4th Grade · 4.MD.C.5

Angle measurement in 4th grade

Angle measurement starts in 4th grade, when your child learns to say how big an angle is in a unit called the degree. An angle is measured as a turn, a part of a full circle. How long its sides are drawn does not change its size.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
4.MD.C.5.a An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through $\frac{1}{360}$ of a circle is called a "one-degree angle," and can be used to measure angles.
4.MD.C.5.b An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

What it means, in plain words

What is an angle? Two rays that start at the same point make an angle. A ray is a straight path that starts at a point and goes on forever in one direction. The shared starting point is called the vertex. The size of the angle is how far one ray has turned away from the other.

To measure angles, picture a circle with its center on the vertex. A full turn around the circle is split into 360 equal small turns. Each small turn is one degree, written $1^\circ$ . That is why there are 360 degrees in a circle. An angle that turns through 50 of those small turns measures $50^\circ$ .

A quarter turn, like the corner of a sheet of paper, is $90^\circ$ . A half turn, which makes a straight line, is $180^\circ$ . The size of an angle has nothing to do with how long its rays are drawn. Angle measurement has no earlier steps in 3rd grade. It is new for every child this year, so a slow start does not mean a gap from last year.

Your child should be able to:

Point to the vertex and the two rays of an angle
Explain that one degree is $\frac{1}{360}$ of a full turn
Say how many degrees are in a quarter turn, a half turn and a full turn
Compare two angles by the turn between the rays, not by the length of the rays

Example

The minute hand of a clock moves from the 12 to the 3. How many degrees does it turn?

A full turn around the clock is $360^\circ$ .
From the 12 to the 3 is a quarter of the way around.
$360 \div 4 = 90$ , so the minute hand turns $90^\circ$ .

Example

A round pizza is cut into 6 equal slices from the center. What angle does each slice make at the center?

All the slices together make one full turn, $360^\circ$ .
Each slice is $\frac{1}{6}$ of the turn.
$360 \div 6 = 60$ , so each slice makes a $60^\circ$ angle at the center.

Where children usually get stuck

Your child says the corner of a big book is a bigger angle than the corner of a sticky note.
The two sides are the most visible thing in the picture, so the child measures their length. Angles are about turning, and both corners are the same quarter turn, $90^\circ$ .
What helps. Lay the sticky note on the corner of the book so the corners match. The turn is the same, and only the sides are longer.
Your child says a straight line has no angle, so a half turn is 0 degrees.
Most pictures of angles show a corner. A straight angle has no corner to see, so it does not look like an angle at all.
What helps. Stand up together and turn: a quarter turn to face the wall, a half turn to face the other way, then all the way around. Name each turn in degrees: 90, 180 and 360.
Your child judges an angle by the little arc drawn at its corner and says the angle with the bigger arc is bigger.
The arc only shows which angle is meant. Whoever drew it chose its size, and it says nothing about the turn.
What helps. Draw the same angle twice, once with a small arc and once with a big arc, and ask which turn is bigger. Then trace each turn with a finger, from one ray to the other.

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Liam is comparing two angles.
Angle P:
Angle Q:
Liam's claim: "Angle P is larger than Angle Q because the sides of Angle P are longer."
Select the statement that best describes Liam's claim.
A
Liam's claim is correct. Longer sides make an angle larger, so Angle P is larger than Angle Q.
B
Liam's claim is correct. Angle Q has shorter sides than Angle P, so Angle Q must be the smaller angle.
C
Liam's claim is incorrect. We cannot compare Angle P and Angle Q because their sides have different lengths.
D
Liam's claim is incorrect. Both angles have the same size opening, so they are the same size. Side length does not change angle size.
Show the answer and how to solve it
Liam's claim is incorrect. Angle P and Angle Q have the same opening, so they are the same size. The longer sides on Angle P do not make it bigger.
1. 1Look at Angle P and Angle Q. Angle P has long sides, while Angle Q has short sides. This is the visual feature Liam focuses on.
2. 2An angle's size depends on the opening between its two rays — not on how long the rays are drawn. You can extend the rays or shorten them, and the angle stays the same size.
3. 3Compare the openings, not the sides. Angle P and Angle Q have the same opening between their rays. So they are the same size, even though Angle P has longer sides.
4. 4Liam's claim is incorrect. He used side length to compare the angles, but side length does not change angle size. The correct way is to compare the openings between the rays.
Common mistake. A common mistake is thinking that longer sides mean a larger angle. Angle P and Angle Q have the same opening between their rays, so they are the same size.
The idea underneath. An angle's size is the amount of opening between two rays meeting at a point. The length of the rays doesn't matter — you can draw the rays as long or as short as you want without changing the angle's size.
2On level
Look at the reference angle below. Then sort each angle card into the correct bin based on whether its opening is larger or smaller than the reference angle.
Drag each item into the group where it belongs, or tap the item, then tap the group.
Larger opening than the reference
Drag or tap items here
Smaller opening than the reference
Drag or tap items here
Keyboard instructions: focus an item and press Enter or Space to pick it up. Move focus to a bin or the bank and press Enter or Space to drop it. Press Escape to cancel.
Show the answer and how to solve it
Cards A and C belong in the 'Larger opening' bin. Cards B and D belong in the 'Smaller opening' bin. The opening of an angle is measured by how far apart its two rays spread, not by which direction the rays point or how the card is rotated.
1. 1Look at the reference angle. Notice how wide the opening is between its two rays. To compare each card to the reference, focus only on the WIDTH of the opening — not which way the rays point or how the card is rotated.
2. 2Card A: This card is in the same direction as the reference, making it easy to compare directly. The two rays spread out much wider than the reference rays. Card A has a larger opening, so it goes in the 'Larger opening' bin.
3. 3Card B: This card is tilted so both rays point toward the upper right. Ignore the tilt and look only at how far apart the two rays are. The rays are very close together — much narrower than the reference. Card B has a smaller opening, so it goes in the 'Smaller opening' bin.
4. 4Card C: This card is rotated so one ray points left and one points downward. Ignore the rotation and compare only the spread between the rays. The rays are spread wider than the reference. Card C has a larger opening, so it goes in the 'Larger opening' bin.
5. 5Card D: Both rays point straight downward, very close together. Ignore the direction and focus only on the spread. The rays are much closer together than the reference rays. Card D has a smaller opening, so it goes in the 'Smaller opening' bin.
Common mistake. A common mistake is sorting Card B into the 'Larger opening' bin because it is tilted upward at a diagonal and may look prominent. But Card B has a very narrow opening — its two rays are close together — so it belongs in the 'Smaller opening' bin. Always look at the spread between the rays, not the rotation.
The idea underneath. An angle's size depends only on how far apart its two rays spread — not on which direction they point or how the angle is rotated. A wide opening always means a larger angle, no matter the orientation.
3On level
Liam is comparing two angles.
Angle A
Angle B
Liam's claim: "Angle B is bigger than Angle A because its rays open wider apart."
Select the statement that best describes Liam's comparison.
A
Liam's comparison is correct; Angle B does have the wider opening.
B
A protractor is needed to confirm which angle is bigger.
C
Liam picked the right angle, but the reason is not correct.
D
Liam is wrong; Angle A is the bigger angle.
Show the answer and how to solve it
Liam's comparison is correct. Angle B does have the wider opening, and saying that 'its rays open wider apart' is exactly the right reason for that conclusion.
1. 1Look carefully at Angle A. Both rays point upward, and the opening between them is relatively narrow.
2. 2Now look at Angle B. One ray points nearly straight to the right and the other points steeply upward. The opening between Angle B's rays is clearly wider than Angle A's opening.
3. 3The size of an angle is the size of its opening, which is how widely the rays spread apart. Liam's reason, 'its rays open wider apart,' is exactly the right way to compare angles. The direction the rays point does not change the size of the opening.
4. 4So Liam is correct. Angle B has the wider opening, and Liam's reason correctly identifies what makes it bigger.
Common mistake. A common mistake is looking for an error in Liam's work when the comparison is actually right. Some students incorrectly think Angle A must be bigger because its rays both point upward, but the direction the rays point does not determine which opening is wider.
The idea underneath. An angle's size is the size of its opening, which is how widely the rays spread apart. The direction the rays point and the length of the rays do not change the size of the angle.
4Stretch
Look at the angle below. About how many degrees is the angle? Type your estimate.
°
Show the answer and how to solve it
Answer. 35
The true measure of the angle is $35^{\circ}$ . Any reasonable estimate from $25^{\circ}$ to $45^{\circ}$ is accepted because estimates within $10^{\circ}$ of the true measure show good benchmark reasoning.
1. 1Look at the angle. To estimate its measure, first compare it to a right angle $(90^{\circ})$ . A right angle is the corner of a square — both rays are perpendicular.
2. 2The angle opens less than a right angle, so it is an acute angle. That means its measure is less than $90^{\circ}$ .
3. 3Now compare to benchmark angles. The angle is between $0^{\circ}$ and $45^{\circ}$ , and it looks quite close to $45^{\circ}$ but noticeably smaller. A reasonable estimate is about $35^{\circ}$ .
Common mistake. A common mistake is mixing up an acute angle with its larger partner and estimating around $145^{\circ}$ instead of $35^{\circ}$ . Always check first whether the angle is smaller than a right angle (acute) or larger (obtuse) before guessing a number.
The idea underneath. Estimating angles works best by comparing to benchmark measures: a right angle is $90^{\circ}$ , half of a right angle is $45^{\circ}$ , and a straight angle is $180^{\circ}$ . Decide which two benchmarks the angle sits between, then choose a number closer to whichever benchmark the angle looks closest to.