The sector, also known as the compass of proportion, is a fun and easy-to-use calculating device that uses the power of geometry to perform a variety of useful math jobs. Although at first glance a sector might seem as quaint and old-school as a slide rule, it can do at least a few maker-style jobs better, faster, and easier than any modern electronic calculator. Once you get the hang of using a sector, you’ll find it a valuable tool in woodworking, drafting, surveying, and even doing mathematical calculations on the fly.

So, what is a sector? The one described here is a pair of rectangular legs, each about 10 inches long, connected by a rotating joint (Figure A).
The numerical scales are inscribed on the legs. My sector is made of plastic, but originally sectors were made from wood, brass, or ivory.

Right up to the late 19th century, using a sector was the way to lay out woodworking or other fabrication projects. It enabled Renaissance scientists to make complex computations and it showed early cannoneers how to aim their guns. So, when you make and use a sector you are figuratively walking in the footsteps of millions of makers of the past.

Usually, the credit for its invention goes to the great 15th-century Italian scientist Galileo Galilei, although some historians of science give invention credit to Englishman Thomas Hood, a Galileo contemporary. In any event, it was certainly Galileo who made the sector popular. Almost immediately after it was invented, Galileo realized that this new gizmo was an extremely useful instrument, and he started teaching others how it worked. Galileo made quite a bit of money by writing and selling a book that explained how to use a sector as an arithmetical tool. For the next 500 years, the sector was one of the most commonly used mathematical instruments in Western civilization.

Its secret lay in the rules of geometry, specifically, in the simple fact that similar triangles have similar length ratios between their various sides. In Figure B, the two triangles ABE and ACD are similar (their angles are congruent), so by definition, the ratio of the length of the sides AB and AC to the ratio of the length of the bases BE and CD is identical. Written mathematically, the ratio of AB / BE = AC / CD.

This geometric truth means you can multiply, divide, find ratios, scale up, scale down, and so on, just by building or measuring similar triangles. And that’s how the sector works; it’s basically just a quick and easy way to build and scale similar triangles.

Making a sector is an easy and inexpensive project and something that parents and children can make together and have fun using even without a specific goal in mind. But don’t let its relative simplicity fool you; the sector is useful to woodworkers, artists, metal workers, and others in a bunch of different ways.

So, let’s get started. First, we’ll fabricate the frame of the sector instrument. Then we’ll make the all-important scales that are written, printed, or inscribed on the frame pieces. Finally, we’ll use the sector to do different types of important mathematical jobs.

## Project Steps

### HOW TO MAKE THE SECTOR

1. Use your saw to cut out the two L-shaped frame pieces as shown in Figure . You can use thin plywood, solid wood, plastic, or any light, non-flexible sheet stock that you can cut into the shape shown. I used 1/8″-thick HDPE plastic and it worked great.

2. Drill a 9/64″ diameter hole in each piece at the spot indicated in Figure C.

3. Insert a #6, ½”-long bolt with two washers into the hole and gently tighten with a wing nut.

### HOW TO MARK THE SECTOR SCALES

The sectors made by the great European instrument makers of the 17th, 18th, and 19th centuries had a lot of different numerical scales etched into them. Each scale was used to perform a different job, and each was given a special name. By far, the most important scale was the one that Galileo called the Line of Lines. The Line of Lines is the one you need to multiply, divide, and find proportions.

Another important scale is called the Line of Circles. With it, you can measure the radius or diameter of any circle and quickly determine its circumference. (It also works in the other direction, allowing you to find the diameter of a circle for a given circumference.)

As Figure D shows, the sectors made by London instrument makers in the 18th century were chock full of scales or “lines.” Many of them were rather specific in what they could do. For instance, the Line of Quadrature allowed engineers to calculate polygons of different side lengths that were equivalent in area. The Line of Inscribed Bodies was used to determine the length of sides of different polygons that were able to fit inside a circle of given diameter. There was even a Line of Metals to calculate the diameter of spheres made of different metals when all the spheres had the same weight.

Since drawing or inscribing accurate scales is time consuming, we’ll limit our scales to just the three most useful: the line of lines, the line of circles, and the line of calculation. Figure E shows the layout of the scales we will use on the frame pieces.

You can draw the scales on the frame pieces using a fine-tipped marker, or if you’re really ambitious, you can etch or engrave the scales. If these options seem like too much fine detail work, fear not because we’ve made a template that you can download, print out, and affix to the frame pieces using glue or spray adhesive.

### FUN WAYS TO USE YOUR SECTOR

The easiest way to learn to use your sector is by looking at a few examples. If you follow along and understand the three examples below, you can substitute whatever numbers or line lengths you need when you use your sector on an actual problem. Also, to use a sector, you’ll need a simple mechanical divider as well (Figure F).

### EXAMPLE 1. HOW TO DIVIDE LINES INTO FRACTIONS OF ANY LENGTH

You can use a sector to easily divide a line into two parts of any ratio. Let’s say you’re building a chest of drawers and for aesthetic reasons you want to place the knobs on the drawer faces 2/7ths (or 20/70ths) of the way down from the top of each face. With a calculator and ruler, this can be time consuming, but with a sector, it’s a snap!

STEP 1: Open your divider so that one tip is on the start of the distance that you want to divide (in this example, the top of the drawer face), and the other tip is on the end of the distance (the bottom of the drawer face), as in Figure G.

STEP 2: Without changing the width of your divider, set one end of the divider on 70 on one of the Line of Lines. Open the jaws of the sector so you can set the other end of the divider on 70 on the other Line of Lines (Figure H).

STEP 3: Without changing the angle of the sector, adjust the tips of the divider so that they touch both Lines of Lines where the scale reads 20 (Figure I). Measure the distance between the divider tips. That distance is exactly 2/7ths of the total distance from the top of the drawer face to the bottom (Figure J).

### EXAMPLE 2. FINDING A FRACTION OF A NUMBER

EXAMPLE 2. FINDING A FRACTION OF A NUMBER
By using the same technique as in Example 1 plus the Line of Calculation, you can find fractions and proportions of numbers. For example, what is 2/7ths of 64?

STEP 1: Place one end of your divider on the zero point of the Line of Calculation. Place the other tip of the divider on 64 (Figure K).

STEP 2: Without changing the width of your divider, open the jaws of the sector so you can set both ends of the divider on 70 on the Line of Lines on each set of jaws of the sector (Figure L).

STEP 3: Place the tips of your divider on the Line of Lines where the scale reads 20 (Figure M). Move the divider to the Line of Calculation. Place one tip on the zero point and the other tip on the line, where it points to the answer to the problem, or 18.3 (Figure N).

### EXAMPLE 3. FINDING A CIRCLE’S DIAMETER OR CIRCUMFERENCE

The Line of Circles makes it easy to find the radius, diameter, and circumference of any circle if you already know any one of those items. For example, what is the circumference of a U.S. quarter coin?

STEP 1: In this example, we use the Line of Circles. Take your divider and open the jaws so the quarter just fits between the tips. This is the diameter of the quarter (Figure O).

STEP 2: Without moving the divider tips, adjust the sector so that you can place the tips of the divider on the dots marked “D” (for “diameter”) on the Line of Circles on both jaws of the sector (Figure P).

STEP 3: Without moving the sector jaws, place the tips of the divider on the points marked “C” (for “circumference”) on the Line of Circles (Figure Q). Place one tip of the divider on the zero point of the Line of Calculation and read the answer from the spot where the other tip touches the Line of Calculation (Figure R) — in this case, about 3 inches.