Example Question - circle diameter
Here are examples of questions we've helped users solve.
Area of a Circle with an Inscribed Square
<p>Let the side of the square be \( s \). Since the area of the square is 4, we have:</p> <p>\( s^2 = 4 \)</p> <p>\( s = 2 \)</p> <p>The diameter of the circle is the diagonal of the square. Using the Pythagorean theorem:</p> <p>\( \text{diagonal}^2 = s^2 + s^2 \)</p> <p>\( \text{diagonal}^2 = 2^2 + 2^2 \)</p> <p>\( \text{diagonal}^2 = 4 + 4 \)</p> <p>\( \text{diagonal}^2 = 8 \)</p> <p>\( \text{diagonal} = \sqrt{8} \)</p> <p>\( \text{diagonal} = 2\sqrt{2} \)</p> <p>Since the diameter is \( 2\sqrt{2} \), the radius \( r \) is \( \sqrt{2} \). The area \( A \) of the circle is:</p> <p>\( A = \pi r^2 \)</p> <p>\( A = \pi (\sqrt{2})^2 \)</p> <p>\( A = \pi \cdot 2 \)</p> <p>\( A = 2\pi \)</p> <p>Therefore, the correct answer is \( B: 2\pi \).</p>
Finding the Diameter of a Circle from a Given Equation
Complete the square for \( x \) and \( y \): \( x^2 + 22x = x^2 + 22x + (22/2)^2 - (22/2)^2 = (x+11)^2 - 121 \) \( y^2 + 30y = y^2 + 30y + (30/2)^2 - (30/2)^2 = (y+15)^2 - 225 \) Rewrite the equation: \( (x+11)^2 - 121 + (y+15)^2 - 225 + 90 = 0 \) \( (x+11)^2 + (y+15)^2 = 121 + 225 - 90 \) \( (x+11)^2 + (y+15)^2 = 256 \) The equation \( (x+11)^2 + (y+15)^2 = 256 \) represents a circle with radius \( r^2 = 256 \), so \( r = 16 \). Diameter \( d = 2r = 2 \times 16 = 32 \). The diameter of the circle is 32.
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