HSG.C.B.5: Arc Length, Radian Measure and Sector Area
In plain English: HSG.C.B.5 is the Common Core geometry standard that asks students to use similarity to show that, for a fixed central angle, arc length is proportional to the radius. The constant ratio arc length ÷ radius is defined as the angle's radian measure, and students derive the sector area formula A = ½r²θ. It is usually taught in Geometry.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Common Core State Standards for Mathematics · Domain: Circles (C) · Cluster: Find arc lengths and areas of sectors of circles Also written as HSG-C.B.5 or G-C.5 · Official standard
Students use similarity to explain a fact they may have only used before: the same central angle cuts off longer arcs in bigger circles, in exact proportion to the radius. Because the ratio arc length ÷ radius is the same in every circle for a given angle, it can serve as a measure of the angle itself. That ratio is the radian measure.
With radians defined, students derive the arc length formula s = rθ and the sector area formula A = ½r²θ, compare them with the degree forms, and apply them to arcs and sectors in real objects such as pizzas, clocks, fans and windshield wipers.
Learning Objectives
By the end of this lesson, students will be able to:
Explain with a dilation why the arc cut off by a fixed central angle is proportional to the radius
Define the radian measure of an angle as the ratio of arc length to radius, and convert between degrees and radians
Use s = rθ to find arc lengths, radii and angles
Derive the sector area formula A = ½r²θ from the fraction of the disk, and use it and its degree form to solve problems
Prior Knowledge Required
Students should already be comfortable with:
Circumference and area of a circle 7.G.B.4
All circles are similar, shown by translation and dilation HSG.C.A.1
Dilations multiply lengths by the scale factor and preserve angle measure HSG.SRT.A.1
Solving proportions and simplifying fractions that contain π
Draw three concentric circles on the board with a right angle at the center. Students work alone for three minutes, then compare with a partner.
Warm-Up Prompt
"A 90° central angle is drawn in circles of radius 2 cm, 4 cm and 10 cm. Find the length of the arc each circle cuts off, as a multiple of π. Then divide each arc length by its radius. What do you notice, and why might it happen?"
The arcs are a quarter of each circumference: π cm, 2π cm and 5π cm. Each ratio arc ÷ radius is π/2. Collect conjectures about why the ratio does not depend on the circle. Students usually say "bigger circle, bigger arc"; push them to say "proportionally bigger" and ask what geometric idea explains proportional lengths. The answer, similarity, is where the lesson starts.
Direct Instruction25 minutes
Part 1: The similarity argument. Use Diagram 1. Place two circles with radii r₁ and r₂ around the same center O, and draw one central angle that cuts arc s₁ from the small circle and arc s₂ from the large one. Walk through the derivation:
All circles are similar (HSG.C.A.1): the dilation centered at O with scale factor k = r₂/r₁ maps the small circle onto the large one.
The dilation keeps the angle: dilations preserve angle measure, and each side of the central angle is a ray from O, which the dilation maps onto itself. So the dilation maps arc s₁ onto arc s₂.
Lengths scale by k: a dilation multiplies every length by k, so s₂ = k · s₁ = (r₂/r₁) · s₁. For students who ask how a curve has a "length", approximate the arc by many short chords; each chord scales by k, so their total does too.
The ratio is fixed: rearranging gives s₁/r₁ = s₂/r₂. The arc length is proportional to the radius, and the constant of proportionality s/r depends only on the angle.
Define radian measure: the radian measure of the angle is θ = s/r, so s = rθ. A full turn cuts off the whole circumference 2πr, so a full turn is 2π radians: 360° = 2π rad and 180° = π rad.
Stress that θ = s/r is a ratio of two lengths, so the units cancel: a radian is a pure number. Part 2: The sector area. Use Diagram 2. A sector with angle θ is the same fraction of the disk as θ is of a full turn, so A = (θ/2π) · πr² = ½r²θ. With degrees, the fraction is n/360 and A = (n/360) · πr². Substituting s = rθ gives the companion form A = ½rs, which students can compare with the area of a triangle with base s and height r.
Arc length is proportional to radius
A 50° central angle is drawn in circles of radius 4 cm and 10 cm. Find both arc lengths and compare the ratio of the arcs with the ratio of the radii.
Equation: s₁ = 10π/9 cm, s₂ = 25π/9 cm; s₂/s₁ = 5/2 = 10/4, and s/r = 5π/18 in both circles
Degrees to radians, then arc length
Convert 135° to radians and find the arc it cuts from a circle of radius 8 m.
Equation: 135° = 135 · π/180 = 3π/4 rad; s = 8 · 3π/4 = 6π ≈ 18.85 m
Radian measure from arc and radius
An arc 15 cm long lies on a circle of radius 6 cm. What is the radian measure of its central angle?
Equation: θ = s/r = 15/6 = 2.5 rad ≈ 143.2°
Sector area in radians
Find the area of a sector with radius 9 in and central angle 2π/3.
Equation: A = ½ · 9² · 2π/3 = 27π ≈ 84.8 in²; check with A = ½rs: s = 6π, ½ · 9 · 6π = 27π
Sector area in context
A windshield wiper blade runs from 6 in to 24 in from its pivot and sweeps an angle of 110°. What area does it clean?
Pairs work three problems on whiteboards and hold them up after each one. (a) A 40° angle is drawn in circles of radius 9 cm and 27 cm. Find both arcs (2π cm and 6π cm) and explain, using the dilation, why the second is 3 times the first. (b) Convert 225° to radians (5π/4) and 7π/10 radians to degrees (126°). (c) Find the area of a sector with radius 5 cm and central angle 1.2 radians (15 cm²). Listen for these errors: using the diameter in place of the radius, putting θ in degrees into s = rθ, and forgetting the ½ in the sector formula.
Independent Practice15 minutes
Students work alone on four problems. (1) An arc 26 cm long lies on a circle of radius 8 cm; find the radian measure of its angle (3.25 rad). (2) A sector has radius 15 cm and angle 4π/5; find its arc length (12π cm) and area (90π cm²). (3) Find the area of a 240° sector of a circle with radius 3 m (6π m²). (4) A sector of radius 4 in has area 10 in²; find its angle in radians (1.25). For problem 3, students write one sentence explaining why the fraction 240/360 appears in the formula.
Closure5-10 minutes
Exit ticket: (1) In two sentences, explain why arc length divided by radius is the same for a given angle in every circle. (2) Convert 300° to radians (5π/3). (3) Find the area of a sector with radius 6 and angle π/4 (9π/2).
Differentiation Strategies
For Struggling Students
Start every problem by writing the fraction of the full turn (for example 90/360 = 1/4) before using any formula
Provide a two-column reference card: degree forms on the left, radian forms on the right, with one worked example in each
Use the concentric-circle drawing from Diagram 1 on a handout so students can see the two arcs while they compute
For Advanced Students
Ask students to explain why the chord approximation of an arc also scales by k under a dilation, and why that justifies scaling arc length
Ask for the angle, in radians, that makes a sector's area equal to the square of its radius (θ = 2)
Ask students to find the area of a circular segment (sector minus triangle) for a 90° angle in a circle of radius 6
Assessment Guidance
What to Look For
The standard says "derive", so check reasoning as well as answers. A complete explanation of proportional arcs names a dilation (or similar circles), says that it keeps the angle, and says that it multiplies arc length and radius by the same factor. For radians, look for the definition θ = s/r, not only the conversion factor π/180. For sectors, students should be able to explain where the fraction θ/(2π) or n/360 comes from. Watch for degrees placed into s = rθ, which is a sign that the definition has not taken hold.
02
Classroom Activities
3 Activities
1
Same Angle, Different Circles
20 minGroups of 3
Groups measure arcs cut by one central angle in three circles and find that the ratio arc ÷ radius is the same in all three. The measured constant becomes the radian measure of the angle.
Procedure
With a compass, draw three circles around one center with radii 3 cm, 5 cm and 8 cm. With a protractor, draw one 70° central angle through all three
Lay string along each intercepted arc, mark it, straighten it and measure it with a ruler
Record each arc length s and compute s/r. The predicted arcs are 7π/6 ≈ 3.67 cm, 35π/18 ≈ 6.11 cm and 28π/9 ≈ 9.77 cm, and each s/r is 7π/18 ≈ 1.22
Write a sentence that uses a dilation to explain why the three ratios agree
Discussion Questions
Your three ratios probably differed slightly. Is that a measuring issue or a mathematical one? How do you know?
If you drew a fourth circle of radius 12 cm, what arc would you predict before measuring?
What would the ratio be for a 140° angle? Why?
Modification for Distance Learning
Students build concentric circles in geometry software, measure the arc lengths with the software's arc tool, and share a table of s, r and s/r in a shared document.
2
Wrap a Radius
15 minPairs
Pairs cut a string as long as the radius of a paper plate and lay it around the rim again and again. The activity makes the radian a physical length and shows why a full turn is about 6.28 radians.
Procedure
Find the center of a paper plate by folding it in half twice. Cut a string equal to the radius
Starting at a marked point, lay the string along the rim and mark its end. Repeat until you return to the start. Count the full string lengths (6) and estimate the leftover part (about 0.28 of a string)
Draw rays from the center to the first two marks. The angle between them is 1 radian. Measure it with a protractor and compare with 180°/π ≈ 57.3°
Mark where half of the rim ends and count: about 3.14 radius lengths
Discussion Questions
Would a larger plate with a longer string give a different angle for one radian? Explain with similarity
Why is the count for a full turn 2π and not a whole number?
3
Sector Area Stations
20 minGroups of 3-4
Groups rotate through four stations. Each station builds one piece of the sector area formula, from fractions of a circle to the general formula A = ½r²θ.
The 4 Stations
Station 1, fractions: a paper circle of radius 10 cm is cut into 10 equal sectors. Find the angle of each (36°) and its area (10π ≈ 31.4 cm²)
Station 2, the proportion: complete A / (πr²) = θ / (2π) and solve it for A. Then write the degree version with n/360
Station 3, slices: cut a paper sector into thin slices and rearrange them, alternating point up and point down, into a shape close to a parallelogram with base about s/2 and height about r. Explain how this suggests A = ½rs
Station 4, context: a 16-inch pizza (radius 8 in) is cut into 6 equal slices. Find the area of one slice (32π/3 ≈ 33.5 in²) and the length of its crust edge (8π/3 ≈ 8.4 in)
Procedure
Spend 5 minutes at each station and record answers on a group sheet
After the rotation, each group presents one station and connects it to the formula A = ½r²θ
Challenge Variation
Groups find the area of a sector of a ring, such as the region between radii 5 cm and 9 cm inside a 1.5-radian angle (½ · 1.5 · (81 - 25) = 42 cm²), and explain why subtracting two sectors works.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: One Angle, Two Radii, Proportional Arcs
Drawn to scale: a 60° angle at O cuts arcs from circles of radius 3 and 6. The dilation centered at O with scale factor 2 maps the inner arc onto the outer one, so the outer arc is twice as long. The ratio s/r is π/3 in both circles; that constant is the radian measure of 60°.
Diagram 2: One Radian and the Area of a Sector
Drawn to scale: in a circle of radius 4, an angle of exactly 1 radian cuts an arc of length 4, equal to the radius. The shaded sector is 1/(2π) of the disk, so its area is ½ · 4² · 1 = 8 square units.
04
Homework Assignment
~30 min
HSG.C.B.5 Homework: Arc Length, Radians and Sectors
Directions: Show all work. Give exact answers in terms of π where possible, then a decimal rounded as stated. For each derivation, write complete sentences and name the property you use.
Part 1: Similarity and Radian Measure (Problems 1-3)
A 72° central angle is drawn in circles of radius 5 cm and 8 cm that share a center. (a) Find both arc lengths as multiples of π. (b) Show that the ratio of the arcs equals the ratio of the radii. (c) Find s/r for each circle and state the radian measure of 72°.
Write a short derivation, in complete sentences, that explains why a fixed central angle cuts off arcs whose lengths are proportional to the radii. Name the transformation you use and the property of it that you need. Then explain how this fact lets us define the radian measure of an angle.
Convert (a) 165° to radians, (b) 7π/4 radians to degrees, and (c) 2 radians to degrees, to the nearest tenth. (d) An arc of length 11 cm lies on a circle of radius 4 cm. Find the radian measure of its central angle.
Part 2: Sector Area (Problems 4-6)
Start from the proportion A / (πr²) = θ / (2π) and derive the formula A = ½r²θ. Then show that A = ½rs is equivalent. Use both formulas to find the area of a sector with radius 10 cm and central angle 3π/5.
A folding hand fan opens into a sector with radius 25 cm and an angle of 150°. Find the area of the fan to the nearest square centimeter and the length of its curved edge to the nearest tenth of a centimeter.
A sector has radius r and central angle θ. A second sector has radius 3r and central angle θ/2. How many times as long is the second arc as the first? How many times as large is the second area? Justify your answers with the formulas s = rθ and A = ½r²θ.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Similarity Argument
Names the dilation, angle preserved and lengths scaled by k
Proportion stated without full reason
No argument
Radian Measure
Defines θ = s/r and converts correctly
Converts correctly without the definition
Incorrect or missing
Sector Area
Formula derived from the fraction of the disk and applied correctly
Formula applied without derivation
Incorrect formula
Accuracy and Units
All answers correct with units and rounding
Minor errors in arithmetic or units
Many errors
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Choose an answer for each multiple-choice question and work the short-answer questions on paper before opening the solution. Your score updates as you go, and Reset quiz starts over.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
An 80° central angle cuts an arc of length 4π/3 from a circle of radius 3. What arc does the same angle cut from a circle of radius 12?
Answer: C
The circles are similar with scale factor 12/3 = 4, and a dilation multiplies arc lengths by 4: 4 · 4π/3 = 16π/3. Check: 12 · (80π/180) = 16π/3. Choice A assumes the arc does not change with the radius, and choice D multiplies by 4² = 16, the factor for areas, not lengths.
Question 2 of 20 · Multiple Choice
Why does a fixed central angle cut off arcs whose lengths are proportional to the radii?
Answer: A
This is the similarity argument of HSG.C.B.5: a dilation centered at O with factor k = r₂/r₁ maps one arc onto the other and scales its length by k. Choices C and D confuse the angle measure of an arc with its length: arcs with the same angle measure have different lengths in different circles.
Question 3 of 20 · Multiple Choice
What is the radian measure of a central angle?
Answer: D
The radian measure is the constant of proportionality θ = s/r, which is the same in every circle for a given angle. Choice C would give half the radian measure. Choice B gives the fraction of a full turn, which is θ/(2π), not θ.
Question 4 of 20 · Multiple Choice
An arc of length 21 cm lies on a circle of radius 7 cm. What is the radian measure of its central angle?
Answer: B
θ = s/r = 21/7 = 3 radians. Choice A divides r by s. Choice C multiplies s by r. Choice D attaches π, but a radian measure is π times something only when the angle is a simple fraction of a turn; here it is the plain number 3.
Question 5 of 20 · Multiple Choice
Convert 210° to radians.
Answer: B
Multiply by π/180: 210 · π/180 = 7π/6. Choice A divides by 360 instead of 180. Choice C inverts the fraction.
Question 6 of 20 · Multiple Choice
Convert 5π/12 radians to degrees.
Answer: D
Multiply by 180/π: (5π/12) · (180/π) = 75°. Choice A multiplies by 360/π instead of 180/π, and choice C multiplies by 90/π.
Question 7 of 20 · Multiple Choice
A circle has radius 10 cm. How long is the arc cut off by a central angle of 0.8 radians?
Answer: A
With θ in radians, s = rθ = 10 · 0.8 = 8 cm. Choice B divides r by θ. Choice C adds a factor of π that belongs only in degree conversions.
Question 8 of 20 · Multiple Choice
Find the length of the arc cut off by a 100° central angle in a circle of radius 18 m.
Answer: C
Convert first: 100° = 5π/9 rad, so s = 18 · 5π/9 = 10π m ≈ 31.4 m. Choice A uses s = rθ with θ in degrees, a common error. Choice D uses the diameter 36 m in place of the radius.
Question 9 of 20 · Multiple Choice
Find the area of a sector with radius 6 cm and central angle 5π/6.
Answer: D
A = ½r²θ = ½ · 36 · 5π/6 = 15π cm². Choice B forgets the ½. Choice A is the arc length s = rθ = 6 · 5π/6 = 5π, a length rather than an area, and choice C doubles that arc length.
Question 10 of 20 · Multiple Choice
Find the area of a 50° sector of a circle with radius 12 in.
Answer: A
The sector is 50/360 of the disk: A = (50/360) · π · 12² = 20π in² ≈ 62.8 in². Choice B is the arc length 12 · 5π/18 = 10π/3, a length rather than an area. Choice C divides by 180 instead of 360, and choice D uses ½r²θ with θ = 50 in degrees.
Question 11 of 20 · Multiple Choice
Which step derives the sector formula A = ½r²θ?
Answer: B
Area of a sector is proportional to its angle, so A / (πr²) = θ / (2π), and simplifying gives A = ½r²θ. Choice C gives 2πrθ, a length times an angle, not an area. Choice D mixes a length with an angle.
Question 12 of 20 · Multiple Choice
The radius of a sector is doubled and its central angle stays the same. What happens to its arc length and its area?
Answer: C
The two sectors are similar with scale factor 2. From s = rθ, the arc length is multiplied by 2; from A = ½r²θ, the area is multiplied by 2² = 4. Choice A treats area like length.
Question 13 of 20 · Multiple Choice
Why is a radian called a unitless measure?
Answer: A
Arc length and radius are both measured in the same unit, so their ratio has no unit. Choice B is false: 2.5 radians is a valid angle with no π. Choice C is false: the definition works in every circle because of similarity.
Question 14 of 20 · Multiple Choice
A sector has radius 8 cm and arc length 5 cm. What is its area?
Answer: D
A = ½rs = ½ · 8 · 5 = 20 cm². Check: θ = 5/8, so ½ · 64 · 5/8 = 20. Choice A forgets the ½, and choice B adds a π that does not belong when s is already a length.
Question 15 of 20 · Short Answer
A 14-inch pizza (diameter 14 in) is cut into 8 equal slices. Find the length of the crust edge of one slice and the area of one slice, to the nearest hundredth.
r = 7 in and each slice has θ = 2π/8 = π/4. Crust edge: s = rθ = 7π/4 ≈ 5.50 in. Area: A = ½ · 49 · π/4 = 49π/8 ≈ 19.24 in². The radius is half the diameter, so using 14 as the radius would give answers 2 and 4 times too large.
Question 16 of 20 · Short Answer
Two circles share a center O and have radii 5 cm and 15 cm. A central angle cuts an arc of length 4 cm from the smaller circle. Find the arc it cuts from the larger circle and explain your reasoning with similarity.
The dilation centered at O with scale factor 15/5 = 3 maps the small circle onto the large one and keeps the angle, so it maps the small arc onto the large arc and multiplies its length by 3: 12 cm. The ratio s/r is 4/5 = 12/15 = 0.8 in both circles, the radian measure of the angle.
Question 17 of 20 · Short Answer
A sector of a circle with radius 12 cm has area 24π cm². Find its central angle in radians and in degrees.
Use A = ½r²θ: 24π = ½ · 144 · θ = 72θ, so θ = 24π/72 = π/3 radians = 60°. Check: (60/360) · π · 144 = 24π.
Question 18 of 20 · Short Answer
Use the definition θ = s/r to explain how many degrees are in 3 radians, to the nearest tenth.
A full turn cuts off the whole circumference 2πr, so its radian measure is 2πr/r = 2π, and 2π radians = 360°. Then 1 radian = 180°/π, and 3 radians = 3 · 180°/π = 540°/π ≈ 171.9°.
Question 19 of 20 · Short Answer
The minute hand of a wall clock is 15 cm long. How far does its tip travel in 20 minutes? Give an exact answer and a decimal.
In 20 minutes the hand turns 20/60 of a full turn: θ = 2π/3 radians. s = rθ = 15 · 2π/3 = 10π ≈ 31.4 cm.
Question 20 of 20 · Short Answer
A lawn sprinkler sprays water up to 12 m and rotates back and forth through a 150° angle. What area of lawn does it water, to the nearest square meter?
The watered region is a sector: A = (150/360) · π · 12² = 60π ≈ 188 m². In radians, 150° = 5π/6 and ½ · 144 · 5π/6 = 60π, the same result.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSG.C.B.5 mean?
HSG.C.B.5 means students can explain, using similar circles, why the arc cut off by a given angle grows in proportion to the radius, and can use that fact to define radian measure as arc length divided by radius. It also asks students to derive the sector area formula, A = ½r²θ in radians or (n/360)πr² in degrees.
Is HSG.C.B.5 taught in Geometry or Algebra 2?
HSG.C.B.5 is usually taught in Geometry, in the circles unit. Radian measure then returns in Algebra 2 or Precalculus with the unit circle and trigonometric functions (HSF.TF.A.1), so a strong Geometry treatment makes that later work easier.
Why do we need similarity to define radians?
Because the definition θ = s/r only makes sense if the ratio is the same in every circle. Similarity guarantees it: the dilation that maps one circle onto another keeps the angle and scales the arc and the radius by the same factor, so their ratio does not change. Without that fact, the "radian measure" of an angle would depend on which circle you drew.
What is the difference between arc measure and arc length?
Arc measure is an angle, in degrees or radians, and is the same for every circle with that central angle. Arc length is a distance, in centimeters or inches, and grows with the radius. A 60° arc has measure 60° in every circle, but its length is π in a circle of radius 3 and 2π in a circle of radius 6.
How do you find the area of a sector?
Multiply the area of the whole circle by the fraction of a full turn the sector covers. In degrees, A = (n/360) · πr². In radians, A = (θ/2π) · πr² = ½r²θ. If you know the arc length s, you can also use A = ½rs.
What mistakes do students make with arc length and sector area?
Common errors are using degrees in s = rθ or A = ½r²θ, using the diameter in place of the radius, forgetting the ½ in the sector formula, and confusing arc measure with arc length. Another frequent slip is scaling area by k instead of k² when the radius changes by a factor k.
Why is a full circle 2π radians?
A full turn cuts off the entire circumference, which is 2πr. Its radian measure is therefore 2πr/r = 2π. That is also why 180° = π radians and 1 radian = 180°/π, about 57.3°.
Is a radian a unit like a degree?
A radian is a ratio of two lengths, so it has no unit in the physical sense: centimeters divided by centimeters. It is still labeled "rad" to avoid confusion with degrees. This is why s = rθ gives an answer in the same unit as r.
How is arc length and sector area tested?
Tests usually ask students to find arc lengths and sector areas from degrees or radians, to convert between the two, and to solve for a missing radius or angle. Because the standard says "derive", teachers should also ask for the similarity explanation and the derivation of the sector formula in words. The digital SAT's Geometry and Trigonometry domain includes circle questions of this kind, such as arc length, sector area and radian measure.
How does HSG.C.B.5 connect to trigonometry?
Trigonometric functions of real numbers use radian measure: on the unit circle, where r = 1, the radian measure of an angle equals the length of its arc (HSF.TF.A.1). That is what lets sine and cosine be graphed as functions of a real number. Angular speed and the arc length formulas in physics also use radians.
07
Related Standards
6 standards
These standards connect to HSG.C.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.G.B.4Prerequisite
Know and use the area and circumference formulas of a circle
