Measuring and Drawing Angles

Open a pair of scissors a little, then a little more. The blades make an angle. Its size changes as one blade turns away from the other.

Find the parts

An angle is made by two rays with the same starting point. A ray starts at one point and goes straight on forever. We draw just a short piece of each ray.

The shared starting point is the vertex. The rays are the angle's arms, also called its sides. On scissors, look at the point where the blades turn. On a clock, look at the center where the hands meet. At a book's corner, follow the two straight edges.

Imagine looking down at a door as it opens. The hinge stays still. The door turns away from the wall. The hinge acts as the vertex, and the door and wall show the arms. Opening the door farther makes a bigger angle.

Use letters to name it

Place B at the vertex. Mark A on one arm and C on the other. Write ∠ABC and say “angle A B C.” The middle letter names the vertex. You can also write ∠CBA. Both names use the same corner and arms.

When just one angle is at B, you can call it ∠B. Three letters help when several angles meet at the same point. They show which two arms you mean.

A small curved arc marks the angle to look at. A number beside it gives the size in degrees. A small square marks a right angle. You can check a right angle with a sheet of paper's square corner. It measures 90°.

Compare the turn, not the drawing

Suppose you draw an angle with arms 2 cm long. Extend both arms to 10 cm with a ruler. The angle stays the same. You have lengthened the lines without changing their directions.

The gap between the drawn ends can fool you. To compare two angles, trace one onto thin paper. Put its vertex on the other vertex. Line up one arm from each angle. Now look at the second arms. The larger angle opens farther from the shared starting arm.

Sliding, turning, or flipping the whole drawing also keeps its size. A tilted square corner is still a right angle. Look at the opening at the vertex, even when neither arm lies flat on the page.

Stand facing one wall. Turn all the way around to face it again. You have made a full turn. A half turn faces the opposite wall. A quarter turn is one right angle.

Split a full turn

We split a full turn into 360 equal parts. Each part is one degree, written 1°. The little raised circle means degrees.

To find a fraction of a turn, share 360° into equal parts. An eighth is ${360}^{\circ }÷8={45}^{\circ }$. A third is ${360}^{\circ }÷3={120}^{\circ }$. Three quarter turns give $3\times {90}^{\circ }={270}^{\circ }$.

Say which way to turn

Clockwise follows the hands of a clock: from 12 toward 3. Counterclockwise goes the other way: from 12 toward 9.

Facing east, turn 90° clockwise and you face south. Turn 90° counterclockwise and you face north. The size is the same, but the direction changes where you finish.

You can reach the same direction by going the long way around. A 90° clockwise turn ends where a 270° counterclockwise turn ends. The two routes together make ${90}^{\circ }+{270}^{\circ }={360}^{\circ }$.

Use the clock's spaces

A clock has 12 equal spaces around its face. Each space is ${360}^{\circ }÷12={30}^{\circ }$. Count spaces between numbers, not the numbers themselves.

In 15 minutes, the minute hand travels three spaces, from 12 to 3. Its turn is $3\times {30}^{\circ }={90}^{\circ }$ clockwise. In 30 minutes it turns 180°. In an hour it turns 360°.

At 8:00, the hour hand points to 8 and the minute hand to 12. Count the shorter route: 12 to 11 to 10 to 9 to 8. Four spaces give $4\times {30}^{\circ }={120}^{\circ }$. The longer route has eight spaces and measures 240°. At 6:00, either route is six spaces, or 180°.

Follow compass points

Picture north at the top of a compass. Clockwise, the eight points are N, NE, E, SE, S, SW, W, NW. The longer names are north, northeast, east, southeast, south, southwest, west, and northwest.

Each step is ${360}^{\circ }÷8={45}^{\circ }$. Maya turns clockwise from north to southwest. Her route is N → NE → E → SE → S → SW. Five steps give $5\times {45}^{\circ }={225}^{\circ }$. Counterclockwise takes three steps, or 135°. These two routes also total 360°.

A square corner gives you a useful starting point: a right angle. Other angles have names that tell you how their turns compare with it.

Match the size to the name

Hold a paper corner beside an angle. If the opening fits inside the corner, it is acute. If it matches exactly, it is right. A wider opening is obtuse until the arms make a straight line.

The exact boundaries matter. An 89° angle is acute. A 90° angle is right. A 91° angle is obtuse. At the next boundary, 179° is obtuse, 180° is straight, and 181° is reflex. A wide-looking angle of 170° is still obtuse.

Look at the marked side

Two arms have an angle on each side. Together, those two angles make 360°. For example, a 60° opening leaves ${360}^{\circ }-{60}^{\circ }={300}^{\circ }$ around the other side.

An arc inside the small opening marks 60°. An arc going the long way around marks the reflex angle of 300°. The arms can be close together even when the marked angle is large.

When a question names ∠ABC without saying “reflex,” use the angle of 180° or less. A reflex angle has an outside arc or the name “reflex ∠ABC.”

Find angle types in shapes

A rectangle has four right angles, even when you turn the page. A triangle shaped like a wide, flat roof can have an obtuse angle at its top. Look inside the shape at that corner.

At a dent in an arrowhead shape, the inside angle goes around more than halfway. That angle is reflex. Follow the inside of the shape to choose the correct side of the arms.

Clock hands give more pictures: at 2:00, their smaller angle is 60°, which is acute. At 5:00, it is 150°, which is obtuse. At 6:00, it is 180°, which is straight.

You do not always need an exact measurement straight away. An estimate is a sensible guess based on an angle you can picture. It also helps you check a protractor reading later.

Build a few benchmarks

A benchmark angle is a familiar size you use for comparison. A right angle is 90° and a straight angle is 180°. Half a right angle is 45°.

You can make 45° by folding a paper corner so its two edges meet. The fold splits the corner into two equal angles. Imagine splitting a right angle into three equal parts: ${90}^{\circ }÷3={30}^{\circ }$. Two of those parts make 60°.

For larger turns, picture three right angles as 270°, and a full turn as 360°.

Choose a nearby size

First decide whether the marked angle is acute, obtuse, or reflex. Then compare it with a nearby benchmark. Ask, “Is it a little bigger, a little smaller, or about halfway?” Give a round-number estimate, often to the nearest 10°.

For example, an opening looks just smaller than half a right angle. Half a right angle is 45°, so “about 40°” is sensible. A guess of 80° would be too close to a whole right angle.

An obtuse opening has its second arm halfway between the 90° and 180° positions. Halfway is $({90}^{\circ }+{180}^{\circ })÷2={135}^{\circ }$. You could estimate about 130° or 140°. Small differences are normal when you judge by eye; a good estimate is usually within about 10°.

Estimate the outside turn

For a reflex angle, judge the smaller opening first. Suppose it looks about 60°. The outside turn is about ${360}^{\circ }-{60}^{\circ }={300}^{\circ }$. Keep looking at the arc so you estimate the side that was asked for.

Compare and check

Order angles by their degree measures, from least to greatest. For example, 95°, 40°, 260°, 180°, and 89° become 40°, 89°, 95°, 180°, 260°. The two acute angles come first, followed by the obtuse, straight, and reflex angles.

Without numbers, trace and line up the vertices and one arm. Compare how far each second arm turns. Longer drawn arms do not make a larger turn.

Suppose you estimate about 50°, then read 130° on a protractor. Those are different angle types. Check the starting zero and follow its scale again. An estimate gives you a reason to stop and check that reading.

A protractor measures angles in degrees. A half-circle protractor runs from 0° to 180°. Look closely at yours before you place it on an angle.

Find the marks that matter

The center point is usually a dot, cross, or small hole. It sits in the middle of the straight side. The baseline is the printed line through that point. It may sit above the plastic's bottom edge.

The two numbered scales run in opposite directions. One begins at zero on the left, and the other begins at zero on the right. Neither scale is always the correct one. Your starting arm chooses which you use.

Many protractors label every 10°. A longer tick shows each 5°, and the smallest ticks are 1° apart. Check the spacing on your own tool.

Measure from zero

1. Estimate the marked angle.
2. Place the center point exactly on its vertex.
3. Turn the tool until its baseline lies along one arm.
4. Find the scale that shows 0° on that arm.
5. Follow that same scale to the other arm.
6. Read to the nearest degree and compare with your estimate.

Keep the center on the vertex while you turn the protractor. If it slips, line it up again before reading.

Choose between two numbers

Suppose the first arm points right. The inner scale starts at zero there. The other arm crosses a mark labeled 50 on the inner scale and 130 on the outer scale.

Follow the inner scale from its starting zero: the angle is 50°. It looks acute, which agrees with your estimate. Reading 130° would mean starting from the zero on the opposite end.

For a finer reading, suppose the arm is three small ticks beyond 50° toward 60°. Each tick is 1°, so the reading is 53°. Count along the scale as its numbers increase.

Make short or awkward arms easier

If an arm does not reach the numbered curve, extend it with a ruler. Draw straight along its original direction. Extending a ray does not change the angle.

You can turn the paper to make the tool easier to hold. You can also measure from the other arm. Start from its zero this time; the angle's size stays the same.

Inside a shape, the corner is the vertex and its two sides are the arms. At a rectangle's corner, both scales read 90°.

Use the printed baseline, not the plastic edge, and check the center before reading. Thick pencil lines can make a careful reading about 1° away from the exact size. Measure again from the other arm if you want to check.

A reflex angle is bigger than 180°. A half-circle protractor cannot reach that far in one reading.

Measure the small opening

Find the smaller angle between the same arms. Center the protractor on the vertex, line up the baseline, and measure that opening. Subtract its size from 360°.

For a 70° opening, the outside angle is ${360}^{\circ }-{70}^{\circ }={290}^{\circ }$. The two parts fit together to make one complete turn.

Try a second example. The smaller opening measures 145°. The reflex angle is ${360}^{\circ }-{145}^{\circ }={215}^{\circ }$. That is just more than a straight angle, which matches its type.

Subtracting 145° from 180° would give 35°. That is the amount beyond a straight turn, not the whole outside angle.

Start with a straight angle

You can also measure a reflex angle in two pieces. Extend the starting arm backward through the vertex with a ruler. Follow the marked reflex arc from the original arm to the extension. This first part is 180°.

Continue along the arc from the extension to the finishing arm. Measure this extra part. Add it to 180°. For an extra part of 110°, the reflex angle is ${180}^{\circ }+{110}^{\circ }={290}^{\circ }$.

The extension must go straight through the vertex. It creates the half turn that makes this method work. For the 215° angle above, the extra part is 35°.

Use a full-circle protractor

A full-circle protractor has a scale reaching 360°. Put its center on the vertex and its zero on the starting arm. Follow the scale in the direction of the marked outside arc. Read where the finishing arm crosses it.

Check the direction as well as the number. Following the short route gives the smaller angle, even though the tool can measure a full turn.

Check the size

Estimate the gap before measuring. A gap a little larger than 90° leaves an outside turn a little smaller than 270°. Your reflex measurement must be above 180° and below 360°.

The same idea works for directed turns. If one route from the first arm to the second is 125°, the opposite route is ${360}^{\circ }-{125}^{\circ }={235}^{\circ }$.

To draw an angle accurately, gather a sharp pencil, ruler, and protractor. First picture the turn you want. An angle of 65° should be smaller than a square corner.

Draw 65° step by step

1. Use the ruler to draw a straight first arm, about 7 cm long.
2. Mark its left end as the vertex.
3. Put the protractor's center point on that end.
4. Line up the baseline with the arm. Find zero on the right.
5. Follow that scale to 65°. Make a small pencil dot there.
6. Remove the protractor. Use the ruler to join the vertex to the dot.
7. Draw a small arc between the arms and label it 65°.

The dot marks a direction from the vertex. Your second arm must start at the vertex and pass through the dot. It does not start at the far end of the first arm.

Check the scale

Before drawing the second arm, look at the marked direction. For 65°, it should make an acute opening. The wrong scale would place the dot at 115° from your arm, because ${180}^{\circ }-{65}^{\circ }={115}^{\circ }$. That makes an obtuse angle.

You may draw the first arm pointing left, up, or sideways. Each time, use the zero that lies on that arm. The inner scale is not always the one you need.

Draw arms at least about 5 cm long when you have room. They are easier to line up and check. Extra length does not change the turn.

Put the labels at the right points

To draw ∠PQR of 40°, start at Q. Draw arm QP, then draw QR at 40° to it. Place P and R on their arms. Q must stay at the shared starting point because it is the middle letter.

Measure your finished angle again. Put the center back on the vertex and read from zero along an arm. A careful drawing is usually within about 1° or 2° of the requested size.

Make a sketch first

A sketch is a drawing made by eye. It helps you plan the opening before using tools. For 30°, picture one third of a right angle. For 150°, picture an opening fairly close to a straight line.

A sketch gives the rough shape. Use the protractor when you need the exact angle, and the ruler when you need straight arms.

For a reflex angle, the arms alone do not show which side you mean. You need an outside arc. A half-circle protractor can help you place the arms by drawing the smaller opening first.

Draw the gap for 240°

The reflex angle and the gap together make 360°. Find the gap:

Draw a first arm and mark its vertex. Line up the protractor, start at zero on that arm, and mark 120°. Join the vertex to your dot with a ruler.

Now draw an arc around the outside, following the long turn between the arms. Write 240° beside this arc. A small inside arc would mark 120°, so the arc's position is part of your answer.

For a reflex angle of 315°, the gap is ${360}^{\circ }-{315}^{\circ }={45}^{\circ }$. Draw the 45° gap and label the outside arc 315°. The large turn leaves just a small opening.

Build on from 180°

You can draw 240° another way. Draw a straight line through the vertex. Choose one half as your starting arm. The other half shows where a 180° turn ends.

Work out how far to continue: ${240}^{\circ }-{180}^{\circ }={60}^{\circ }$. From the opposite half of the line, draw another 60° in the same turning direction. The new ray is the finishing arm. Mark one arc showing the whole 240° turn from your starting arm.

Keeping the direction matters. Turning back toward the starting arm would undo some of the first half turn.

Draw with a full-circle tool

Place the full-circle protractor's center on the vertex and zero on the first arm. Follow its scale in your chosen turning direction to the required reflex size. Make a dot, remove the tool, and join the dot to the vertex.

Finish with the outside arc and label. Check by measuring the smaller gap and adding it to your reflex label. Together they must make 360°.

Make a rough sketch before using your tools. Put each letter, side length, and angle in place. The sketch helps you see which measurements belong at each corner.

One side and two end angles

Draw triangle ABC with AB = 6 cm, angle A = 50°, and angle B = 60°.

1. Draw AB 6 cm long. Start at the ruler's 0 mark, not its plastic end.
2. Center the protractor on A. Line up the baseline with AB, starting at zero toward B.
3. Mark 50° above AB. Draw a ray from A through the mark.
4. Move the center to B. The baseline now follows BA, so zero points toward A.
5. Mark 60° above AB and draw a ray from B through it.
6. Label the place where the rays cross C.

Both new rays must be on the same side of AB and open toward each other. If they point away on opposite sides, check your sketch and redraw. Extend short rays with a ruler until they meet.

Measure the angle at C. A careful drawing should give close to 70°. This measurement helps check your drawing.

Two sides and the angle between them

Now draw AB = 5 cm, AC = 4 cm, and ∠BAC = 40°.

Draw AB first. Put the protractor at A and draw a ray at 40° to AB. From A, measure 4 cm along that ray and mark C. Join C to B with the ruler.

The 40° angle is at A, between the two given sides. The 4 cm length starts at A too. Marking 4 cm from B would put C in a different place.

Make a rectangle

For a rectangle 6 cm by 4 cm, draw a 6 cm base. At each end, draw a 90° ray on the same side of the base. Measure 4 cm along each ray from the base. Join those two new endpoints.

For other shapes, work around the corners in order. Draw the next given length along the direction set by its corner angle.

Check the finished shape

Use your ruler to check lengths to the nearest millimeter. Check the given angles with the protractor, moving its center to each corner. Compare your measurements with the labels on the sketch. A wrong length needs a ruler check; a wrong opening needs an angle check.

Think of an angle as a turn made of one-degree steps. If you split it into parts, each step belongs to one part. Adding the parts gives you the whole angle.

Add the pieces

Ray BD sits inside ∠ABC. It splits the angle into ∠ABD of 35° and ∠DBC of 50°. They share vertex B and arm BD. Together, they fill the whole angle without gaps or overlap.

So ∠ABC measures 85°. The same rule works for three or more parts. Make sure each part is counted once. Adding two angles that overlap would count some of the turn twice.

Find a missing piece

Suppose ∠ABC measures 120°. One part is 45° and the other is unknown. Call the unknown $x$. Write an equation showing how the parts make the whole:

Take the known part away:

Check by putting it back: ${45}^{\circ }+{75}^{\circ }={120}^{\circ }$. Adding 45° to 120° would give 165°, larger than the whole you are splitting.

A box can stand for the unknown instead of a letter. The job is the same: find the number that makes the equation true.

Use a right-angle whole

A right angle measures 90°. Split it into 35° and one missing part. The missing part is ${90}^{\circ }-{35}^{\circ }={55}^{\circ }$. You can use the square corner mark as the whole's measurement, even if 90° is not written.

Add turns and equal slices

A robot turns 30° clockwise, then another 45° clockwise. Its total turn is ${30}^{\circ }+{45}^{\circ }={75}^{\circ }$ clockwise. Both turns go the same way, so they add.

A round pizza has eight equal slices meeting at its center. One slice has an angle of ${360}^{\circ }÷8={45}^{\circ }$. Three slices together make $3\times {45}^{\circ }={135}^{\circ }$. Find the size of one equal part before grouping several parts.

You can also check angle addition by measuring a drawing. Measure each part, add them, and measure the whole. The totals should be close. Each reading can be a little off, so several small measuring errors can add up.

A straight line and a full turn give you two useful totals. Before calculating, look at the space the angles fill.

On one side of a straight line

Angles that fill one side of a straight line at a point add to 180°. That is a half turn.

If one angle is 130°, the angle beside it is ${180}^{\circ }-{130}^{\circ }={50}^{\circ }$. Check: ${130}^{\circ }+{50}^{\circ }={180}^{\circ }$.

Three angles can share the same half turn too. Suppose they are 40°, 75°, and $y$. Add the known parts first:

Subtract that total from the half turn:

Putting all three back gives ${40}^{\circ }+{75}^{\circ }+{65}^{\circ }={180}^{\circ }$.

All the way around a point

Angles that fill the entire space around a point add to 360°. There must be no gaps or overlap.

Around a point, two parts measure 90° and 150°. Call the remaining part $x$. The known total is ${90}^{\circ }+{150}^{\circ }={240}^{\circ }$. The missing part is $x={360}^{\circ }-{240}^{\circ }={120}^{\circ }$.

Using 180° here would choose only half of the space. Trace around the vertex: do the parts fill a half turn or a full turn?

Use the total in a story

Leo's laptop screen makes a 105° angle with the keyboard. To lie flat in line with the keyboard, it must reach 180°. The extra opening is ${180}^{\circ }-{105}^{\circ }={75}^{\circ }$. The question asks how much farther it moves, so 105° is not the answer.

A spinner has three sections. Two measure 90° and 160°. Their total is ${90}^{\circ }+{160}^{\circ }={250}^{\circ }$. The remaining section measures ${360}^{\circ }-{250}^{\circ }={110}^{\circ }$ because the spinner fills a circle.

Trust the labels on a rough drawing

A picture labeled “not drawn to scale” may show angles that look different from their stated sizes. Work from the numbers and the straight-line or full-turn fact. Measuring that picture will not give the exact missing angle.

Finish by adding all the parts. The answer must complete exactly 180° on a line or 360° around a point. You can also check its type: beside an obtuse angle on a straight line, the missing angle is acute.