Lesson Proof of the formula for the radius of the circumscribed circle

This Lesson (Proof of the formula for the radius of the circumscribed circle) was created by by ikleyn(52747)

: View Source, Show
About ikleyn:

Proof of the formula for the radius of the circumscribed circle

In this lesson you will learn the proof of the formula for the radius of the circumscribed circle about a triangle:

,

where

and

are the measures of the triangle sides,

is the area of the triangle and

is the radius of the circumscribed circle about the triangle.

Let

ABC be a triangle with the side measures

and

(Figure 1),
and let the point O be the center of the circumscribed circle.

As you know, the area of the triangle DELTA

ABC is equal to

(1)

where alpha

is the angle between the sides

and

(see the lesson
Formulas for area of a triangle under the current topic in this site).

Next, sin%28alpha%29

in accordance with the Law of sines Theorem (see
the lesson Law of sines and the lesson Law of sines - the Geometric Proof
in this site).

Figure 1a. To the formula for the
radius of the circumscribed circle

Figure 1b. To the proof of the formula
for the radius of the circumscribed circle

By substituting this expression for sin%28alpha%29

into the expression (1) for area you get the required formula

.

The proof is completed.

Example 1

Find the radius of the circumscribed circle about the triangle with the side measures of 4 cm, 13 cm and 15 cm.

Solution

The semiperimeter of the triangle is

.
The area of the triangle, according to the Heron's formula (see the lessons
- Proof of the Heron's formula for the area of a triangle and
- One more proof of the Heron's formula for the area of a triangle
in this site), is

sqrt%2816%2A%2816-4%29%2A%2816-13%29%2A%2816-15%29%29

.

Now, using the formula

proved above, you can calculate the radius of the circumscribed circle. It is

= 8

This lesson has been accessed 6148 times.