What Is the Formula for Triangle Circumradius?

TL;DR

The formula for the circumradius of a triangle is the product of its sides divided by four times its area. This relationship shows that the circumradius connects the triangle's geometry to its circum-circle, indicating how far the triangle reaches from its center to the vertices.

Transcript

What i want to do in this video is to come up with a relationship between the area of a triangle and the triangle's circumscribed circle or circum-circle. So before we think about the circum-circle let's just think about the area of the triangle. So let's say that the triangle looks something like this. Actually I don't want to make it look isoscel... Read More

Key Insights

• 😃 The area of a triangle can be calculated using the formula A = 1/2 * b * h, where b is the base and h is the height.
• ⭕ A circum-circle is a circle that passes through all of the vertices of a triangle.
• 🔺 An inscribed angle subtended by the same arc in a circle is congruent in two different triangles.
• 🙃 Similar triangles have proportional corresponding sides.
• 🥳 The ratio of a side to the diameter of a circum-circle is equal to the ratio of the height to the hypotenuse in a right triangle.

Questions & Answers

Q: What is the formula for calculating the area of a triangle?

The area of a triangle can be calculated as half the base multiplied by the height, using the formula A = 1/2 * b * h.

Q: What is a circum-circle?

A circum-circle is a circle that passes through all of the vertices of a triangle.

Q: How can you identify a right triangle in an inscribed circle?

If one of the sides of the triangle is a diameter of the circle, then the triangle is a right triangle with the angle opposite the diameter measuring 90 degrees.

Q: How can the area of a triangle be related to the radius of its circum-circle?

The radius of the circum-circle is equal to the product of the side of the triangle divided by four times the area of the triangle.

Summary & Key Takeaways

• The video explores the relationship between the area of a triangle and its circum-circle.

• By dropping an altitude, the area of the triangle can be calculated as half the base times the height.

• By constructing a triangle with a diameter through the circum-circle, it is shown that the ratio of the side to the diameter is equal to the ratio of the height to the hypotenuse.