triangle inscribed and circumscribed about the same circle radius of circle is 12. Find the area of the circle by taking the average of the triangles

Question

We are doing lower and upper approximations in geometry which is why you cannot use pi r^2. Please reply ASAP. Thanks.

Answer 1

To do this problem, are you using inscribed equilateral triangles and circumscribing equilateral triangles?

 

If so, I suggest that you make a decent drawing, separately, for each triangle. You can then break the problem down into a series of 30-60-90^o triangles, one of which will have a radius of the circle as its hypotenuse (this occurs for inscribed circle) or the short leg (this occurs for the circumscribed circle). From these, using the classic hypotenuse:short leg:longer leg ratio of 2:1:√3, you'll gradually find the sides of the inscribed & circumscribing triangles, etc.

 

Recall that the area of any 30-60-90 triangle is ¼s².

 

Do this & you'll have the two areas that you seek.

 

As in most geometry problems, the key is a geometric diagram, made large enough for that labeled measurements can be marked (and checked).