Math Problem Statement

Consider a square of side length 5. Place four equal circles in the corners of the square, each circle touching the two adjacent sides of the square. Then, add a fifth circle inside the square such that it touches all four corner circles but does not overlap with them. What is the area of the inner circle?

Solution

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Square Properties
Algebra

Formulas

Diagonal distance = 5√2 - 2r
Distance between centers: 2(r + R)
Area of a circle: A = πR^2

Theorems

Pythagorean Theorem
Properties of Tangent Circles

Suitable Grade Level

Grades 10-12