There is only one equilateral triangle inscribed in a circle whose diameter is 20 cm. Now, the triangle of the largest area inscribed in a circle is equilateral.
Having said that, let's calculate the area of such a triangle. Look at the following picture:
To calculate the area of the (green) triangle, we will follow the following procedure:
0. To find the value of a/2,
,1. to find the value of a,
,2. To use Heron's formula for the area of a triangle.
Step 1)
r denotes the radius of the black circle, which is exactly half its diameter. Since the diameter of the black circle is 20 cm, we get that
[tex]r=\frac{20}{2}cm=10\text{ cm.}[/tex]Then, we have
By the cosine trigonometric relation, we have
[tex]\cos (30)=\frac{\frac{a}{2}}{10}\text{.}[/tex]Then,
[tex]\begin{gathered} \frac{\sqrt[]{3}}{2}=\frac{\frac{a}{2}}{10}, \\ 10\cdot\frac{\sqrt[]{3}}{2}=\frac{a}{2}, \\ 5\cdot\sqrt[]{3}=\frac{a}{2}, \\ \frac{a}{2}=5\cdot\sqrt[]{3}. \end{gathered}[/tex]Step 2)
[tex]\begin{gathered} \frac{a}{2}=5\cdot\sqrt[]{3}, \\ a=2\cdot5\cdot\sqrt[]{3}, \\ a=10\cdot\sqrt[]{3}\text{.} \end{gathered}[/tex]Step 3)
For our triangle is equilateral, all of its sides have the same length (a). Then, the semi-perimeter of the triangle (s), which is the sum of all lengths divided by 2, is
[tex]s=\frac{a+a+a}{2}=\frac{3}{2}\cdot a\text{.}[/tex]Now, let's recall Heron's formula for the area (A) of a triangle:
[tex]A=\sqrt[]{s(s-a)(s-a)(s-a)}\text{.}[/tex]In our particular case, it becomes
[tex]A=\sqrt[]{\frac{3}{2}a(\frac{3}{2}\cdot a-a)^3}\text{.}[/tex]Simplifying it, we get
[tex]\begin{gathered} A=\sqrt[]{\frac{3}{2}a(\frac{3}{2}\cdot a-a)^3}, \\ A=\sqrt[]{\frac{3}{2}a(\frac{1}{2}a)^3}, \\ A=\sqrt[]{\frac{3}{2}a\cdot\frac{1}{8}a^3}, \\ A=\sqrt[]{\frac{3}{16}a^4}, \\ A=\frac{\sqrt[]{3}}{4}a^2. \end{gathered}[/tex]Replacing the value of a in the last expression, we get
[tex]\begin{gathered} A=\frac{\sqrt[]{3}}{4}(10\cdot\sqrt[]{3})^2, \\ A=\frac{\sqrt[]{3}}{4}\cdot100\cdot3, \\ A=25\cdot3\cdot\sqrt[]{3}, \\ A=75\cdot\sqrt[]{3}\text{.} \end{gathered}[/tex]AnswerThe area of the equilateral triangle inscribed in a circle of diameter 20 cm is
75√3 cm².
