Wait – wait – wait! Don’t just skip this story because the word “Math” appears in the title. You may regret not learning some cool ways to use simple middle-school arithmetic and, okay, some high school geometry to learn something new. Trust us, it will be easy stuff.
Our formulas are easy. Sure, there are dozens of automotive equations that are useful, but they’re also complex and difficult to execute. The most obvious example is the formula for compression ratio. We’re old enough to remember sharpening our pencils, finding an eraser and large legal pad, and spending an hour or two crunching out multiple variations on piston dome, chamber volume, deck height, and head gasket combinations to find an ideal compression ratio. Thankfully, in the 21^{st} Century, there’s no reason to waste time doing that. Several free online compression-ratio programs will crunch the numbers for you.
The long-hand version of the compression ratio equation is cumbersome. That’s why it’s not in our Top 5, but it doesn’t mean you shouldn’t know how it works. Having tossed out that caveat, we’ll make it easy for you. Performance Trends offers access to a convenient compression ratio calculator that you can download for free. Just click on the Performancetrends.com website, find the “downloads” button, and look for the compression ratio calculator.
Short of that, we’d like to think these short-but-sweet equations are worth saving for future reference. We use at least one of these almost every day for technical articles. Some gearheads recoil at the thought of even simple math, but this stuff is elementary and we promise no Ibuprofen will be required to run these numbers.
1. How to Calculate Engine Displacement
How big is it? That’s the classic car guy question, and we can show you how to quickly calculate this based on three simple inputs: bore, stroke, and the number of cylinders. This can be useful, for example, when considering adding a stroker crank.
Way back in high-school geometry class (assuming you were paying attention and not ogling the cute girl in the second row), the volume of a cylinder is calculated using the formula of an area of a circle (bore) times the length of the cylinder (stroke). As our example, we’ll use the ubiquitous 350ci small-block Chevy that is fitted with a 4.00-inch bore and a 3.48-inch stroke (displacement = bore radius x bore radius x π x stroke).
To do this, you must first calculate the area of the bore by finding the radius of the bore. Half of the diameter is the radius, so in this case, it is 2 inches. Next, multiply that by itself (2 x 2), then multiply that times π (3.14159).
Area = Radius x Radius x π (or Radius Squared x π)
2 x 2 x 3.1416 = 12.5664
Now, you are ready to calculate the displacement:
Displacement = Area x Stroke
For our example 350 Chevy engine:
12.5664 x 3.48 = 43.73 cubic inches
All that’s left to do is multiply that volume times the number of cylinders.
43.73 x 8 cylinders = 349.84 cubic inches
There is also a handy shortcut:
Displacement = Bore x Bore x Stroke x 0.7854 x Number of Cylinders
4 x 4 x 3.48 x 0.7854 = 349.8 (rounded to 350)
There’s also an easy way to remember 0.7854 (a simplified constant to convert the bore squared into Pi-R squared.) If you study the four numbers in the upper left-hand quadrant of any hand calculator, you will see these numbers in clockwise sequence. Once you’ve used it a few times, it becomes second nature, and calculating displacement just got really simple.
2. The Horsepower Equation
Every gearhead and car guy should know this formula by heart. It’s simple, and there’s more than 100 years of internal-combustion effort and a Scotsman’s work on steam engines for this formula.
We won’t get into the entire history, but suffice to say, Scottish-inventor James Watt came up with this formula in the late-1700s to relate the power of his new steam engine to draft horses. The term horsepower was born. We’ll save you the details of how the 5,252 denominator was created. If you really want to know, Google can fill in the details.
Horsepower = (Torque x RPM) / 5,252
Here’s the inside information on making power. All internal engines make torque, defined as the twisting motion of the crankshaft. If you can make the same torque in less time (measured in revolutions per minute – RPM), then your engine will do more work and make more horsepower.
Let’s look at two examples:
Example A
A 454ci big-block Chevy makes 425 lb-ft of torque at 5,500 rpm. If we calculate the horsepower, it looks like this:
HP = (425 x 5,500) / 5,252
HP = 2,337,500 / 5,252
HP = 445
Example B
A much smaller 302ci small-block Chevy makes 333 lb-ft of torque but at a much higher 7,000 rpm.
HP = (333 x 7,000) / 5,252
HP = 2,331,000 / 5,252
HP = 443.8 rounded to 444
These two engines make close to the same horsepower, even though they are radically different. The key is engine speed. Of course, the big-block will make a ton-more torque than the little 302. But, you can see that if the engine is durable enough to live at a higher engine speed, it is a great way to make more power. This is no secret – engine builders have known this from the beginning of the internal combustion engine.
There are drawbacks to this high-RPM equation. Engines with long-duration camshafts don’t like to run at low engine speeds, while large displacement engines can rely on size to make monster torque and often offer much greater reliability.
3. Estimate HP
This is our favorite formula, and the one we use most often. If you think it might be fun to be able to amaze your friends with the ability to predict – with surprising accuracy – how much power a normally-aspirated street engine makes, then you want to commit this simple formula to memory. But first, we need to lay the foundation on how this works.
The formula is based on two estimates: one for torque per cubic inch, and the other for the peak horsepower RPM point. The first estimate is relatively simple. For street engines on pump gas with a good cylinder head, intake, and exhaust systems, our buddy Steve Brule’ at Westech Performance likes to use 1.25 lb-ft of torque per cubic inch. So let’s say we have a 383ci small-block with good AFR heads, 10:1 compression, a decent cam, headers, and an Edelbrock Performer RPM dual plane intake.
Displacement x 1.25 = Peak TQ
So let’s use this first part of the formula on our theoretical 383ci:
383 x 1.25 = 478.7 lb-ft of torque
Over years of looking at engine power curves, Brule’ has noticed that street engines generally lose 10 percent of their torque at peak horsepower. This new number is the torque the engine will make at the peak power RPM point.
Peak TQ x 0.90 = torque at peak HP
So now let’s plug our torque number in:
478.8 x 0.90 = 430.8 lb-ft – let’s call this number Torque 2 or TQ2
Now, we employ the full horsepower equation just learned in the previous example. But, this is where the second estimate number comes into play. We must estimate the RPM point where the engine will make its peak horsepower number. Camshaft timing generally has the greatest effect on this number with a longer duration cam pushing this peak RPM point higher.
If we happen to know the exact peak torque RPM, then we can roughly add 1,500 to 1,800 RPM to the peak torque rpm to estimate the peak horsepower RPM. For example, if peak torque occurs at 4,000, then we can expect the peak horsepower RPM point at somewhere between 5,500 and 5,800 rpm.
For the purposes of this example, we’ll choose 5,700 rpm as our horsepower peak number. So now, we can just plug our TQ2 number into the horsepower equation:
HP = (Torque x RPM) / 5,252
HP = (430.8 x 5,700) / 5,252
HP = 467.5 – we’ll round that off to 467.
Just for fun, we dug up the dyno specs on a mild-383, pump-gas small-block from a previous story and charted the engine’s power curve. Our estimate of torque is almost perfect, but we’ll admit that we worked backward from this test to choose the peak RPM point. Given that, you can see the horsepower estimate is very close with 467, while the engine actually produced 453. Our assessment was high by 14 horsepower, which is only a 3-percent error factor.
This formula will work with any normally aspirated internal-combustion engine, but as you can see, it requires accurate estimates for both peak torque and the peak horsepower RPM point. Race engines with higher compression will make upwards of 1.55 to 1.6 lb-ft of torque per cubic inch. An NHRA Pro Stock engine will be even higher.
This formula can be used to estimate a race engine’s potential, regardless of the engine speed. Again, the key is carefully estimating torque per cubic-inch and peak engine speed.
As an example, with 427 cubic inches and 1.50 lb-ft per cubic inch, that equals 640 lb-ft of torque. According to Ben Strader at EFI University, race engines tend to lose 12-percent torque at peak horsepower, as opposed to 10 percent, so we’ll use his factor. Calculating 640 x .88 = 563 lb-ft, and if the peak horsepower occurs at 11,000 rpm, it equals to 1,179 hp. That’s pretty stout.
Now, you can use this formula to amaze your friends with your engine acumen and horsepower expertise!
383 Small-Block Chevy Power Curve
RPM |
TQ1 |
HP1 |
3100 |
449 |
265 |
3200 |
456 |
278 |
3300 |
460 |
289 |
3400 |
464 |
300 |
3500 |
466 |
310 |
3600 |
468 |
321 |
3700 |
471 |
332 |
3800 |
473 |
343 |
3900 |
475 |
353 |
4000 |
475 |
362 |
4100 |
474 |
370 |
4200 |
469 |
375 |
4300 |
465 |
381 |
4400 |
463 |
388 |
4500 |
462 |
396 |
4600 |
461 |
404 |
4700 |
460 |
412 |
4800 |
459 |
419 |
4900 |
457 |
426 |
5000 |
455 |
433 |
5100 |
452 |
439 |
5200 |
448 |
443 |
5300 |
443 |
447 |
5400 |
437 |
449 |
5500 |
431 |
451 |
5600 |
424 |
452 |
5700 |
417 |
453 |
5800 |
409 |
451 |
5900 |
399 |
448 |
6000 |
389 |
445 |
4. How to Convert 1/8-mile E.T. and Speed to ¼-Mile Numbers
There’s much more emphasis on 1/8-mile drag racing now that cars are running so quickly, and it is often difficult to relate 1/8-mile (660 feet) times to 1,320 numbers. Many moons ago, a good friend Dr. Dean Hill along with his friend Dr. D. Craig Hane, published a reference book called the Pocket Dyno. This was published long before the days of home computers, and the book is full of conversion tables to convert 1/8-mile to 1/4-mile elapsed times (e.t.) and other useful tables and charts.
We used Dr. Hill’s information and converted his numbers to a simple conversion equation.
1/4-mile E.T.= 1/8-mile E.T. x 1.54
As an example, our Chevelle recently ran a 7.051 in the 1/8-mile.
7.051 x 1.54 = 10.85 e.t. in 1/4 mile
We also have a simple conversion for M PH. This is a bit more generalized but seems to hold up for MPH estimations.
¼-mile MPH = 1/8^{th} mile MPH x 1.25
In our case, we ran a 98.98 mph trap speed in the 1/8-mile.
1/4-mile MPH = 98.98 MPH x 1.25
1/4-mile MPH = 123.72 MPH.
There are numerous reasons why MPH may not always be accurate. Several variables come into play starting with examples where, at any speed above 90-100 MPH, the aerodynamic effects of older ’60s cars will lower the quarter-mile speeds compared to a slippery third-generation Camaro, for instance. But this formula still works as an estimate for quarter-mile times and speeds.
5. Gear Ratio Calculations and Effects of Tire Diameter
Let’s start this with a simple gear computation. To determine gear ratio, simply divide the ring gear teeth by the number of teeth on the pinion. If we have 40 teeth on the ring gear and 10 teeth on the pinion: 41 / 10 = 4.10:1 ratio. This also works for any gear or even a blower drive. Merely divide the driven pulley tooth count by the drive tooth. This also works for belt pulleys – just use diameters instead of a tooth count.
When we look into the effect of tire size on the gear ratio, it gets a little more complicated. Let’s start by finding engine RPM from a given vehicle speed. We’ll need to know the rear gear ratio, the rear tire diameter, and the vehicle speed. Let’s use a Camaro with a Muncie four-speed and 4.10 gears running a 28-inch tall tire at 70 mph. This equation assumes no slippage as with a manual transmission. With a torque converter not in lockup, there will be some slippage in the converter.
Engine RPM = (MPH x Gear Ratio x 336) / Tire Diameter
Engine RPM = (70 x 4.10 x 336) / 28
Engine RPM = (96,432) / 28
Engine RPM = 3,444
Now let’s use that same Camaro, but we’ll plug in a TKO five-speed with a 0.64:1-overdrive ratio. This requires the same formula but we have to first determine something called Final Drive Ratio.
With the Muncie four-speed, Fourth gear is 1:1 with the output shaft turning the same as the input. But with an overdrive, this uses a gear inside the transmission to increase the speed of the output shaft compared to the input. The formula for Final Drive is simple:
Final Drive Ratio = Rear Gear Ratio x Overdrive Ratio
Final Drive Ratio = 4.10:1 x 0.64:1
Final Drive Ratio = 2.62:1
Now, our numbers for cruising engine speed at 70 mph will be drastically reduced with the overdrive ratio:
Engine RPM = (70 x 2.62 x 336) / 28
Engine RPM = 61622 / 28
Engine RPM = 2,200
There are three variations on the RPM equation that solve for the other variables. To keep this story brief, we won’t offer examples for each since they all execute similarly.
MPH = (RPM x Tire Diameter) / (Gear Ratio x 336)
Gear Ratio = (RPM x Tire Diameter) / (MPH x 336)
Tire Diameter = (MPH x Gear Ratio x 336) / RPM
One variable that is fun to look at is solving for the effect of tire diameter on gear ratio. So, using our same Camaro, let’s say we want to know the Effective Gear Ratio if we changed from a 28-inch tall tire to a shorter 26-inch-tall rear tire.
Effective Gear Ratio = (Old Tire Diameter / New Tire Diameter) x Gear Ratio
Effective Gear Ratio = (28 / 26) x 4.10
Effective Gear Ratio = 1.0769 x 4.10
Effective Gear Ratio = 4.41:1
Conversely, if the Camaro initially ran a 26-inch-tall rear tire and we wanted to add a 28-inch-tall tire, logic tells us that the taller tire will reduce the effective gear ratio.
Effective Gear Ratio = (26 / 28) x Gear Ratio
Effective Gear Ratio = 0.9285 x 4.10
Effective Gear Ratio = 3.80:1