The figure shows a path (the shaded portion) laid around a rectangular park EFGH.
(i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area.
An example of a formula — Area of a rectangle = length × width.
[Hint: There is a relation between the areas of EFGH, the path, and ABCD.]
(ii) If the width of the path along each side is

Question

Bright Tutorials · Accepted Answer

(i) Measurements needed:
Length of outer rectangle ABCD = A
Width of outer rectangle ABCD = B
Length of inner rectangle EFGH = a
Width of inner rectangle EFGH = b
Let A = 10, B = 8, a = 6, b = 4.
Calculation: Area of path = Area of ABCD – Area of EFGH
= 10 × 8 – 6 × 4
= 80 – 24
= 56 m
2
Formula:
Area of path = (A × B) – (a × b)
(ii) Yes, if the width of the path is uniform along each side, we can find its area, but we also need the dimensions of either the outer or inner rectangle.
If the width of the path d = 2 m (Uniform on all sides)
Length of inner path EFGH = l = 16 m
Width of inner path EFGH = w = 11 m
Length of outer rectangle = l + 2d
= 16 + 2(2)
= 20 m
Width of outer rectangle = w + 2d
= 11 + 2(2)
= 15 m
Breaking the path into rectangles
Now there are 4 rectangles
Left rectangle = w × d
= 11 × 2
= 22 m
2
Right rectangle = w × d
= 11 × 2
= 22 m
2
Top rectangle = (l + 2d) d
= 20 × 2
= 40 m
2
Bottom rectangle = (l + 2d) d
= 20 × 2
= 40 m
2
Total area of path = 22 + 22 + 40 + 40 = 124 m
2
Formula:
Area of path = 2d(l + w) + 4d
where d – width of path
l – length of inner path
w – width of inner path
(iii) No, the area of the path does not change.
Reason: The area of the path depends only on:
Area of outer rectangle ABCD.
Area of inner rectangle EFGH.

Area — Question 2