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### Area of Triangles When Altitudes Are Given

```Date: 12/16/2002 at 02:39:51
From: Ridhi Kashyap
Subject: Area of Triangles when altitudes are given

Dear Dr. Math,

I was wondering how to find the area of a triangle when the lengths of
all three altitudes are given.

An example: Find the area of a triangle with altitudes of lengths 20
cm, 28 cm. and 35 cm.

Thank you,
Ridhi
```

```
Date: 12/16/2002 at 09:52:38
From: Doctor Floor
Subject: Re: Area of Triangles when altitudes are given

Hi, Ridhi,

Let us denote the area of the triangle ABC by D, the sidelengths by
a, b, c, and the corresponding altitudes by ha, hb, hc, respectively.
Then we know that

a*ha = b*hb = c*hc = 2D

and thus

a/2 = D/ha
b/2 = D/hb
c/2 = D/hc.

We can use this to rewrite Heron's formula

D = sqrt(s(s-a)(s-b)(s-c))

(where s = (a+b+c)/2) into

D = sqrt[(D/ha + D/hb + D/hc)*(-D/ha + D/hb + D/hc)*
(D/ha - D/hb + D/hc)*(D/ha + D/hb - D/hc) ]

= D^2*sqrt[(1/ha + 1/hb + 1/hc)*(-1/ha + 1/hb + 1/hc)*
(1/ha - 1/hb + 1/hc)*(1/ha + 1/hb - 1/hc) ]

so that we finally find the formula

1/D = sqrt[(1/ha + 1/hb + 1/hc)*(-1/ha + 1/hb + 1/hc)*
(1/ha - 1/hb + 1/hc)*(1/ha + 1/hb - 1/hc) ]

or

D = 1/sqrt[(1/ha + 1/hb + 1/hc)*(-1/ha + 1/hb + 1/hc)*
(1/ha - 1/hb + 1/hc)*(1/ha + 1/hb - 1/hc) ].

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Triangles and Other Polygons
High School Triangles and Other Polygons

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