The exact perimeter of a square inscribed within a circle is 4·r·√2, where 'r' represents the radius of the circle. This formula provides a precise measurement, leveraging the fundamental geometric relationship between an inscribed square and its circumscribing circle.
Understanding the Perimeter of an Inscribed Square
When a square is inscribed in a circle, all four of its vertices lie on the circle's circumference. This specific arrangement creates a direct and calculable relationship between the dimensions of the square and the circle.
Key Geometric Relationship
The most crucial insight for determining the perimeter of an inscribed square is that its diagonal is precisely equal to the diameter of the circle.
- Circle's Diameter (D): The diameter of a circle is twice its radius (D = 2r).
- Square's Diagonal (d): For a square inscribed in a circle, its diagonal
dis equal to the circle's diameterD. Thus,d = 2r.
Deriving the Perimeter Formula
To find the perimeter, we first need to determine the side length (s) of the square. We can use the Pythagorean theorem, as a square's diagonal divides it into two congruent 45-45-90 right-angled triangles.
- Diagonal of the Square: As established, the diagonal
dof the inscribed square is equal to the circle's diameter, sod = 2r. - Pythagorean Theorem: In a square with side length
s, the diagonaldrelates tosbyd² = s² + s², which simplifies tod² = 2s². - Solving for Side Length (s):
- Substitute
d = 2rinto the equation:(2r)² = 2s² 4r² = 2s²s² = 2r²- Taking the square root of both sides:
s = √(2r²) = r√2. - Therefore, each side of the inscribed square measures
r·√2.
- Substitute
- Calculating the Perimeter (P): The perimeter of a square is
4times its side length.P = 4s- Substitute
s = r√2:P = 4(r√2) - P = 4r√2
This formula directly integrates the radius of the circle, offering a straightforward way to calculate the square's perimeter without needing intermediate measurements beyond the circle's radius.
Practical Example
Let's illustrate with an example to clarify the calculation:
| Circle Radius (r) | Square Diagonal (2r) | Square Side Length (r√2) | Square Perimeter (4r√2) | Approximate Perimeter (√2 ≈ 1.414) |
|---|---|---|---|---|
| 5 units | 10 units | 5√2 units | 20√2 units | ≈ 28.28 units |
| 10 cm | 20 cm | 10√2 cm | 40√2 cm | ≈ 56.57 cm |
Key Steps and Concepts
To recap the process of finding the perimeter of an inscribed square:
- Identify the diameter: The diameter of the circle is twice its radius (
2r). - Relate diameter to square's diagonal: The square's diagonal is equal to the circle's diameter (
2r). - Use the Pythagorean theorem: Apply
a² + b² = c²to the square's sides and diagonal, or simply remember that for a square,diagonal = side × √2. - Calculate the side length: Solve for the side length, which will be
r√2. - Determine the perimeter: Multiply the side length by 4 (
4 * r√2).
This geometric relationship is a fundamental concept in mathematics and finds applications in various fields, from design to engineering. For more details on geometric figures and their properties, you can explore resources like Khan Academy's Geometry section.
