The 5 Best Automotive Math Formulas With Jeff Smith

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 21st 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.

We recently built this big-block Chevy street engine using a 4.500-inch bore and a 4.250-inch stroke. How big is it? If you got 540ci – you win a gold star.

If you look at the keypad on our well-used calculator, note the numbers in the upper left quadrant. Start with a decimal point then plugin 7-8-5-4. You’ll note this in a square clockwise pattern. You can use this as a way to help you remember the shortcut.

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.

All engine dynos use this basic equation to calculate horsepower. A dyno only measures torque and computes horsepower using this same equation. Automated dynos like this SuperFlow at Westech Performance do the math for you very quickly.

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

We use this formula to estimate power for street engines, and it is often very close but, there are some exceptions. We built this pump-gas, 10:1-compression 6.0L (364ci) that was finalized with a set of TFS heads and a COMP cam, and it made 500 lb-ft of torque. That equates to 1.37 lb-ft per cubic inch. The engine produced 557 hp at 6,700 RPM, which was shy of the estimated 572 hp.

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 = 1/8th 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.

This is our Chevelle at Irwindale running low 7’s at 98 MPH. Those times convert to high-10’s at 123 mph for the quarter-mile.

This is the book written by Dean Hill and Craig Hane. It was published in 1974 and is now-long out of print. The information goes beyond just converting 1/8-mile times to 1/4-mile. It also contains dozens of charts to estimate flywheel horsepower based on weight and trap speed.

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

Not sure of the ratio of that set of gears sitting on the shelf? Dividing the ring gear teeth by the pinion count will offer an immediate answer.

Many of the equations used in this story can be found in the John Lawlor Auto Math Handbook. As you can see, ours is well-used. It has recently updated material with input from our friend and former Chrysler engineer, Bill Hancock. It’s a great reference manual located on the quick-access side of our library.

About the author

Jeff Smith

Jeff Smith, a 35-year veteran of automotive journalism, comes to Power Automedia after serving as the senior technical editor at Car Craft magazine. An Iowa native, Smith served a variety of roles at Car Craft before moving to the senior editor role at Hot Rod and Chevy High Performance, and ultimately returning to Car Craft. An accomplished engine builder and technical expert, he will focus on the tech-heavy content that is the foundation of EngineLabs.
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