Perimeter & Area

Problem

Perimeter as a multiple of a unit length

A big rectangle ABCD is built from 5 identical small rectangles. One small rectangle stands upright on the left, and the other 4 lie sideways and are stacked on the right side. The whole outline has a perimeter of 54 inches, and I need to find how many square inches the big rectangle covers.
A B C D
Measurement & data
Your answer
How to solve
Strategy Draw a Diagram — From the picture I let the short side of one small rectangle be one 'unit length.' Marking the diagram shows the long side equals 4 units, so I can write every side of the big rectangle, and its whole perimeter, as a count of unit lengths. Then finding the unit length is one small subproblem, and the area is a second subproblem.
1STEP 1

Name a unit length from the picture

Four short sides stacked reach the height of one long side, so a long side is 4 units.

long side = 4 × short side
2STEP 2

Write each side of the big rectangle in units

The big rectangle stands 4 units tall and 1 + 4 = 5 units wide.

height = 4 units, width = 1 + 4 = 5 units
3STEP 3

Express the perimeter as a multiple of the unit length

Its perimeter is 2 times (5 + 4) = 18 unit lengths.

P = 2×(5 + 4) = 2 × 9 = 18 units
4STEP 4

Find the unit length

Those 18 units make 54 inches, so one unit is 54 divided by 18 = 3 inches.

18 × □ = 54 → □ = 54 ÷ 18 = 3 in
5STEP 5

Compute the area

The big rectangle is 15 inches wide and 12 inches tall, so its area is width times height = 15 times 12 = 180 square inches.

A = 15 × 12 = 180 in²
Answer
180 square inches
The unit length 3 in gives a 15 in by 12 in rectangle, perimeter 2 times (15 + 12) = 54 in, which matches the given perimeter. The 5 small rectangles are each 3 in by 12 in with area 36 in², and 5 times 36 = 180 in², the same as 15 times 12. Both ways agree, so 180 square inches is consistent.
Takeaway

Call the short side '1 unit,' count how many units go around the border, and the perimeter becomes easy sharing: 54 in over 18 units = 3 in each.

  • Name a unit length from the picture
  • Write each side of the big rectangle in units
  • Express the perimeter as a multiple of the unit length
  • Find the unit length
  • Compute the area
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
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1 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 12 problems