# Chapter 4. Wasan as a Practical Science

## Column Heron’s Formula (Level 1)

The area of a triangle can be calculated when the lengths of its three sides are found. Assuming that the area of a triangle is S and let where a, b, and c are the lengths of the three sides of the triangle, then it follows that . The formula can be proved as follows:

Looking at the figure, we know that 、h2=c2-x2=b2-(a-x)2 holds true, and therefore we obtain

from c2-x2=b2-a2+2ax-x2.

Then, using

it follows that

4a2h2=4a2c2-(c2+a2-b2)2
=(2ac+c2+a2-b2)(2ac-c2-a2+b2)={(c+a)2-b2}{b2-(c-a)2}
=(c+a+b)(c+a-b)(b+c-a)(b-c+a)

Here, letting a+b+c=2s, we obtain 4a2h2=2s(2s-2b)(2s-2a)(2s-2c)=16s(s-a)(s-b)(s-c).
Therefore, is derived from a2h2=4s(s-a)(s-b)(s-c).

On the other hand, as , we obtain
.

Using this formula, we can easily calculate the area of any triangle with the lengths of the sides of which are all represented with natural numbers. As described in Column "Pythagorean theorem", when the area of this triangle is also a natural number, it is called a Heronian triangle. Triangles whose three sides are (4,13,15), (3,25,26), (9,10,17), (7,15,20), (6,25,29), (11,13,20), (5,29,30), (13,14,15) and so on, are Heronian triangles; please confirm this using Heron’s formula.

The radius of the inscribed circle of a triangle can be calculated when the lengths of its three sides are found. From the figure, it follows that △ANO≡△AMO, △CMO≡△CLO and △BLO≡△BNO.So, we obtain

;

as a+b+c=2(l+m+n) , we obtain

.

Therefore, we obtain

by Heron’s formula.
Finally, let us present the formula for calculating the inscribed quadrangle of a circle.

As shown by the figure, assuming the lengths of the four sides are a, b, c, d, and letting , the area of the quadrangle ABCD is . We would like you to try to prove the formula. For your reference:
If C and D overlap, the quadrangle becomes a triangle with d=0 and the formula changes to Heron’s formula itself.

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