Perimeter of polygons with an inscribed circle

Key Notes :

🟢 What is an Inscribed Circle (Incircle)?

• A circle drawn inside a polygon that touches all its sides.
• The point where the circle touches a side is called the point of tangency.
• The center of the circle is called the incenter (🔺 for triangles).

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🟣 Key Property of Tangents from a Point

👉 If two tangents are drawn from the same vertex to the circle,
then they are equal in length.

Example in a quadrilateral:

• From vertex A, the two tangent lengths are AP = AS.
• From vertex B, the two tangent lengths are BP = BQ, and so on.

This property is the secret behind finding the perimeter! 🔑✨

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🔵 How Perimeter Is Found Using Tangent Lengths

Each side of the polygon is formed by adding two tangent segments,
one from each of its endpoints.

Example for a quadrilateral ABCD:

• Side AB = AP + BP
• Side BC = BQ + CQ
• Side CD = CR + DR
• Side DA = DS + AS

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🟠 Amazing Result: Perimeter = Sum of All Tangent Lengths × 2

Since every tangent length repeats twice (once for each side),
the perimeter = (AP + BP + CQ + DR + …) × 2.

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🟢 Special Case — Triangle

For a triangle, if tangent lengths are x, y, z at three vertices,
then:

• Perimeter = x + y + z + x + y + z = 2(x + y + z)
• Or simpler:
  ➤ Perimeter = sum of the three sides (as usual!)
  because each side is formed by adding tangent lengths.

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🔴 Formula Summary ✨

⭐ Perimeter of polygon with an inscribed circle

P = 2(x1 + x2 + x3 + …)

where x1 , x2 , x3… are tangent lengths from each vertex.

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🟣 Visual Shortcut

• Tangent lengths from each vertex are equal.
• Add all tangent lengths.
• Multiply by 2.
• 🎉 You get the perimeter!

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🟡 Why Useful?

This method is useful when:
✔ Side lengths are not given
✔ Tangent lengths (from vertices to points of tangency) are provided

Learn with an example

let’s practice!