Close this search box.

Nerd Out with Archery Math

There’s a lot to love about archery. It improves concentration, provides physical exercise and fosters new friendships, plus it’s a ton of fun. Math, on the other hand, might not be quite as exciting in practice, but you can’t have archery without it. Whether it’s scoring, calculating the weight of an arrow or figuring out its kinetic energy, math is built into archery. 

Breaking down the numbers gives important insight into the physics behind how everything works. If you’re on the fence about math but an archery enthusiast, use it to put purpose into math practice. For parents, adding some archery-related math figures makes working with numbers more fun for students. 

Here are a few archery problems to solve. 

Measuring Draw Length

This simple math formula is great for younger mathematicians. The draw length is the distance you pull back the string for comfortable and proper shooting form. Measuring your draw length is an easy but important formula. 

To measure draw length you’ll need two people and a tape measure. 

1. Have one person stand relaxed and raise their arms into a “T” position. 

2. The other person measures the wingspan from the tip of one middle finger, across the back, to the tip of the other middle finger. 

3. Take that number and divide it by 2.5. 

4. The draw length is the result.  

Calculating Arrow Weight (Without a Scale)

Arrow weight is one of the most important calculations in archery. An arrow’s weight determines the physics behind how it flies. Weight influences the arrow’s speed, trajectory and penetration into a target. It’s important to shoot the right weight for each bow setup. Shooting an arrow that’s too light, for example, can damage equipment and even harm the archer. 

To calculate an arrow’s weight without a scale: 

1. Measure the length of the arrow’s shaft (distance from the base of the nock groove to the shaft’s end).

2. Multiple it by the grains per inch (GPI) written on the side of the arrow; this information can also be found on the manufacturer’s website. 

3. The resulting number is approximately the arrow’s weight in grains, though you’ll also need to consider the weight of the point, insert, nock and fletch to determine the arrow’s true finished weight. 

The Front-of-Center Formula

Front-of-center is another important arrow calculation. The FOC formula indicates how the weight is distributed throughout the arrow. The FOC number describes the percentage of an arrow’s total weight in its front half. It’s good to know for accurate shooting, especially at long range. 

Here’s how to calculate your arrow’s FOC, according to Easton Archery:

  1. Divide the arrow’s length (distance from the base of the nock groove to the shaft’s end) by 2.
  2. Find the balance point. That’s where the arrow balances perfectly. Mark the point, and measure from there to the nock’s throat.
  3. Subtract the center of the arrow measurement (calculated in Step 1) from the balance point (calculated in Step 2).
  4. Multiply Step 3’s answer by 100.
  5. Divide the answer from Step 4 by the arrow’s overall length. That is the arrow’s FOC percentage.

Understanding Momentum

An arrow’s momentum is important to know. The more momentum an arrow has, the greater the force is required to stop it. That translates into the arrow’s penetration into the target. Momentum is equal to mass times velocity. 

To calculate momentum:  

Mass of arrow (grains) x arrow speed (fps) / 225,400 (gravitational constant) = momentum. 

If you don’t know the arrow’s speed you can visit an archery shop and shoot arrows through a chronograph. 

Determining Kinetic Energy

Kinetic energy is the amount of energy possessed while the arrow is in motion. This happens when the string is released, and the potential energy is converted into kinetic energy with the movement of the arrow. When the arrow hits its target, the energy is transferred into the target. 

To calculate kinetic energy: 

Mass of arrows (grains) x arrow speed (fps)2 / 450,240 = kinetic energy 

Archery Scoring

The Tokyo Olympics start July 23, 2021, with the archery competition taking place July 23 to 31. The Olympics are a fun opportunity to incorporate a little math practice by keeping track of scores. 

Olympic archery follows the rules of target archery. Targets have 10 concentric scoring rings, separated into five colors: yellow, red, blue, black and white. The yellow rings score 10 and nine points, the red rings score eight and seven points, the blue rings score five and six points, the black rings score four and three points and the white rings score two points and one point. Missing the target means zero points. 

The tournament’s qualification phase consists of 72 arrows shot for a total score and seeds the archers for matches. Archers shoot head-to-head in brackets with the winner of each advancing and the loser being eliminated, until there’s a champion. 

Learning to score is a great skill to acquire because it helps develop archery skills and makes spectating much more fun. For more practice scoring, here are some other indoor tournament formats

Archery is one of the most versatile pastimes. Whether it’s for exercise, camaraderie or spicing up your education through DIY calculations, archery has something for everyone. Archery consistently proves it has something for everyone, even people who love math. 



If you liked this one, read these next




Learn the basics here, from the different styles of archery to how to choose the bow that’s right for you.


Stay Up to date on everything archery with our newsletter


Locate archery stores and ranges in your neck of the woods.