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Problem 535

# Show that the perimeter of an equilateral triangle which is inscribed in a circle with radius of length $$\mathrm{r}$$ is $$3 \sqrt{3} \mathrm{r}$$.

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The short answer to the question is that when an equilateral triangle is inscribed in a circle of radius r, its perimeter can be calculated by first splitting the triangle into two 30-60-90 right triangles by dropping a perpendicular from a vertex to the base. Using the Pythagorean theorem and the properties of those right triangles, the side length of the equilateral triangle is found to be $$\sqrt{3} r/2$$. Then, by multiplying this side length by 3, the perimeter of the triangle is found to be $$3\sqrt{3} r$$.
See the step by step solution

## Step 1: Understanding Equilateral Triangle and Circle Properties

An equilateral triangle has all its sides of equal length. When such a triangle is inscribed in a circle, each vertex of the triangle touches the circumference of the circle and the center of the circle is equidistant from the three vertices of the triangle. The distance from the center of the circle to any point on the circumference is known as the radius (r) of the circle.

## Step 2: Drawing Perpendicular from Vertex to Base in Triangle

For a clear demonstration, consider one of the vertices of the triangle. Draw a perpendicular line from this vertex to the base of the triangle. This line forms the height (h) of the triangle, splitting the triangle into two 30-60-90 right triangles. The line drawn is also a radius of the circle.

## Step 3: Applying Properties of 30-60-90 Triangle

In the 30-60-90 triangle, the side opposite the 30° angle is half the length of the hypotenuse, which in this case is the radius of the circle. Hence, it equals to r/2. The side opposite the 60° angle, which is a side of the equilateral triangle, can be found using Pythagoras theorem. Since, the hypotenuse is r and base is r/2, the side of the equilateral triangle (a) is $$\sqrt{r^2 - (r/2)^2} = \sqrt{3} r/2$$.

## Step 4: Calculating Perimeter of the Triangle

The perimeter (P) of an equilateral triangle is just three times the length of one side, hence: $$P = 3 * (\sqrt{3} r/2) = 3 \sqrt{3} r/2 * 2 = 3\sqrt{3} r$$. This shows that an equilateral triangle inscribed in a circle of radius r does have a perimeter of $$3 \sqrt{3} r$$.

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