# Right triangles and height **URL:** \ **Category:** Factual Questions\ **Created:** [September 29, 2005, 1:25am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956 "2005-09-29T01:25:43Z")\ **Posts on this page:** 19\ **Page:** 1
**Author:** ![deadeyesdad](https://avatars.discourse-cdn.com/v4/letter/d/dec6dc/32.png) [@deadeyesdad](https://boards.straightdope.com/u/deadeyesdad)\ **Post date:** [September 29, 2005, 1:25am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/1 "2005-09-29T01:25:43Z")
We searched here and goggled anything we thought might give us the answer and came up empty. We need to find the height of an about a 150 foot tall tree. If we measure a known distance from the base we can shoot the angle to the top. So how do we use this info to get the height of the tree? 180-90-(x angle)=y angle. Can we figure out the length of the hypotenuse(sp) with 1 side and the angles and then use that to figure out the height of the tree. I seem to remember some formula for this ---
**Author:** ![Valgard](https://avatars.discourse-cdn.com/v4/letter/v/7feea3/32.png) [@Valgard](https://boards.straightdope.com/u/Valgard)\ **Post date:** [September 29, 2005, 1:35am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/2 "2005-09-29T01:35:04Z")
> [@deadeyesdad](#): > > We searched here and goggled anything we thought might give us the answer and came up empty. We need to find the height of an about a 150 foot tall tree. If we measure a known distance from the base we can shoot the angle to the top. So how do we use this info to get the height of the tree? 180-90-(x angle)=y angle. Can we figure out the length of the hypotenuse(sp) with 1 side and the angles and then use that to figure out the height of the tree. I seem to remember some formula for this Basic trig - the height will be d\*tan(x) where d = distance out from the base and x = angle to the top. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [September 29, 2005, 1:35am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/3 "2005-09-29T01:35:50Z")
Easy cheesy. Use the Cosine Rule. Divide the known distance you measure on the ground by the cosine of the angle to get the height. Make sure you set your calculator to use the units of angular measure you measured the angle in. ---
**Author:** ![LiveOnAPlane](https://avatars.discourse-cdn.com/v4/letter/l/3d9bf3/32.png) [@LiveOnAPlane](https://boards.straightdope.com/u/LiveOnAPlane)\ **Post date:** [September 29, 2005, 1:38am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/4 "2005-09-29T01:38:43Z")
Well, you could do that, and I leave it up to the more mathematically-inclined for the answer. I’d suggest thinking about ratios, though. Let’s say you put a 1-ft. stick in the ground and then measure its shadow. Say for the sake of argument that it is 9 inches. That is, .75 of the true length. Now, measure the shadow of the tree (if you can, of course). Say that it is 200 feet. You know that the shadow-length of your “control” is 75% of the true length; so .75 X 200= 150 feet, and that is the height of your tree. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [September 29, 2005, 1:39am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/5 "2005-09-29T01:39:06Z")
Or just hold up a meter stick perfectly straight, measure its shadow, measure the shadow of the tree, compute from there. ---
**Author:** ![tofergregg](https://avatars.discourse-cdn.com/v4/letter/t/bbe5ce/32.png) [@tofergregg](https://boards.straightdope.com/u/tofergregg)\ **Post date:** [September 29, 2005, 1:39am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/6 "2005-09-29T01:39:24Z")
> [@deadeyesdad](#): > > We searched here and goggled anything we thought might give us the answer and came up empty. We need to find the height of an about a 150 foot tall tree. If we measure a known distance from the base we can shoot the angle to the top. So how do we use this info to get the height of the tree? 180-90-(x angle)=y angle. Can we figure out the length of the hypotenuse(sp) with 1 side and the angles and then use that to figure out the height of the tree. I seem to remember some formula for this The tangent of the angle between the height and the base equals the height divided by the base: tan (angle) = height/base The height, therefore, is equal to the tangent of the angle times the base: height = tan (angle) \* base You don’t need the length of the hypotenuse, nor the other angle. If you want to do the calculation on a calculator and have the angle in degrees, make sure the calculator is in “degree mode” instead of radian mode. If you want to use the Google Calculator, you need to specify “degrees.” I.e., let’s say your base is 50ft, and the angle from the base to the top is 60º. In google, you would type: [tan (60 degrees) \* 50ft in feet](http://www.google.com/search?hl=en&lr=&c2coff=1&client=safari&rls=en-us&q=tan+%2860+degrees%29+*+50ft+in+feet&btnG=Search) Google would spit out the answer (86.6 feet). -Tofer ---
**Author:** ![SomeGuy](https://avatars.discourse-cdn.com/v4/letter/s/ac8455/32.png) [@SomeGuy](https://boards.straightdope.com/u/SomeGuy)\ **Post date:** [September 29, 2005, 1:39am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/7 "2005-09-29T01:39:54Z")
> [@Q.E.D.](#): > > Easy cheesy. Use the Cosine Rule. Divide the known distance you measure on the ground by the cosine of the angle to get the height. Make sure you set your calculator to use the units of angular measure you measured the angle in. That will get you the (non-vertical) distance from point on the ground away from the base to the top. Valgard is right about how to get the height. ---
**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\ **Post date:** [September 29, 2005, 1:40am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/8 "2005-09-29T01:40:14Z")
Of course, my post is just a shadow of his. ---
**Author:** ![Q.E.D](https://avatars.discourse-cdn.com/v4/letter/q/51bf81/32.png) [@Q.E.D](https://boards.straightdope.com/u/Q.E.D)\ **Post date:** [September 29, 2005, 1:42am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/9 "2005-09-29T01:42:01Z")
> [@Q.E.D.](#): > > Use the Cosine Rule. Divide the known distance you measure on the ground by the cosine of the angle to get the height. Ah, Hell’s bells. I’m calculating the length of the hypotenuse. You don’t want that, so do the TAN(x) thing the others gave you. Do still make sure you’re calculating in the right units, though. That’s bitten me in the ass a few times. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [September 29, 2005, 11:06am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/10 "2005-09-29T11:06:55Z")
…or… move away from the tree until the angle subtended by it is 45 degrees, then the height of the tree is equal to the distance you are from it. ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [September 29, 2005, 11:59am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/11 "2005-09-29T11:59:24Z")
Don’t forget that alll the suggestions so far assume perfecly flat ground between the base of the tree and the point you’re measuring the angles or distances from. If the ground’s not flat you’ll need some more math of the same sort to determine the altitude difference between the tree base and the base of your measurment point. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [September 29, 2005, 12:23pm UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/12 "2005-09-29T12:23:49Z")
As a corollary to that, mine also assumes eye-level = ground level. ---
**Author:** ![deadeyesdad](https://avatars.discourse-cdn.com/v4/letter/d/dec6dc/32.png) [@deadeyesdad](https://boards.straightdope.com/u/deadeyesdad)\ **Post date:** [September 29, 2005, 6:17pm UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/13 "2005-09-29T18:17:02Z")
Thanks to all of you. This afternoon we will get it done. ---
**Author:** ![MaryEFoo](https://avatars.discourse-cdn.com/v4/letter/m/ba8739/32.png) [@MaryEFoo](https://boards.straightdope.com/u/MaryEFoo)\ **Post date:** [September 29, 2005, 9:16pm UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/14 "2005-09-29T21:16:17Z")
Wait, if you use ratios and the ruler is 12" long, and the shadow is 9" long; and the tree shadow is 200’ long, then it’s 200 **divided by** 0.75 = 266.7 feet tall. ---
**Author:** ![AskNott](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/asknott/32/5790_2.png) [@AskNott](https://boards.straightdope.com/u/AskNott)\ **Post date:** [September 30, 2005, 12:39am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/15 "2005-09-30T00:39:07Z")
There’s a much simpler boy-scout answer. Stand back from the tree a ways. Hold a yardstick vertical at arm’s length, and see how many inches from the ground to the treetop. Then, with your thumb on that mark, turn the yardstick horizontal with your thumb aligned with the base of the tree. Have an assistant go to the spot indicated by the end of the stick, and measure, on the ground, the distance from there to the tree. ---
**Author:** ![LiveOnAPlane](https://avatars.discourse-cdn.com/v4/letter/l/3d9bf3/32.png) [@LiveOnAPlane](https://boards.straightdope.com/u/LiveOnAPlane)\ **Post date:** [September 30, 2005, 12:44am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/16 "2005-09-30T00:44:41Z")
> [@MaryEFoo](#): > > Wait, if you use ratios and the ruler is 12" long, and the shadow is 9" long; and the tree shadow is 200’ long, then it’s 200 **divided by** 0.75 = 266.7 feet tall. Huh? Then it is 200 **multiplied** by .75 = 150 feet. ---
**Author:** ![crowmanyclouds](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crowmanyclouds/32/19884_2.png) [@crowmanyclouds](https://boards.straightdope.com/u/crowmanyclouds)\ **Post date:** [October 1, 2005, 11:44am UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/17 "2005-10-01T11:44:24Z")
> [@Mangetout](#): > > …or… move away from the tree until the angle subtended by it is 45 degrees, then the height of the tree is equal to the distance you are from it. Saw Mr Wizard do this one,Mangetout’s almost halfway there. You place a container of water (large bowl works perfect) on the ground. while looking at the reflection of the tree that you’ll see on the waters surface (water gives you a LEVEL reflection, ya can’t use a mirror (unless it floats) slowly walk backwards. You measure the distance from where you can see the top of the tree (the reflection)to the bowl and the distance from the bowl to the base of the tree.The baseline from you to the bowl and the distance from your eyes to the ground (the height of your eyes) gives you two side of a triangle just calculate the angle. The angle and the reflection to tree baseline gives you the hight of the tree. ---
**Author:** ![Jman](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jman/32/4416_2.png) [@Jman](https://boards.straightdope.com/u/Jman)\ **Post date:** [October 1, 2005, 12:29pm UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/18 "2005-10-01T12:29:14Z")
> [@LiveOnAPlane](#): > > Huh? Then it is 200 **multiplied** by .75 = 150 feet. No, Mary’s right. If the ruler is longer than its shadow, how can the tree not be longer than its shadow? ---
**Author:** ![chappachula](https://avatars.discourse-cdn.com/v4/letter/c/d2c977/32.png) [@chappachula](https://boards.straightdope.com/u/chappachula)\ **Post date:** [October 1, 2005, 5:40pm UTC](https://boards.straightdope.com/t/right-triangles-and-height/323956/19 "2005-10-01T17:40:48Z")
> [@deadeyesdad](#): > > … we can shoot the angle to the top. just curious --how are you going to measure the angle?