How do I Find the Radius of a Circle Intersecting the Apexes of a Square's Four Corners?
Drawing the largest square that will fit inside of a circle was the beginning part of the method Archimedes used in his calculation for the area of a circle. Using this method the squares apexes or corners each come in contact with the circle. Given the symmetry of the two shapes, he was able to conclude that the Pythagorean Theorem could be used to find the diameter of the circle, which in turn can be used to find the radius of the circle.
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Instructions
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Draw the diagram of a square inside of a circle, using a protractor for the circle and a pencil and ruler to form the square. Draw the square so each corner or apex touches the circle. Draw one straight line from one corner of the square to the opposite corner forming a diagonal that creates 2 triangles.
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Draw a second straight diagonal connecting the other two opposing corners of the square. The diagonals of the square are equal to each other, showing that each diagonal is congruent to the diameter of the circle and that they must pass through the center of the circle. Once this is determined, erase the second diagonal drawn, leaving the first diagonal intact.
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Label the width of the square "A", the length of the square "B" and the diagonal drawn down the center of the square and circle, the hypotenuse, "C". Notice the diagonal line passing through the center of the circle splits the square in half to form two right triangles. A right triangle is a triangle with one 90-degree angle.
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Write down the Pythagorean Theorem which states that C² = A² + B² for the length, width and hypotenuse of a right triangle. The formula is already set to solve for the hypotenuse, or the diameter of the circle.
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Measure the sides of the square using a ruler and record the length as "A" and the width as "B". The measurements A and B should be equal.
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Substitute the measurements taken into the Pythagorean Theorem formula and solve the equation for C. For example if A and B are measured to be 6 inches in length then the hypotenuse C² = (A² + B²) = (6 inches)² + (6 inches)² = (72 inches)². Taking the square root of 72 will give you C = 8.5 inches, which is not only the hypotenuse of both right angles but also the diameter of the circle.
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Write down the equation to calculate radius r = d/2. In the formula the symbol r stands for the radius of the circle and d stands for the diameter of the circle.
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Determine the radius equation using the previously calculated hypotenuse or diameter. The diameter of the circle in the example was calculated to be d = 8.5 inches. Solving for the radius r gives you r = d/2 = 8.5 inches/2 = 4.25 inches.
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- Photo Credit geometry image by Alexey Klementiev from Fotolia.com
