Scale Drawings, Maps and Dilations

A scaled copy keeps the shape of the original. Every length changes by one common multiplier, called the scale factor. Matching sides, angles, and points are called corresponding parts. The point matching A is often labeled A′, read “A prime.”

Find the multiplier

Start by matching the parts. A short side usually matches a short side, even when a copy is turned. Divide a copy length by its original length:

A rectangle measures 4 cm by 6 cm. Another measures 10 cm by 15 cm. Match 4 with 10 and 6 with 15. Then calculate:

Both sides have the same factor. Both rectangles also have right angles. The second rectangle is a scaled copy with factor 2.5. Their width-to-length ratios agree too: 4 : 6 and 10 : 15 both simplify to 2 : 3.

The direction matters. Going back from the larger rectangle gives a different factor:

The reverse factor is the reciprocal of the first: 1 divided by 2.5. Name the original and copy before dividing.

Check the whole shape

One matching pair gives a possible factor. Check all the corresponding sides and angles to confirm it. Compare a 4 cm by 6 cm rectangle with a 6 cm by 8 cm rectangle. The short sides give 1.5, but the long sides give about 1.33. These rectangles are not scaled copies.

Adding 2 cm to each original side caused that mismatch. Scaling means multiplying. A genuine factor-1.5 copy of the 4 cm by 6 cm rectangle measures 6 cm by 9 cm.

Angles keep their size. A triangle with angles 30°, 60°, and 90° still has those angles after scaling by 3. Its lengths triple; its angles do not. For polygons, checking only side ratios can miss a changed shape. A square and a slanted rhombus can have equal sides but different angles.

Use a known factor

A factor greater than 1 makes a larger copy. A factor between 0 and 1 makes a smaller copy. A factor of 1 keeps the size. Fractions and decimals work like whole numbers: 1/2 halves each length, and 2.5 makes each length two and a half times as long.

Suppose a triangle has sides 5 cm, 7 cm, and 9 cm. Its copy has a longest side of 13.5 cm. The longest sides correspond, so:

Multiply each remaining original side by 1.5. The missing copy lengths are 7.5 cm and 10.5 cm. To recover an original length from a copy length, divide by the factor. For example, 10.5 divided by 1.5 gives the original 7 cm side.

Grid paper helps you keep a figure's proportions. Work with horizontal and vertical moves, including the moves that make a slanted side. Choose a starting point with enough space for the new figure.

Scale the grid moves

A rectangle 3 squares wide and 2 squares tall becomes 6 squares wide and 4 squares tall with factor 2. Both dimensions double. Keeping the height at 2 while doubling the width would stretch the shape.

For a slanted side, record how far it goes across and up. A move of 2 squares right and 1 square up becomes 6 right and 3 up with factor 3. Draw a straight line between the new endpoints. The matching slope stays the same because both step counts use the same factor.

Do not count diagonal grid squares as units of length. A square's diagonal is longer than one of its sides. Scaling the horizontal and vertical steps locates the endpoint without measuring that diagonal.

Follow a composite outline

An L-shape can be traced using these moves:

1. Go 4 right, then 1 up.
2. Go 3 left, then 2 up.
3. Go 1 left, then 3 down.

These moves close the outline. To draw its factor-2 copy, double each move:

1. Go 8 right, then 2 up.
2. Go 6 left, then 4 up.
3. Go 2 left, then 6 down.

Join each corner in order. You should finish at the starting point. A gap means a move was missed or counted incorrectly. This L-shape has perimeter 14 units and area 6 square units. The doubled copy has perimeter 28 units and area 24 square units. The inside corner scales along with the outside edges.

Make a smaller copy

For a factor of 1/2, halve every move. A right triangle with legs of 6 squares and 4 squares becomes one with legs of 3 squares and 2 squares. A length of 5 squares becomes 2.5 squares. An image corner can fall halfway between grid lines; it need not sit on a grid intersection.

After drawing, check the angles as well as the lengths. Right angles remain right angles. Slanted sides should have the same proportions between their horizontal and vertical moves.

Draw without a grid

Use a ruler for lengths and a protractor for angles. A right triangle with legs of 3 cm and 4 cm has a third side of 5 cm. With factor 1.5, draw the legs 4.5 cm and 6 cm long, keeping their 90° angle. Joining their ends makes a third side of 7.5 cm.

A scaled copy may be moved, turned, or flipped on the page. Those changes do not alter its size or shape. When a problem also gives a center of dilation, that center fixes where the image points must go.

Scaling changes lengths and areas in different ways. Perimeter measures a boundary in length units. Area measures a surface in square units. Keeping those jobs separate helps you choose the right multiplier.

Perimeter uses the length factor

Every side is multiplied by the scale factor, so their sum is too. If the scale factor is $k$, then:

A triangle with sides 5 cm, 7 cm, and 9 cm has perimeter 21 cm. With factor 1.5, its perimeter is:

You could also scale each side and add 7.5, 10.5, and 13.5. Multiplying the original perimeter takes fewer steps.

Area uses the square of the factor

A 3-by-2 rectangle has area 6 square units. Scaling by 3 makes it 9 by 6, with area 54 square units. Each original unit square becomes a 3-by-3 block containing 9 unit squares. The area multiplier is 9 because both dimensions triple:

The rule applies to every figure. A triangle with base 4 cm and height 3 cm has area 6 cm². With factor 2, its base becomes 8 cm and its height 6 cm. Its area becomes 24 cm², four times the original. A circle of radius 3 cm has area $9\pi$ cm². Doubling its radius gives area $36\pi$ cm², also four times as much.

Doubling lengths does not double area. For a quick check, compare a 2-by-3 rectangle with its 4-by-6 copy. Their areas are 6 and 24. Both the width and the height have changed, so the area gains two factors of 2.

Reductions and reverse problems

An 8 cm by 6 cm rectangle has area 48 cm². Halving the lengths makes a 4 cm by 3 cm rectangle, with area 12 cm²:

The area is one fourth, even though each length is one half. Angles still keep their original size.

Sometimes the areas are known first. A 4 in by 6 in photo has area 24 in². A similar print has area 96 in². First find the area multiplier: 96 divided by 24 is 4. The positive length factor is 2 because 2 squared is 4. The print therefore measures 8 in by 12 in. An area multiplier of 25 would mean a length factor of 5.

Convert square units carefully

Area conversions also act in two directions. Since 1 m is 100 cm, a square meter is 100 cm by 100 cm:

Since 1 ft is 12 in, a square foot is 12 in by 12 in, or 144 in². Use these square-unit conversions when a plan's area is measured in cm² or in² but the real area is needed in m² or ft².

A scale drawing shows an object or place with all lengths changed by one common factor. Plans and maps usually shrink reality. A drawing of a tiny insect may enlarge it. The scale connects the drawing lengths to the actual lengths.

Scales with units

“1 cm represents 2 m” means that each centimeter on the drawing stands for 2 meters in reality. Two drawing centimeters stand for 4 meters. The units on the two sides can differ because they are named.

In US customary units, a plan might say “1 inch represents 8 feet.” An architect's plan may use “1/4 inch = 1 foot.” Here the equals sign means “represents.” A quarter inch and a foot are not equal actual lengths.

Ratio scales

A scale of 1 : 200 is read “1 to 200.” One drawing unit stands for 200 of the same actual unit. It can mean 1 cm represents 200 cm or 1 in represents 200 in. It does not mean 1 cm represents 200 m.

To change a word scale to a ratio, convert both lengths to the same unit and simplify. Since 1 m is 100 cm, “1 cm represents 5 m” becomes 1 cm to 500 cm, or 1 : 500.

The same method works with inches and feet. There are 12 inches in a foot, so “1 inch represents 8 feet” becomes 1 : 96. For the architect's scale:

To change a ratio to a convenient word scale, convert the actual length. At 1 : 50,000, one drawing centimeter represents 50,000 cm. That is 500 m or 0.5 km. At 1 : 2,500, one drawing centimeter represents 25 m. Useful facts are 1 km = 1,000 m = 100,000 cm, 1 yd = 3 ft, and 1 mi = 5,280 ft = 63,360 in.

Read or build a scale bar

A scale bar is a line with marks labeled by real distances. At 1 : 50,000, 2 cm represents 1 km. Draw a bar with marks every 2 cm and label them 0, 1, 2, and 3 km. A distance spanning two of those gaps represents 2 km.

A scale bar stays useful when a whole map is resized evenly. The bar and the map grow or shrink together. The printed ratio does not change its wording, so it no longer describes the resized map correctly.

Compare equivalent scales

“1 cm represents 5 m,” “2 cm represents 10 m,” and 1 : 500 describe the same scale. A bar with 5 m steps spaced 1 cm apart shows it too. For “3 cm represents 12 m,” divide both parts by 3. Each centimeter represents 4 m, giving 1 : 400.

A larger second number makes a smaller drawing of the same object. At 1 : 1,000, each paper centimeter represents more ground than at 1 : 100. A ratio scale also gives a scale factor: from reality to a 1 : 500 drawing it is 1/500; from drawing to reality it is 500.

A scale works in both directions. It can turn a drawing measurement into an actual measurement, or tell you how long to draw a real object. Decide which direction you need before calculating.

From drawing to reality

For a ratio scale 1 : 50, multiply each drawing length by 50. A wall drawn 9 cm long represents:

The multiplication gives the actual length in the same unit as the drawing measurement. Convert afterward to a useful unit.

For a scale with units, use its named relationship directly. At “1 cm represents 2.5 m,” a 7 cm line represents 17.5 m. A ratio table can keep the units clear: 1 cm matches 2.5 m, 2 cm matches 5 m, and 4 cm matches 10 m.

From reality to the drawing

To draw a 6 m wall at 1 : 50, first change meters to centimeters. There are 100 cm in a meter, so:

At “1 cm represents 2.5 m,” a 20 m wall needs 20 divided by 2.5, or 8 cm. With “1 inch represents 4 feet,” a 5.5 in plan length represents 22 ft, while a 30 ft wall needs a 7.5 in drawing length.

Find an unknown scale

A 12 m wall is drawn 6 cm long. Write drawing length to actual length in one unit:

The scale is 1 : 200, or “1 cm represents 2 m.” Simplifying 6 : 12 would give 1 : 2 and mix centimeters with meters. The conversion must come first.

The method also works for maps. A 4 cm distance representing 10 km gives 4 : 1,000,000, which simplifies to 1 : 250,000.

Find a floor area in two ways

A room is drawn 8 cm by 6 cm at 1 : 50. Convert its sides first. The actual lengths are 400 cm and 300 cm, or 4 m and 3 m. Its floor area is:

You can also scale the drawing area. It is 48 cm². Each length grows by 50, so area grows by 50 squared:

Divide by 10,000 to convert cm² to m². This gives the same 12 m². Multiplying the area by only 50 would use the length factor once and give an area that is too small.

On a plan labeled “1/4 inch = 1 foot,” each whole inch represents 4 ft. A bedroom drawn 3 in by 2.5 in is therefore 12 ft by 10 ft. Its area is 120 ft². Converting the side lengths first makes the final area unit easy to see.

Check whether the size makes sense. A bedroom is a few meters wide. An answer such as 0.24 m² for a whole room points to a missed scale factor or a square-unit conversion error.

A map can show the straight gap between two places or a route that bends between them. Measure the length that matches the journey. A road or trail usually follows a longer path than the direct line between its ends.

Straight and curved distances

For a straight distance, put the ruler's zero mark at one point. Read the distance at the other. On a 1 : 25,000 map, a 6.4 cm line represents:

For a curved route, lay string along it and mark the ends. Straighten the string and measure the marked length. You can also divide the route into short, nearly straight pieces. Add their lengths, or transfer them one after another onto a paper edge, turning the strip at each bend.

Pieces of 2.1 cm, 3.4 cm, and 1.5 cm total 7 cm. At 1 : 50,000, this represents 350,000 cm, or 3.5 km. With a scale bar, transfer the total length onto the bar and read the real distance directly.

Compare two maps

A 1 km road measures 4 cm at 1 : 25,000 and 2 cm at 1 : 50,000. The first map draws lengths twice as long. The same ground area therefore occupies four times as much paper area.

On equal-sized sheets, the 1 : 25,000 map covers a smaller region in more detail. The 1 : 50,000 map covers more ground. Compare the fractions, or test one real distance, if the scale numbers feel confusing.

Turn distance into time

Use the route distance with the journey's average speed:

A footpath measures 7.5 cm at 1 : 20,000. It represents 150,000 cm, or 1.5 km. At 4 km/h, the walking time is:

Multiply hours by 60 to get 22.5 minutes. A decimal hour is part of 60 minutes; 0.375 hours is not 37.5 minutes.

A road map uses “1 inch represents 25 miles.” The complete road route measures 3.5 in, so it is 87.5 miles. Driving at an average of 50 mph takes 1.75 hours, or 1 hour 45 minutes.

Use other rates

A road measures 12 cm at 1 : 500,000. That is 60 km. A car using 6.5 liters per 100 km needs:

The distance is 0.6 of 100 km, so the fuel is 0.6 of 6.5 liters. Keep the distance unit consistent with the rate.

There are 100,000 cm in 1 km. Dividing 150,000 cm by 100,000 gives 1.5 km, not 15 km. Read north from the map's north arrow before describing directions; north is often at the top, but the arrow confirms it.

A useful scale makes the whole object fit while leaving its features easy to read. Start with a rough sketch and real measurements. Compare both the length and width with the available paper space.

Choose a scale that fits

A garden is 18 m by 12 m. The paper space is 20 cm by 15 cm. At “1 cm represents 1 m,” the plan is 18 cm by 12 cm. It fits and uses the space well.

At “1 cm represents 50 cm,” the plan would be 36 cm by 24 cm, too large. At “1 cm represents 2 m,” it would be 9 cm by 6 cm. That fits, but the smaller drawing makes details harder to measure. Scales using 1, 2, 5, or 10 real units per centimeter often keep the calculations simple.

Convert before drawing

For a room 5.4 m by 3.6 m with a 0.9 m door, choose 1 : 50. Convert the real measurements to centimeters, then divide:

• Length: 540 divided by 50 gives 10.8 cm.
• Width: 360 divided by 50 gives 7.2 cm.
• Door opening: 90 divided by 50 gives 1.8 cm.

Draw the rectangle with a ruler. Use a protractor or drafting triangle to keep the corners at 90°. Mark the door gap at its scaled width. Write the scale on the plan and label the actual measurements when needed. Aim to draw and measure lengths to the nearest millimeter.

Small paper errors can mean large real differences. At “1 cm represents 2 km,” a 1 mm error represents 200 m. A measured 8.3 cm stands for about 16.6 km. Use “about” for a result read from a drawing.

Redraw at a new scale

A 10 cm line on a plan using “1 cm represents 2 m” represents 20 m. At “1 cm represents 5 m,” draw that same length as 4 cm.

You can also find one multiplier for the whole drawing:

Multiply every old drawing length by 0.4. Paper areas become 0.16 of their old sizes because 0.4 squared is 0.16. A bigger number of real meters per centimeter makes a smaller drawing.

Work with models and enlarged drawings

A scale model is three-dimensional. Its length, width, and height all use the same factor. A 1 : 18 model of a car 4.5 m long has model length:

A 1 : 200 model building 30 cm tall represents a building 6,000 cm, or 60 m, tall.

Some drawings enlarge small objects. At 20 : 1, drawing lengths are twenty times actual lengths. A 4 mm ant is drawn 80 mm long. A 30 mm detail on that drawing represents 1.5 mm in reality, found by dividing 30 by 20.

A dilation scales a figure and fixes where its image goes. It needs a center and a positive scale factor. The starting figure is the original, or preimage. The result is the image. In many countries, a dilation is called an enlargement, including when it makes the figure smaller.

Measure from the center

Let the center be O and the factor be $k$. For each point P, draw the ray starting at O and passing through P. Its image P′ lies on that same ray, with:

If P is 3 cm from O, factor 2 puts P′ 6 cm from O. Do not add another 6 cm beyond P: that would put the image 9 cm from O. The multiplied distance is the whole distance from the center.

With factor 1/3, the same P goes to a point 1 cm from O. For a positive factor, an image point other than the center stays on the same side of O as its original.

What the factor changes

The center itself stays fixed. When the factor differs from 1, changing the center changes the image's position, even if the factor stays the same. With factor 1, every choice of center leaves the figure unchanged.

Every length uses the factor. If AB is 4 cm and the factor is 1.5, A′B′ is 6 cm. Perimeter uses the same factor, and area uses its square. Angles keep their size, so a dilation makes a scaled copy.

Lines and the direction of the figure

A line through the center maps to itself. Points can move along it, but the whole line stays in place. A line away from the center maps to a parallel line. Corresponding sides are therefore parallel or lie on the same line through O.

With a positive factor, the image faces the same way as the original. It is not turned or flipped. A rotation or reflection can be combined with a dilation when a similar figure faces another way.

Where the center can be

The center may lie outside, inside, or on the figure. If it is a vertex, that vertex stays put. The two sides meeting there keep their lines. With the center inside a convex shape, such as a triangle or rectangle, an enlargement with factor greater than 1 surrounds the original.

Translations, rotations, and reflections are rigid transformations. They keep lengths and angles. A dilation with factor different from 1 keeps angles but changes lengths. This gives a way to change size before moving a figure into the required position.

To dilate a figure, locate each image vertex from the center. Join those vertices in the original order. The same method works for triangles, rectangles, and composite outlines.

Use rays and measured distances

Draw a ray from the center through each original vertex. Measure each distance from the center, multiply it by the factor, and mark the image vertex at that distance on the ray. For a reduction, the image point lies between the center and its original point.

On a grid, count horizontal and vertical steps instead of measuring slanted distances. With center (2, 1), P(4, 3) is 2 right and 2 up. For factor 2, take 4 right and 4 up from the center. The image is P′(6, 5).

Center at the origin

When the center is (0, 0), the coordinates already give the steps from the center. Multiply both coordinates:

Triangle A(1, 1), B(3, 1), C(1, 2), dilated by 3, has image A′(3, 3), B′(9, 3), C′(3, 6). The horizontal side AB is 2 units long; A′B′ is 6 units long.

Negative coordinates use the same rule. With factor 1/2, (6, 4) becomes (3, 2), and (−4, 2) becomes (−2, 1). Halving a negative coordinate keeps its sign and halves its distance from zero. The coordinates may be negative while the scale factor is positive.

Center away from the origin

With another center, first find the steps from that center. Scale the steps, then add them back to the center's coordinates. For center (a, b), this gives:

Use center (2, 0), factor 1/2, and P(6, 4):

1. From (2, 0) to (6, 4) is 4 right and 4 up.
2. Halve those steps to get 2 right and 2 up.
3. Start at (2, 0) and take those steps to reach (4, 2).

Multiplying the original coordinates directly would give (3, 2). That would use the origin as the center, changing the transformation.

Keep a center vertex fixed

A square has vertices (1, 1), (3, 1), (3, 3), and (1, 3). With center (1, 1) and factor 2, its image vertices are (1, 1), (5, 1), (5, 5), and (1, 5). The center vertex needs zero steps and stays fixed. The side length doubles from 2 to 4, and area grows from 4 to 16 square units.

Check each vertex away from the center on the same ray as its original, at the correct multiple of its original distance. Also check matching sides: they should be parallel or on the same line, and their lengths should use the given factor.

A full dilation description tells both its center and its factor. “Twice as big” gives a size change but leaves the position unclear. Use corresponding sides for the factor and corresponding points for the center.

Find the factor first

Triangle P(3, 2), Q(5, 2), R(3, 3) has image P′(5, 3), Q′(9, 3), R′(5, 5). The side PQ is 2 units long, and P′Q′ is 4 units long. Divide image length by original length:

Check another side. PR is 1 unit and P′R′ is 2 units, also giving factor 2. The larger image fits a factor greater than 1. Dividing original by image would give 1/2, the reverse transformation's factor.

Locate the center

Draw a line through P and P′. Draw another through Q and Q′. Extend the lines until they cross. For this example, they meet at (1, 1). The line through R and R′ passes through that same point, providing a check.

Use pairs whose joining lines are distinct. If a vertex stays fixed while other vertices move, that fixed vertex is the center. When factor 1 leaves every point fixed, there is no unique center to find: any center gives the same unchanged figure.

Test the candidate center

Count steps from (1, 1) to the original points and then their images:

• P is 2 right and 1 up; P′ is 4 right and 2 up.
• Q is 4 right and 1 up; Q′ is 8 right and 2 up.
• R is 2 right and 2 up; R′ is 4 right and 4 up.

Each step pair doubles. Each original and image point lies on the same ray from (1, 1). The complete description is “a dilation with center (1, 1) and scale factor 2.”

Without coordinates, compare measured distances from the center. If OP is 2.4 cm and OP′ is 6 cm, the factor is 6 divided by 2.4, or 2.5. Use the full center-to-image distance, not just the gap from P to P′.

Undo the transformation

Use the same center and the reciprocal factor. Factor 1/2 about (1, 1) takes the image triangle back to its original. For example, P′ is 4 right and 2 up from the center. Halving those steps gives 2 right and 1 up, reaching P(3, 2).

Check whether one dilation is enough

A single positive dilation keeps the figure facing the same way. Corresponding sides are parallel or share a line, and joining lines for matching points meet at one center when the factor differs from 1.

A turned or flipped copy needs a rotation or reflection as well. A same-size copy that has only shifted is a translation. Match the point labels before joining lines; joining nonmatching vertices can produce a false center.

Two figures are similar when a dilation combined with translations, rotations, or reflections can map one onto the other. They have the same shape. They may have different sizes and may face different directions.

Match the parts in order

The symbol $\sim$ means “is similar to.” In:

A matches D, B matches E, and C matches F. Therefore AB matches DE, BC matches EF, and CA matches FD. Use the letter order even if one triangle is turned on the page. Matching the bottom sides just because both are at the bottom can pair the wrong lengths.

Corresponding angles are equal, and corresponding side lengths share one factor. Shortest sides match shortest sides; longest sides match longest sides. Congruent figures are similar too, with factor 1.

Check size and shape

Rectangles 4 by 6 and 6 by 9 have factor 1.5 for both dimensions. Their angles are all right angles, so they are similar. Rectangles 4 by 6 and 6 by 8 give different factors and are not similar.

For polygons, equal angles alone are not enough. A 2-by-2 square and a 2-by-5 rectangle both have four right angles, but their side ratios do not agree. Equal side ratios alone are not enough either: a square and a slanted rhombus can have the same side lengths while their angles differ. Any two squares are similar, as are any two circles.

Find a missing length

Suppose ABC is similar to DEF, with AB = 4, BC = 6, CA = 8, and DE = 6. The correspondence gives:

Use the same factor for the other sides. EF is 9, and FD is 12. Check the ratios 9 divided by 6 and 12 divided by 8; both give 1.5. If a length belongs to the smaller figure, divide a larger matching length by 1.5 to recover it.

Describe a sequence of transformations

Start with triangle vertices (1, 1), (2, 1), and (1, 3). Dilate about the origin by factor 2 to get (2, 2), (4, 2), and (2, 6). Then translate 3 units right to get (5, 2), (7, 2), and (5, 6).

This sequence maps the first triangle onto the last. The dilation changes its size; the translation places it at the target position. Describe the center and factor of the dilation, then the distance and direction of the translation. For a rotation or reflection, also state the details needed to locate that move.

Photos give a familiar shape check. A 4 in by 6 in photo scales to 8 in by 12 in with factor 2. An 8 in by 10 in print gives factors 2 and about 1.67. Keeping the whole photo in that size would stretch it; keeping its proportions would require cropping or leaving unused space.

A bearing describes a direction using an angle. Start from north, turn clockwise, and write three digits. This gives a precise direction even when no named compass point lies exactly that way.

Compass points and three digits

The main compass points in clockwise order are north, east, south, and west. They are 90° apart. Northeast, southeast, southwest, and northwest lie halfway between them, making 45° steps around an eight-point compass.

Write 8° as 008° and 45° as 045° when giving bearings. North is written 000° here; you may also see 360° for a full turn back to north.

Read the starting point

“The bearing of B from A” is measured at A. Draw a north line at A and a line from A toward B. Measure the clockwise angle between them. The word “from” tells you where to put the protractor's center.

North lines at different points on a map are parallel. Follow the map's north arrow. When north points toward the top edge, the north lines can run parallel to the vertical grid lines or side edge.

Use a 180° protractor

For a direction east of the north line, a bearing up to 180° can be read directly clockwise from north. Put the protractor's center at the starting point, align zero with north, and read the scale that increases clockwise toward the destination.

A direction west of the north line has a bearing greater than 180°. Measure its smaller counterclockwise angle from north, then subtract that angle from 360°. A 70° counterclockwise angle gives:

You can also measure clockwise from south and add 180°.

Draw a given bearing

To draw 230° from P, draw a north line at P. The direction is 50° clockwise past south because 180° plus 50° is 230°. Mark that angle and draw a ray from P through the mark. Another method is to measure 130° counterclockwise from north.

Estimate the region before reading a protractor. A northeast direction belongs between 000° and 090°; southeast between 090° and 180°; southwest between 180° and 270°; northwest between 270° and 360°. For example, reading 060° for a northwest direction signals that you measured the wrong way or used the wrong scale.

Bearings can describe a return direction or compare several directions. First identify whether you are reversing a journey, turning from a heading, or comparing rays from one point. Each uses a different angle relationship.

Find the back bearing

The back bearing points from the destination toward the start. Opposite directions differ by a half turn, or 180°. The north lines at the two places are parallel.

• For a bearing below 180°, add 180°.
• For a bearing of 180° or more, subtract 180°.

Thus 060° reverses to 240°, 300° reverses to 120°, and 180° reverses to 000°. If a lighthouse is on a bearing of 135° from a harbor, the harbor is on a bearing of 315° from the lighthouse.

For 060°, subtracting from 360° gives 300°, the reflection across the north line. The correct opposite direction is 240°.

Find the smaller angle between directions

When two bearings start at the same point, subtract the smaller number from the larger. If the result is at most 180°, it is the smaller angle between the directions.

A point B lies on bearing 040° from A, and C lies on bearing 110° from A. Angle BAC is:

If the difference exceeds 180°, subtract it from 360°. Bearings 350° and 010° differ numerically by 340°. Their directions are only 20° apart across north, because 360° minus 340° is 20°.

Follow a turn

A clockwise turn adds to the current bearing. A counterclockwise turn subtracts. A walker heading 070° who turns 90° clockwise heads 160°. Turning 90° counterclockwise from 070° gives −20°; add 360° to write the heading as 340°.

Wrap around after a full turn too. From 300°, a 90° clockwise turn gives 390°. Subtract 360° to write the new bearing as 030°. The turn is measured from the old heading; the new bearing is measured from north.

Locate a point from two bearings

Place A and B on a scale drawing with B 10 km due east of A. A boat is on bearing 045° from A and 315° from B.

Draw a north line at each point. From A, draw the northeast ray. From B, draw the northwest ray. Their crossing locates the boat, 5 km north of the midpoint of AB. Measuring from A to the crossing gives about 7.1 km.

Each bearing has its own starting point. Drawing both rays from A would lose the information supplied by B. Extend rays in their stated directions and mark the intersection clearly before measuring distances.

A navigation journey can have several legs. Each leg has a distance and a bearing. A scale drawing puts those legs together so you can measure the final position, the direct return distance, and the return direction.

Draw the first leg

A ship sails 8 km on bearing 070°, then 5 km on bearing 160°. Choose “1 cm represents 1 km,” so the lengths become 8 cm and 5 cm. Leave enough room for both legs.

Mark start A and draw its north line. Measure 070° clockwise from north. Draw an 8 cm segment in that direction to B. This locates the turning point.

Start the next bearing at the turning point

At B, draw another north line parallel to the first. Measure 160° clockwise from this new line. Draw the second leg 5 cm long to C.

Do not measure the second bearing at A. That would describe a second direction from the start, not a journey continuing from B. Also keep the bearing separate from the turn: the heading changes by 90°, but the second bearing is 160° from north.

Read the direct distance and bearing

Join A to C. Measuring the scale drawing gives about 9.4 cm, so C is about 9.4 km from A. At A, measure clockwise from north to AC. The bearing is about 102°.

The direct route back is CA. Its bearing is about:

A to B to C covers 13 km. The direct distance AC is shorter because it cuts across the bend.

Check a right-angled journey

The change from heading 070° to 160° is 90°. The journey makes a right-angled triangle. For a right triangle, the square of its longest side equals the sum of the squares of its other two sides. This is the Pythagorean theorem:

That agrees with the measured 9.4 km. Measurements from a drawing are approximate. Measure angles to the nearest degree and lengths to the nearest millimeter, then report a sensible precision.

Combine the route with time

At 12 km/h, the two original legs take 13 divided by 12 hours, or 1 hour 5 minutes. The direct return takes about 9.4 divided by 12 hours. Multiplying by 60 gives about 47 minutes.

To place a campsite 2.5 km from a station on bearing 300° at 1 : 50,000, first convert 2.5 km to 250,000 cm. Divide by 50,000 to get 5 cm. From the station, draw bearing 300° and mark the campsite 5 cm along it.

For longer journeys, use a scale that still fits. Legs of 30 km to 40 km may suit “1 cm represents 5 km.” Check the whole route's space and write the chosen scale beside the drawing.