7 Construction Math Hacks: Estimating & More

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Math is power... especially in construction. You can dive as deep as you want into mathmatics
as a carpenter and many times, the more you know, the more you make. The math shortcuts in this video are not complicated. Anyone can do them and every carpenter needs to be able to.
Here's another fun math hack: • How to “Quarter in an ...
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Опубликовано:

26 янв 2024

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Комментарии : 226

@Kasey_Duke 3 месяца назад

I run a small carpentry business with 2 younger (early 20's) employees. I had one employee tell me he was watching one of your videos the other day. I immediately felt comfort knowing that he was learning from a man of great knowledge and experience. Somebody who has a heart of a teacher and wise enough to have a RU-vid channel to share that wealth of knowledge. Thank you sir for that. You are grasping the right audience that way.

@edmelisky774 3 месяца назад

He isn't teaching just the young!@ ones

@mattfinchum348 2 месяца назад

Where are you located kasey lol, need another 23 year old???

@ruidadgmailcanada8508 2 месяца назад

This guy and Stumpy Nubs are my go to for teaching. You lucked out with that employee. Self-motivation is hard to teach. 👍

@masonmansitomargiela1399 2 месяца назад

Here a student of this great teacher

@seansenter6530 Месяц назад

This makes me feel good. I’m 25 I wish my father was like this he does carpentry and knows a lot of every trade and has taught me a lot but this guy is what I aspire to be.

@jaylewis8789 3 месяца назад

I was good at math in school. On my first construction job we were laying out lines on the site for a lazy el shape that had piles of fill around it that didn't allow for clear measurements and straight lines. (this was in the ground for a foundation). My boss was getting frustrated trying to lay it out and I finally opened my mouth and tried to tell him how to use Pythagorean's theorem. After a couple more failures he finally let me do what I was saying and laid I it out in perfectly. This was 1971 so I had to do the calculations on a paper bag. I immediately became his lead man. Math is good.

@andrewsackville-west1609 3 месяца назад

That's ago, some guys were trying to build the architect's stream, the builder's nightmare, of curved walls. This one with a 70' radius, that one with 112' radius, etc. They spent hours with take measures and line pieces of stuff wire and couldn't make it work. Finally, when it came to the wall that was a 250' radius, they were ready to throw in the towel. I built them a spreadsheet that converted the radius into a table of offsets and they could lay it out in no time with a dry line and a flexible batten. Math

@neby_nebs 3 месяца назад

when young people like me say "where would this kind of math help" in school, NONE of my teachers have been as articulate as this. the hard part about "enjoying" math is seeing the fruit of all your memorization, but this REALLY helps!

@TonberryV 3 месяца назад

This, so many times, this exact thing is what teachers need to understand. In school I didn't get this from a math teacher, I got it from a chemistry teacher making a comment about how they used trig to find the slope of a roof (small structure) and then taking those results to find area for sheeting and subsequent insulation and the like. It was a lightbulb moment for a lot of kids in the class.

@KingBigBabyJesus 3 месяца назад

I just use a2+b2=c2

@nickmangia-forestry 3 месяца назад

I was no good at math in high school. Things didn’t start making sense till I went to college to learn forestry and a little bit of surveying. Every day I use pretty basic algebra and geometry and I wish teachers could have gotten me more into it earlier

@coyotech55 3 месяца назад

I took a calculus course aimed at technical people. Practical calculus, and it was actually fun. No proving theorems, no graphing of equations, just take this number and that number and do this with them, and there's your answer. I was a civil engineering / survey tech, and I still use it for my building projects and handyman work sometimes. I had calculated many things by the "iterative" process, which means calculating a thing over and over until you move in on it and get close enough to the number you're looking for: for instance the high or low point of a ditch or berm. You can do it quickly and easily with calculus. I never thought of calculus as something practical that would make calculations easier. Same thing with geometry and trigonometry. They're not mysteries to be afraid of, they're handy tools to use when you need to plan and build something.

@OptimisteProductions 3 месяца назад

@@coyotech55 same. I never understood the point of trigonometry for example. Then I became an electrician and I realised it was needed for the triangle of power

@JacobGreeno66 3 месяца назад

I'm a carpenter here in Northern California been watching your channel for years now so many tricks and tips I've learned I always tell the guys I learned this from uncle Scott. Thanks the continued gift of wisdom.

@JOHN-fx3xo 3 месяца назад

I can’t say it enough. You amaze me. I would love to be your shadow. You have so many life skills that I and the rest of the world would benefit from. I am not in the same trade as you. But the knowledge you share translates across life. Thank you for being you. Thank you for sharing. Please never stop recording.

@castorriverfarm5236 3 месяца назад

When I wrote my carpentry exam there were many questions where it asked you to measure quantities. The real skill as a carpenter is “what should I order”

@jeremiahjackson117 3 месяца назад

Never been to school for my trade, but in my opinion, no matter how good you are at estimating, if you can’t work the tools, you ain’t no carpenter.

@blakecoles2902 3 месяца назад

30% more

@ajwhittem3 3 месяца назад

3 4 5 rule. 43 years as a plumber/plumbing contractor. Invaluable. I am retired now but I volunteer at a vocational training school here in Medford and we teach this to all. Keep up the good work brother

@jackheinemann1994 3 месяца назад

The next step from that is memorizing the standard triangles

@willf9779 2 месяца назад

Bigger areas use the 6,8,10

@rehoboth_farm Месяц назад

1.4142...

@mathematicalmuscleman 2 месяца назад

Mathematics is most definitely Power. It is the fabric of this Universe. Having earned a Bachelors Degree and a Masters Degree in Applied Mathematics, I can truly state the sheer elegance and power of Mathematics.

@DITH85 3 месяца назад

I'm not even in construction. I'm a Mechanic but I find your videos absolutely fascinating and I've learned things I never knew I never knew and even put a few of them into action. And I have to say thats one of the most succinct descriptions of Pythagoras Theorum I've ever heard, brilliant.

@thomassilvis8764 3 месяца назад

I enjoy hanging around people smarter than me. Having a cup of coffee and your videos is truly a pleasure. Thank you.

@northkaskazini721 3 месяца назад

So do I and so am I 😂

@johnsrabe 3 месяца назад

(That makes you both smarter - or at least wiser - than most people!)

@igotskunkyfunky 3 месяца назад

You're not hanging around him though

@PhotonFlightTeam 3 месяца назад

beginning 2 AM, had a stressful morning. this..........calming, informative, worthy. Scott, gotta be lots and lots of builders who needed this and will use it. well done! ciao

@thewholls7176 3 месяца назад

PS - I forgot to mention as a DIY person. The amount of percentage waste I allow for depends on the material. Reason I say that is, because if you’re dealing with timber or aluminium, you can get away with 5% But if you’re dealing with something like floor, tiles or roof tiles You’re better off with 15% Because at the end of the day, when you have those extras left over, you now have spares that will 100% colour match the original And as you know, tiles are baked in batches with glaze, and if you need to replace a few tiles here and there a few years later, the colour will be different But if you were to allow 15% overage that covers the breakages on the job, the odd normally here and there, that gives you a good amount to stack in the corner of your garage as spares …….. And if you have a damaged floor tile that you cut out with an angle grinder and drop in one of the spares because the concrete slab has settled and cracked a few tiles, you will be glad you had 15% overage…… I would suggest for your viewers, it depends on the material

@andrewwebb5878 3 месяца назад

This is my favorite sort of video you produce. Explaining trade practices, or methods of how to use basic hand tools etc.

@Brian-Hansen 3 месяца назад

I’ve never built anything. In fact, I don’t even own a power drill, but for some reason, I really enjoy your videos. Keep up the great work!

@mark_luna 3 месяца назад

You oughta change that! Building stuff, ANYTHING, is very rewarding.

@williammrdeza9445 3 месяца назад

Great tips, Scott! This was very informative for those of us who occasionally build stuff.

@floridabeardedwoodworker 3 месяца назад

This might be the most important you tube channel ever

@James-ts5nb 3 месяца назад

You sir are always a wealth of knowledge. Keep up the excellent work

@thefireiron3575 3 месяца назад

I used to think the trades were for stupid people, but I've used more math as a builder than I ever expected to in my whole life. Arithmetic, fractions, and enough context to look up semi-complex formulas when you need them will give you a serious advantage in life and in the trades.

@CliffsideStables 3 месяца назад

WOW, you packed a lot of wisdom/knowledge in this short episode. “Keep on Keeping on!” Tim in northern TN

@Oo7Hola 3 месяца назад

This man is a Gem to us that are always learning. Thank you for your time sir.

@daiblaze1396 3 месяца назад

Not a contractor. At time I will skip some parts but it is always a pleasure to watch your advice. Thanks for taking time to share your passion.

@calvinclark3416 3 месяца назад

Story time: Math DID get me a raise and some job security too. Set up: I was putting up tar paper with a GC on the second story of a building with a nasty grade beneath. We needed a length of about 20 feet. Without thinking too much, I peeled off a length and started a "new roll," a circle with a one foot diameter--> rolled it 7 thick.... Here's the math: a circles circumference is pi if the diameter is 1-->7X3.14... is about 21, so about 20 (with a little bit to play around with). No multiple trips up the ladders or resets, no seams or waste, no wrestling around with a bulky roll and trying to manage it while positioning the piece and stapling... GC's face.. He asked me later if I would be happy with an X (about 20%) increase in pay (duh), and offered me one of the most genuine complements I've received in the trads or otherwise; said clearly wasn't another s*** head and he to keep me around. TLDR: Use math to impress you boss.....

@mazzg1966 3 месяца назад

Passing on wise knowledge....thank you for the video and for always trying to teach us!

@mopreppinmolivin 3 месяца назад

I am glad you mentioned the pythagorean theorem. I have worked with and around many carpenters through the years that knew the "3-4-5" rule but didnt actually know what it was or the equation for it.

@rpruneau68 3 месяца назад

Doesn't matter as long as they knew how to use it. Just like Plumber can determine the length of any 45° offset by measuring the distance between the two pipes and multiplying by 1.41. Where 1.41 is the square root of 2 which is the hypotenuse of a 1 X 1 unit triangle. Is this information important to understand while attempting to determine other TRUTHS or LAWS if you will, yes! But certainly isn't important for the Plumber that is just concerned with HOW to solve for the field issues quickly on the fly.

@danielahart3204 3 месяца назад

As a machinist always working with angles- a young man showed me a trig app. Wow! What a game changer. In 30 seconds I have an answer.

@esp1818 3 месяца назад

Care to share the name of the app?

@danielahart3204 3 месяца назад

@@esp1818 horitech trigonometry calculator

@claudew2899 3 месяца назад

I use Trig Solver. I worker my career as a Machinist and used up truck load of pencils calculating angles. The app saves a lot of time but takes away the skill.

@thomasgeorge815 3 месяца назад

Thank you for sharing these skills here for future generations. I appreciate all the hard work you do to put out these videos and I'm sure I'm not alone. Keep up the great work sir.

@tree_carcass_mangler 3 месяца назад

This is great stuff. Thanks for posting. Thumbs way up!

@rolandomadrigal8910 3 месяца назад

You’re the best, thank you for all of your wisdom

@eliinthewolverinestate6729 3 месяца назад

A good laser level helps. I hate green colored lasers. I red laser is still easier to see than green on red brick. Can't we get lasers in blue and orange some of the most contrasting colors. We order extra 2x4's so we can send junk wood back.

@ugofernandez8748 3 месяца назад

Muchas GRACIAS señor carpintero...👍🏼👌🏼

@Attemptedvelocity 2 месяца назад

I love these videos. Really a great source for information

@randywarnock2937 3 месяца назад

I started as a carpenter in 1976 building custom homes. Eventually I joined the Carpenter Union and started heavy construction. Eventually I started in WA State Department of Natural Resources in Engineering Division with the Carpenter Shop. We built fire lookouts and other projects statewide. Eventually I was appointed as Construction and Maintenance Superintendent. I started with DNR 1983 and retired in 2012. I still work in construction.

@briantaylor9266 3 месяца назад

Good useful tips. Hopefully the underlying message to young folks is "Yes, you will use arithmetic and math in your everyday life." I find that there is a disappointing number of people that see the simple math you describe as an insurmountable challenge.

@danielahart3204 3 месяца назад

I wish math teachers had presented these formulas in real life scenarios.

@awaterproofing 2 месяца назад

What a smart guy! True craftsmen!

@rjphoto66 3 месяца назад

Great concise and very helpful I hope to the young folks coming up

@psidvicious 3 месяца назад

Most people learn the 3-4-5 rule pretty quick in their constuction careers, but really grasping and understanding the formula for the reason it works (A²+B²=C²) is a real eye-opener for so many other applications.

@051biohaz 3 месяца назад

Another great one, thanks for all the knowledge..:)

@TheDAVE858 3 месяца назад

Thank u for the service you provide.

@maximuzc.6549 3 месяца назад

That was just great video! Thank you 🤯

@rustyshackleford5060 3 месяца назад

Keep up the good work

@OregonWingnut 3 месяца назад

Math memory tricks are useful in aviation too. You’ve given some great ones here. Your line about comfort with math leading to higher income is spot on…. I am proof.

@wayneswonderarium 3 месяца назад

The only difference between pros and newbies is the number of shortcuts we know - use the alligator to remember less than/greater than, i before e, etc.

@michaelgarrow3239 3 месяца назад

I went into piloting from construction.

@thomasvarney723 3 месяца назад

Very helpful and timely, thank you!

@dtemp132 3 месяца назад

I think EC should do a video on valley/hip rafters when there are two different pitches coming together. I had a devil of a time figuring out the math when I had to do a rafter between 12/12 pitch and a 4/12 pitch.

@scottgates4616 3 месяца назад

Great tips! The hip or valley angle of rise/17 can also be understood using A squared + B squared = C squared. The hip or valley runs the hypotenuse of the 2 adjoining roof runs. Since those 2 adjoining roof runs are based on 12" run, you can calculate the Pythagorean theorem of 12 x 12 which equals about 16.97, close enough to 17.

@codykanz1350 2 месяца назад

You're a hero. Thank you.

@dans4900 3 месяца назад

I figure 1 stud per foot and that gives you the trimmers and corners. That's even easier

@jchapman0920 3 месяца назад

Fantastic video! Thank you!

@onetruewater 3 месяца назад

Thank you for these crucial tips

@mitchbledsoe5562 3 месяца назад

Awesome thanks

@carlmcdaniel5681 2 месяца назад

Absolutely great knowledge

@74stevedc 3 месяца назад

I have a Construction Master app on my phone it figures every square including fractions, concrete for sauna tubes, walls footings floors roof rafters stairs and way more than you could ever imagine.

@krenwregget7667 3 месяца назад

Excellent video full of usefull information. Another good channel that doesn't really teach but opens the books on estimates for lumber, concrete, sheeting, etc is Ken's Carpentry which is a crew that mainly builds garages in Maine. He has lots of solid info about bulk materials purchasing and forecasting materials costs.

@mikewatson4644 3 месяца назад

My general rule on concrete is to order a little extra. And have a place to go with the extra.

@rpruneau68 3 месяца назад

That's the Law of Material for any large quantity of material

@psidvicious 3 месяца назад

Concrete estimating rule that never failed me: 1) Do the math exactly 2) Round up (no matter the remainder) 3) Add 1 more yard.

@waltermatthewberg 3 месяца назад

I use the 3-4-5 rule when I need a parallel line to start running metal panels on a roof. Sometimes the eave is all wavey even on new construction so I make a bunch of 6 inch marks on the bottom then hold a chalk line and snap it when it looks like a good average. Then I do the 3-4-5 marks off that line to chalk a perpendicular line that will be parallel with the first panel. I also measure for roof penetrations so hopefully none land in the overlap of the panel.

@theartistone5860 3 месяца назад

Math never lies. I was bad at math in high school but had some old guy teach me tricks (and I listened) when I was building the B2 and the F-22 for Boeing at 19 years old.

@Fnu-ex1tc 3 месяца назад

This is one useful video hack. I am not a math genius, so I'll have to refer back to this video consistently.

@thegreatestever4887 Месяц назад

In those two mathematical examples you showed with the squares saved me six months of studying and study fees. Blew my mind😲

@DaisyLewis-fr7tb 2 месяца назад

Good man to explain this thank you

@markmeidra7759 3 месяца назад

I love your channel. You’re like the Rick Steves of the construction industry 💪🏽💪🏽💪🏽

@dtx817 2 месяца назад

Dam good food for thought‼️ Ty🍻👍

@chrisfisher1100 3 месяца назад

THE 3 4 5, 3x3=9 4x4=16 9+16=25 the SQUARE ROOT of 25=5 PERFECTION!!!

@kendavis8046 3 месяца назад

Amateur "home pro" here on a couple of things that have turned out okay, mostly flooring, but a few odds and ends that turned into pieces of furniture and other things. But on flooring, with one exception, I have always added about 15% to the amount that I measure. Except for my first entry into putting a travertine floor in the guest bathroom. I measured the floor, and found a closeout on a stone that my wife and I likes, and they had exactly the amount I needed, per the drawings. Bottom line, I ran out by "just a tiny bit" behind the toilet, but I was able to fill in the void with an offcut and some construction adhesive, and I will never tell anyone to look at the spot I know is a fill. From that point on, I have always preferred to have a buffer for badly cut material or material that I don't want to put up with for whatever reason. Thanks as always for your channel, your content, and your whole positive attitude.

@mboykin6047 3 месяца назад

Nice lighting

@eM-ed5pz 3 месяца назад

Great information

@AlanTheBeast100 3 месяца назад

For us amateurs building, eg, a garden shed, the tool of choice is Excel with every part in the sketch accounted for in the spreadsheet. For my project there was very little waste material and no "missing" material. (Also made cost estimating very easy to do).

@clevernamehere7559 3 месяца назад

Made your own Lego set 😂

@AlanTheBeast100 3 месяца назад

@@clevernamehere7559 ?

@codykanz1350 2 месяца назад

Thanks!

@bc65925 3 месяца назад

Another thing about that 3,4,5 square... it has an area of 6. So it's a 3,4,5,6 triangle. The rules of trig are very handy also. Always remember So... Cah... Toa. SOH, CAH, TOA. Sin of the angle = opposite/hypotenuse and so on.

@paulhallgren7001 3 месяца назад

Very wise video!

@americanangler94559 3 месяца назад

Great episode

@CanadianMason85 3 месяца назад

For estimating, especially for precision I use ÷144 alot to get an acurate square footage. Measure to the inch, multiply and divide the answer by 144.

@tomcor2 3 месяца назад

you have probablly already gotten many comments on this but just in case- i've been a carpenter since 1964, and enjoy your utube shows, but an easier way to figure concrete as told to me many years ago by a cement finisher)= feet x feet x inches x .0031 fast and easy. thanks for all rhe info, keep up the good work tom c western PA

@willamettehops 3 месяца назад

Interesting, that seems a lot easier!

@leviharper4694 3 месяца назад

Awesome video

@claudew2899 3 месяца назад

Enjoyed your explanation of trigonometry and used it many times. BUT, now there's a App for that. I use Trig Solver. Just fill in the blanks and the solution is pop's up. Works great.

@CerebralAilment 2 дня назад

These are cool gotta sub for this

@hpo3409 3 месяца назад

Many years ago. I try messaging you about my unique work, with different running metal fabrics and dandy roll covers. I never heard back. Still here if you care to hear any of it.

@johnfisher9969 3 месяца назад

I'm a drywall pro, you can rough figure the wall and ceiling footage in an 8' ceiling house at floor space times 3.5, a 9' house is 3.75. That'll give a good enough idea for a footage quote. Common applications only of course.

@rjtumble 3 месяца назад

These would be great formula's to put on a single page on the Academy site for people to print out or include as a note on their phone.

@JohnComeOnMan 3 месяца назад

This guy gives off big "It's the right thing to do" Wilford Brimley vibes.

@castorriverfarm5236 3 месяца назад

I hear Wendell berry vibes myself

@reidflemingworldstoughestm1394 3 месяца назад

Would that be the same as the Wilford Brimley vibes from The Firm?

@JohnComeOnMan 3 месяца назад

@@reidflemingworldstoughestm1394 Wholesome contractor content by day, cold-blooded enforcer by night...

@scottaddison8071 3 месяца назад

I love your videos

@spudboyQ 2 месяца назад

The right triangle solver app is free and priceless.

@DennisSnider-vv1ob 2 месяца назад

I just square A hit add then square B and hit the square root button on me pocket calculator and there's my hypotenuse like you do as well another thing is handy to learn is converting tenths and hundredths into inches and fractions on the fly in your head

@joeb7508 3 месяца назад

I don’t have to remember cause I just saved this video! Thanks 😊

@taylorvonwinckelmann904 3 месяца назад

if you are lucky enough to live in oregon and run in to him he really is an absolute legend of a man and great person he spreads great knowlegde much respect

@toolsaddiction 3 месяца назад

I have a degree in computer science, and work with math every day. My hobby is building and reno, this was still very useful.

@dylanmounsey7515 2 месяца назад

Just the thumbnail alone got the gears in my head working

@punkys-tz7gw 5 дней назад

Goldmine!!!!!!!!!!!!!!!!!!!!!!!

@tristanconnolly5675 3 месяца назад

Nice.

@Jminenna 3 месяца назад

literally ran and got my notebook and pen and took notes from this video

@GijsbertNielsTerlaakPoot 3 месяца назад

Very usefull! Unfortunately you don’t have a lookalike in the metric world.

@tacklebox6239 3 месяца назад

Glad the old man is back. It was a bear listening to Nate on the last video.

@williamdemilia6223 3 месяца назад

Hey ... Scott is not an old man. no way!!

@andromedajacobson2520 3 месяца назад

A 3-4-5 triangle is called a pythagorean triple. 5-12-13 is another one. There are so many of them. The law of cosines applies to any triangle, regardless of the interior angles. It is a more complex form of the pythagorean theorem. It is, for any three sides of a triangle, a, b, and c, and their opposing angles, A, B, and C, then c^2 = a^2 + b^2 - 2ab*cos(C). When Cosine(C) equals 0, that is, when C=90 degree, then it becomes the pythagorean theorem. This assumes you can measure the angle C, between sides a and b. Any triangle will work for this formula if you have the angle included in two measured sides. There is also a law of sines that is really easy. Also, if you really want to go hardcore on math, don't use degrees. Use radians, look them up. For example, if I know the angle of an arc in radians, I just have to multiply the radius of the arc by the angle in radians to obtain the arc length. Radians simplify a lot of arc maths. Of course, lines that aren't straight and angles that aren't 90 degrees (or pi/2 radians) aren't common so it makes sense not to need to know this stuff.

@donwilliams3626 3 месяца назад

Very useful information that works for a one room addition, to a whole house lumber package. So I guess you don't plan to recommend an estimator app for our cell phone!

@mikalius344 3 месяца назад

Hip and valley cuts -- using Pitch divided 17 only works if both sides of the hip and valley are the same pitch. If they're different (we call them bastard hips/valleys out east here), then that divisor has to change. The 17 is derived from the fact you're implementing a 3rd dimension into your slope calculation. For a hip or valley with an 8/12 on each side, picture two right triangles with legs of 8 (vertical) and 12 (horizontal) at right angles to one another, with the vertical 8 leg in common. Looking top-down on this arrangement, you'll see that with the hip or valley (not the jack rafters) you'll be connecting the top of the common 8" leg with the perpendicular projections of the two ends of the 12" legs. Essentially trying to find the 3-dimensional diagonal of a box 12w x 12l x 8h. (12 * 12) *2)) = 16.971

@essentialjudge2279 3 месяца назад

Substitute 17 for 12 if the hip or valley is at 45°. Because the run is now the diagonal ( hypotenuse) of 12,12, which is 17 (16.96), but the rise remains the same.

@williamdemilia6223 3 месяца назад

yes. if the hip / valley rafter is 45 degrees. Be careful of the Bastard Hip! my question .. 2779 . How did you get/find the degree symbol?? thanks.

@RambozoClown 3 месяца назад

@@williamdemilia6223 Press and hold the ALT key and type 0 1 7 6 on the numeric keypad of your keyboard.

@rpruneau68 3 месяца назад

ASCII code from way back. Alt + 248 is degree symbol. 90° forms a right angle. @@williamdemilia6223

@PimpDaddyDisco 2 месяца назад

I need this so bad

@michaelgarrow3239 3 месяца назад

The hip/valley rafter is 12• square root of 2 for the run. To be perfectly accurate it works out to 16.97056 but 17 is fine. Keep up the good work.

@rpruneau68 3 месяца назад

Square Root of 2 can be approx @ 1.414 as this is considered a 1 x 1 unit triangle with 2 being the hypotenuse, so a common use number irrelevant of units or quantity to be multiplied by. For example, a plumber can determine the length of pipe required for a 45° offset by measuring the distance between the pipes and multiplying by SQRT of 2 approx 1.414.

@robstephens 3 месяца назад

Pythagorean's theorem is a special case of the Law of Cosines in which the angle between a and b is 90 deg. There are online calculators to calculate the angles and lengths without having to use a calculator.