𝐴𝐵𝐶 is an equilateral triangle of side length 12 centimeters that is inscribed in a circle. Find the radius of the circle, giving the answer to two decimal places.
Let’s begin by sketching out a diagram. It doesn’t need to be perfectly to scale, but it should be roughly in proportion so we can check the suitability of any answers we get. Since the triangle is inscribed in a circle, that means the vertices must all lie on the circumference of the circle itself. We can construct the radii of the circle as shown. Now let’s add some angles. We know that the angles in an equilateral triangle are all 60 degrees. This means that the measure of the angle 𝑂𝐴𝐵 must be half of this. That’s 30 degrees.
Similarly, the measure of the angle 𝑂𝐵𝐴 must also be 30 degrees. Finally, since angles in a triangle sum to 180, we can calculate the measure of angle 𝐴𝑂𝐵 by subtracting 30 and 30 from 180 to get 120 degrees. If we look solely at the non-right-angled triangle 𝐴𝑂𝐵, we’ll see that we know the measure of three angles and the length of one of its sides. We can use the law of sines to calculate the missing lengths. We know it’s not the law of cosines since that requires at least two known sides.
Remember, the law of sines can be used in either form. Since we’re trying to find a missing length, we choose to use the first form. This form requires us to do less rearranging to solve any equations that we get. If we we’re trying to find a missing angle however, we would choose the second form. Let’s label the sides of our triangle. The side opposite the angle 𝐴 is denoted by lowercase 𝑎. The side opposite angle 𝑂 is denoted by lowercase 𝑜. And the side opposite angle 𝐵 denoted by lower case 𝑏.
We’re trying to calculate the length of the radius of this circle. So we’re trying to find either 𝑎 or 𝑏. Let’s calculate the length of the side 𝑎. We know the measure of the angle at 𝑂 and the length of the side 𝑜, so we’ll use these two parts of the formula: 𝑎 over sine 𝐴 equals 𝑜 over sine 𝑂. Notice that we’ve changed the letters to suit the triangle we’ve drawn. The next logical step is to substitute any values we have from our triangle into the formula for the law of sines.
That gives us 𝑎 over sine 30 is equal to 12 over sine of 120. We can solve this equation by multiplying both sides by sine of 30. That gives us that 𝑎 is equal to 12 over sine of 120 multiplied by sine 30. Putting that into our calculator and we get a value of 6.9282. Correct to two decimal places, that gives us 6.93 centimeters as the radius of the circle. It’s useful to know that we can check our answer with something called the extended law of sines.
This says that for a triangle inscribed in a circle, its ratio of its side length to the sine of the opposite angle is equal to two times the radius. That’s 𝑎 over sine 𝐴 equals 𝑏 over sine 𝐵 equals 𝑐 over sine 𝐶, which is equal to two 𝑟. If we just choose two parts of this formula, let’s say 𝑐 over sine 𝐶 equals two 𝑟, we can quickly calculate the value of the radius. The measure of the angle at 𝐶 is 60 degrees. And the length of the side is 12.
So our formula will become 12 over sine 60 equals two 𝑟. We can solve this equation by dividing both sides by two. That gives us that the radius is six divided by sine 60, which is once again 6.9282 as we found out earlier. The radius of this circle then is 6.93 centimeters.