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A circle with centre (X, Y) and radius r has formula:

(x - X)² + (y - Y)² = r²

As the points (6, 3), (-5, 2) and (7, 2) are on the circle:

  1. (6 - X)² + (3 - Y)² = r²
  2. (-5 - X)² + (2 - Y)² = r²
  3. (7 - X)² + (2 - Y)² = r²

Subtracting (2) from (3) gives:

(7 - X)² - (-5 - X)² + (2 - Y)² - (2 - Y)² = r² - r²

→ 49 - 14X + X² - (25 +10X +X²) = 0

→ 49 - 25 -14X -10X +X² - X² = 0

→ 24 - 24X = 0

→ 24X = 24

→ X = 1

Substituting this into (1) and (2) gives:

  1. (6 - 1)²+ (3 - Y)² = r² → 5² + 9 - 6Y + Y² = r² → 34 - 6Y + Y² = r²
  2. (-5 - 1)² + (2 - Y)² = r² → (-6)² + 4 - 4Y + Y² = r² → 40 - 4Y +Y² = r²

Subtracting (1) from (2) now gives:

40 - 34 - 4Y - (-6Y) +Y² - Y² = r² - r²

→ 6 + 2Y = 0

← 2Y = -6

→ Y = -3

→ The equation of the circle is (x - 1)² + (y - -3)² = r² → (x - 1)² + (y + 3)² = r²

Using one of the points (6, 3), say, and substituting for x and y gives:

(6 - 1)² + (3 + 3)² = r²

→ 5² + 6² = r²

→ 25 + 36 = r²

→ r² = 61

The area of a circle is given by area = π × radius²

→ the area of the circle through those three points is π × 61 = 61π units² ≈ 191.64 units²

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Points: (0, 0) and (3, 4) Using Pythagoras: distance fro (0, 0) to (3, 4) = 5 cm which is the radius Circumference: 2*pi*5 = 31.416 cm rounded to three decimal places Area: pi*5^2 = 78.540 square cm rounded to three decimal places


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No, a circle's circumference is not the same as its width. The circumference refers to the total distance around the circle, while the width of a circle is typically described by its diameter, which is the distance across the circle through its center. The relationship between the two is that the circumference is equal to the diameter multiplied by π (pi).

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