Problem
of the Week
Problem
C and Solution
Teddy
Was Framed

Problem

Pat has four rectangular pieces of wood, each \(30\) cm long and \(3\) cm wide. She arranges the four pieces
of wood to form the border of a picture frame for a picture of a teddy
bear, as shown.

Four identical rectangles are arranged to form a square frame as described below.

One rectangle is placed along the top with its length
horizontal.

Another rectangle is placed on the right, with its length
vertical, so its top side is aligned with the top side of the top
horizontal rectangle.

Another rectangle is placed along the bottom, with its length
horizontal, so its right side is aligned with the right side of the
right vertical rectangle.

Another rectangle is placed on the left, with its length
vertical, so its bottom side is aligned with the bottom side of the
bottom horizontal rectangle. Also, its left side is aligned with the
left side of the top horizontal rectangle.

Determine the area of the region enclosed by the wooden frame.

Solution

Solution 1

The region enclosed by the wooden frame is a square with side length
\(30 - 3 = 27\) cm.

Thus, the area of the region enclosed by the frame is equal to \(27\times 27 = 729\) cm\(^2\).

Solution 2

The outer perimeter of the frame forms a square with side length
\(30 + 3 = 33\) cm.

The area of the outer square is therefore \(33\times 33 = 1089\mbox{ cm}^2.\)

The area of the region enclosed by the wooden frame is equal to the
area of the outer square minus the areas of the four wooden
rectangles.

Each wooden rectangle has area \(30 \times
3 = 90\) cm\(^2\).

Therefore, the area of the region enclosed by the frame is equal to
\(1089-4 \times 90 = 729\) cm\(^2\).