Mensuration Problem generalization

In this article, I want to talk about a generalization that I developed a few days ago. To explain the generalization better, we shall first start with a sample problem:-

Problem) Consider a triangle with sides of 3cm, 6cm, and 8cm. A man runs around the triangle in such a way that he is always at a distance of 1cm from the sides of the triangle, then the distance travelled by him is-

The solution to the above is as follows-

First, let the take the triangle ABC. Then we draw parallel lines to all the sides to form an exterior triangle with all its sides at a distance of 1cm from the smaller triangles’ corresponding sides and curve them to form a connecting arc at the edges.

We then drop perpendiculars from each of the points ABC onto the bigger triangle on both sides of the points. Then, as we can see, since we dropped perpendiculars on both sides, the sum of the angles of angle A and the angle opposite A equals 180°.

Similarly, angle opposite B = 180° – B and angle opposite C = 180° – C.

Now the distance travelled in an arc = radius x angle subtended

In the above problem, since the man travels at a distance of 1cm from the triangle, the radius = 1cm and the angle at each corner point is 180°-A, 180°-B, 180°-C.

Therefore, the distance travelled by the man = (3+6+8) + (1)( 180°-A) + (1)( 180°-B) + (1)( 180°-C)

                                                                                 = 17 + 1(180°-A+180°-B+180°-C)

                                                                                 = 17 + 3π – (A+B+C) = 17 +3π – π = 17 + 2π

The above problem however can be generalized for an n-sides polygon, with a man running at a distance of k from the sides.

Let us take an n-sides polygon ABCD…..N. The sides of the polygon are X₁, X₂……, X_P. A man runs around the polygon at a distance k from the sides of the polygon.

Then the angles opposite to each point become; (π-A), (π-B), (π-C)…… (π -N).

Therefore, the distance travelled = (X₁+ X₂+……+ X_P) + (k) (π-A) + (k)(π-B) + (k)(π-C)+……+ (k)(π –N)

                                                            = (X₁+ X₂+……+ X_P) + (k)( π-A + π-B + π-C +…… +π –N)

                                                             = (X₁+ X₂+……+ X_P) + (k)( Nπ – (A+B+C+……+N))

                                                             = (X₁+ X₂+……+ X_P) + (k)( Nπ – (N-2)π)  [Since the sum of the

interior angles of a n-polygon = (n-2)π]  

                                                             = (X₁+ X₂+……+ X_P) + (k)(2π)

Therefore, the distance travelled by a person around any n-sided polygon at a distance k from its sides

= (X₁+ X₂+……+ X_P) + (k)(2π)