so L = 25, and the area is 13 × 25 = 325 square units.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to find volume and surface area of cubes
• How to solve percent of change
• How to solve multi-step word problems

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