# Area of Inscribed Triangle

The area of an inscribed triangle, a concept in geometry, holds intriguing mathematical significance. When a triangle is inscribed within a circle, its vertices touching the circle's circumference, a unique relationship emerges between the triangle's sides and angles. This geometric arrangement gives rise to a concise formula for calculating the area of such a triangle based on the radius of the circumscribing circle. Exploring the properties of inscribed triangles not only deepens our understanding of geometric principles but also unveils elegant connections between seemingly disparate elements in mathematics.

Questions

- A triangle has corners at #(3 ,3 )#, #(2 ,9 )#, and #(8 ,4 )#. What is the area of the triangle's circumscribed circle?
- A triangle has corners at #(5 ,7 )#, #(2 ,1 )#, and #(3 ,6 )#. What is the area of the triangle's circumscribed circle?
- if a circle has a radius of 6.2 feet .what is the area of the circle in square feet rounded to the nearest tenth foot?
- A triangle has corners at #(2 , 4 )#, #(8 ,2 )#, and #(1 ,3 )#. What is the radius of the triangle's inscribed circle?
- A triangle has corners at #(6 ,4 )#, #(7 ,6 )#, and #(3 ,8 )#. What is the area of the triangle's circumscribed circle?
- A triangle has corners at #(5 ,8 )#, #(2 ,6 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/2 #, vertex B has an angle of #( pi)/4 #, and the triangle's area is #25 #. What is the area of the triangle's incircle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(5pi)/12 #, and the triangle's area is #28 #. What is the area of the triangle's incircle?
- A triangle has corners at #(6 ,4 )#, #(6 ,5 )#, and #(3 ,3 )#. What is the area of the triangle's circumscribed circle?
- A triangle has corners at #(9 ,7 )#, #(2 ,1 )#, and #(5 ,2 )#. What is the area of the triangle's circumscribed circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(pi)/2 #, and the triangle's area is #2 #. What is the area of the triangle's incircle?
- A triangle has corners at #(6 , 3 )#, #(3 ,5 )#, and #(2 ,9 )#. What is the radius of the triangle's inscribed circle?
- A triangle has corners at #(2 , 1 )#, #(3 ,4 )#, and #(1 ,2 )#. What is the radius of the triangle's inscribed circle?
- What is the area of a circle which diameter is 8.5m?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/6 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?
- A triangle has corners at #(5 , 2 )#, #(4 ,7 )#, and #(5 ,6 )#. What is the radius of the triangle's inscribed circle?
- A clock tower has a circular clock face with an area of #144pi# square feet. What is the diameter of the clock face?
- A triangle has corners at #(7 ,5 )#, #(2 ,3 )#, and #(1 ,4 )#. What is the area of the triangle's circumscribed circle?
- Suppose a circle of radius r is inscribed in a hexagon. What is the *perimeter* of the hexagon? Thanks!
- What is the area, in square inches, of a regular octagon with 9-inch sides if the distance from the center to any vertex is 10 inches?