If a triangle is inscribed in a circle of radius 10 cm, what is the maximum area

If a triangle is inscribed in a circle of radius 10 cm, what is the maximum area of the triangle?

If a triangle is inscribed in a circle of radius 10 cm, what is the maximum area of the triangle?

Step 1: Understand that a triangle can be inscribed in a circle, meaning all its vertices touch the circle.
Step 2: Know that the radius of the circle is given as 10 cm.
Step 3: Recall the formula for the area of a triangle inscribed in a circle: Area = (1/2) * r² * sin(θ), where r is the radius and θ is the angle opposite the base.
Step 4: Since we want to find the maximum area, we need to maximize sin(θ). The maximum value of sin(θ) is 1, which occurs when θ is 90 degrees.
Step 5: Substitute the radius into the formula: Area = (1/2) * (10 cm)² * 1.
Step 6: Calculate (10 cm)², which is 100 cm².
Step 7: Multiply by (1/2): (1/2) * 100 cm² = 50 cm².
Step 8: Conclude that the maximum area of the triangle inscribed in the circle is 50 cm².

Inscribed Triangle Area – The area of a triangle inscribed in a circle can be maximized when the triangle is equilateral, utilizing the formula for area based on the radius and sine of the angle.
Circle Geometry – Understanding the relationship between the radius of the circle and the properties of triangles inscribed within it.
Trigonometric Functions – Using the sine function to determine the maximum area based on the angles of the triangle.