Area of an inscribed triangle

TL;DR

Learn how to derive the expression for the area of a triangle inscribed in a circle using the formula 1/2 * base * height.

Transcript

okay this video I'm going to show you guys how to get an expression for the area of a triangle and I'll just cut a triangle to be ABC and that triangle is inscribed it in a circle with radius R okay so first of all I'll just show you guys a picture and in this video I'll just show you guys how to write that expression and the main goal is actually ... Read More

Key Insights

• ⚾ The area of a triangle can be calculated using the formula 1/2 * base * height.
• 🔺 The height of the triangle can be derived using the sine of the angle opposite the height.
• 😑 The expression for the area of an inscribed triangle can be obtained by using the law of cosines and sine.

Questions & Answers

Q: How is the area of a triangle related to its base and height?

The area of a triangle is equal to 1/2 times the base of the triangle multiplied by its height, which is the perpendicular distance from the base to the opposite vertex.

Q: How can the expression for the height of the triangle be derived?

The height of the triangle can be derived by using the sine of the angle opposite the height, which is equal to the height divided by the side opposite the angle. Rearranging the equation gives the expression for the height as C * sin(A).

Q: How can the expression for the area of the triangle be derived using the law of cosines?

By using the law of cosines, we can relate the base of the triangle to the radius of the circle and the angle opposite the base. This can be rearranged to find the expression for the base as the square root of 4R^2 * sin^2(B). Multiplying the base, height, and the sin(A) term gives the expression for the triangle's area.

Q: Why is the expression for the area of an inscribed triangle useful?

The expression for the area of an inscribed triangle allows us to calculate the maximum possible area for a given inscribed triangle, by maximizing the values of the angles A, B, and C.

Summary & Key Takeaways

• The video explains how to find the expression for the area of a triangle inscribed in a circle with radius R.

• The base of the triangle can be any of the three sides, and the height is the perpendicular line from the base to the opposite vertex.

• By using the law of cosines and sine, the expression for the area of the triangle can be written as 1/2 * B * C * sin(A), where B is the base, C is the side opposite angle A, and A is the angle.