# A rectangle of sides 3 cm and 4 cm is inscribed in a circle. Find the radius of the circle.

bridgetrbcs | Certified Educator

If this rectangle is inscribed in the circle then it is drawn into the circle with each corner of the rectangle lying on the edge of the circle.

If the sides of the rectangle are 4 and 3cm, you can label your longer side 4 and your shorter side 3.  Draw one diagonal of your rectangle.  Remember that a diagonal is drawn across the rectangle from one corner to the opposite corner.  We have created a right triangle based on the rule that all corners of a rectangle are 90 degrees.  Now that we have a right triangle AND we know two sides of our right triangle (3 and 4) we can find the final side.

• a^2 + b^2 = c^2
• 3^2 + 4^2 = c^2
• 9 + 16 = c^2
• 25 = c^2
• 5 = c

Notice that c, as our diagonal, cuts straight through the center of our circle.  A line that starts on the edges of the circle and goes through the center of a circle is a diameter.  Therefore, our diameter is 5.   Radius is 1/2 of diameter so 1/2 of 5 is 2.5cm.

cburr | Certified Educator

bridgetrbcs' answer was great.  The only thing I would add is that you should always be on the alert for a 3-4-5 right triangle.  This special relationship comes up ALOT in geometry problems!!  Basically, in a 3-4-5 triangle, the two sides are multiples of 3 and 4, and the hypotenuse is a multiple of 5.  NOTE that they all have to be multiplied by the same thing, to keep the ratio of 3:4:5.  So, if you have a problem where you know the sides of a right triangle are 6 and 8, you know the hypotenuse will be 10 without doing anything more.