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### Distance to an Object

```
Date: 04/07/2000 at 01:50:17
From: Brent Pattison
Subject: Easy way to calculate distance using trigonometry

I'm a middle school teacher who wants to give his students some
hands-on application work in geometry. One idea I had was to get
students to measure the distance from one object to another in our
schoolyard using geometry. I remember that if one knows the
measurement of a base line and both angles leading towards the object,
the distance can be calculated. Is there an easy way I could show my
students to do this, without having to teach them all about
trigonometry? I think they'd get a kick out of watching math work this
way!

Thanks.
```

```
Date: 04/07/2000 at 13:14:16
From: Doctor Rob
Subject: Re: Easy way to calculate distance using trigonometry

Thanks for writing to Ask Dr. Math, Brent.

You can use similar triangles to do this, instead of trigonometry. Use
the baseline and the two angles:

C
o._
/.  `-._a
b/ .d     `-._
/  .          `-._
o---+--------------`o
A    X    c           B

Let AB be the baseline with length c. Let the object be C, and let
the two angles be A = <BAC and B = <ABC. You want to find the lengths
a and b.

Construct a small triangle with the same shape as ABC, that is, a
similar triangle. You can do this because you know angles A and B (and
thus C = 180 - A - B degrees). Measure its sides a', b', and c', and
use the known value of c to find a = a'*(c/c') and b = b'*(c/c').

If you want to find d, the altitude of the triangle, you can do that
by using Hero's or Heron's Formula to get the area K of triangle ABC,

K = sqrt(s*[s-a]*[s-b]*[s-c]),

where  s = (a+b+c)/2, and then d = 2*K/c.

Not only does this use similar triangles, it uses A+B+C = 180 degrees,
Hero's formula, and K = base*height/2. You can also talk about the
accuracy of the result and significant figures, based on the accuracy
of the angle and length measurements.

This kind of mathematics goes back to the Egyptians, who invented
surveying to reestablish the boundaries of fields flooded during the
annual floods of the Nile. You can even bring in the history of
mathematics in this way.

Sounds like fun to me!

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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