Exercises: Derive arc length formula - Common Core Math - HS Geometry

A

Recall / Warm-Up

1.

All circles are similar. Which transformation maps any circle onto any other circle of different radius?

A.

A reflection across the x-axis

B.

A translation followed by a dilation

C.

A rotation of 90°

D.

A horizontal stretch

2.

A central angle of 60° is drawn in a circle with radius 3 cm and in a circle with radius 9 cm. Which statement is true about the two arcs?

A.

Both arcs have the same length because the angle is the same

B.

The arc in the larger circle is three times as long as the arc in the smaller circle

C.

The arc in the larger circle is nine times as long

D.

Arc length depends only on the central angle, not the radius

3.

The circumference of a circle with radius $r$ is $C = 2\pi r$ . What fraction of the circumference is cut off by a central angle of 120°?

A.

$\dfrac{1}{4}$

B.

$\dfrac{1}{3}$

C.

$\dfrac{1}{2}$

D.

$\dfrac{2}{3}$

B

Fluency Practice

1.

Find the arc length $s$ of a circle with radius $r = 6$ cm and central angle $\theta = \dfrac{\pi}{3}$ radians. Express your answer in terms of $\pi$ .

2.

A circle has radius $r = 10$ m and a central angle of $\theta = \dfrac{\pi}{4}$ radians. What is the arc length $s$ ? Express your answer in terms of $\pi$ .

3.

A central angle of $\theta = \dfrac{2\pi}{3}$ radians is drawn in a circle of radius $r = 8$ cm. Determine the arc length $s$ in terms of $\pi$ .

4.

Find the area $A$ of a sector with radius $r = 4$ m and central angle $\theta = \dfrac{\pi}{2}$ radians. Express your answer in terms of $\pi$ .

5.

A sector has radius $r = 6$ cm and central angle $\theta = \dfrac{\pi}{3}$ radians. What is the sector area $A$ ? Express your answer in terms of $\pi$ .

C

Varied Practice

$A circle with center O, radius 9 cm, and a highlighted arc of length 6π cm showing the central angle theta$

1.

In a circle with radius $r = 9$ cm, a central angle intercepts an arc of length $s = 6\pi$ cm. Find the central angle $\theta$ in radians, then convert to degrees.

$A circle showing a 60-degree sector with radius 5 cm and arc length s to be found$

2.

A circle has radius $r = 5$ cm and central angle $\theta = 60\degree$ . First convert $60\degree$ to radians, then find the arc length $s$ . Express your answer in terms of $\pi$ .

3.

Explain in your own words why radian measure is not arbitrary — that is, why it arises naturally from the geometry of circles. Reference the similarity of circles in your explanation.

4.

A sector has radius $r = 10$ cm and central angle $\theta = \dfrac{\pi}{4}$ radians. Which expression gives the area of the sector?

A.

$10 \cdot \dfrac{\pi}{4}$

B.

$\dfrac{1}{2} \cdot 100 \cdot \dfrac{\pi}{4}$

C.

$\dfrac{\pi/4}{2\pi} \cdot 2\pi \cdot 10$

D.

$\pi \cdot 100 \cdot \dfrac{\pi}{4}$

D

Word Problems

$A Ferris wheel diagram showing a 5π/6 radian sector with radius 20 m and arc length s to be found$

1.

A Ferris wheel has a radius of 20 meters. A passenger travels along an arc that subtends a central angle of $\dfrac{5\pi}{6}$ radians.

How far does the passenger travel along the arc? Express your answer in terms of $\pi$ .

$A pizza divided into 6 equal slices with one slice highlighted, showing a diameter of 16 inches$

2.

A circular pizza has a diameter of 16 inches. It is cut into 6 equal slices.

1.

What is the arc length of the curved crust on one slice? Express your answer in terms of $\pi$ .

2.

What is the area of one pizza slice? Express your answer in terms of $\pi$ .

E

Error Analysis

1.

Diego solved this problem: "A circle has radius 5 cm and central angle 60°. Find the arc length."

Diego's work:
$s = r\theta = 5 \times 60 = 300 \text{ cm}$

What mistake did Diego make?

A.

Diego used the wrong formula — he should have used $A = \frac{1}{2}r^2\theta$

B.

Diego used the angle in degrees directly in the formula $s = r\theta$ , which requires radians

C.

Diego squared the radius when he should not have

D.

Diego forgot to multiply by $\pi$

2.

Priya solved this problem: "A sector has radius 6 m and central angle $\frac{\pi}{3}$ radians. Find the area of the sector."

Priya's work:
$A = r\theta = 6 \cdot \frac{\pi}{3} = 2\pi \text{ m}^2$

What mistake did Priya make?

A.

Priya forgot to convert the angle to degrees

B.

Priya used the arc length formula $s = r\theta$ instead of the sector area formula $A = \frac{1}{2}r^2\theta$

C.

Priya used the fraction-of-circle method with the wrong full value: she multiplied by the circumference $2\pi r$ instead of the area $\pi r^2$

D.

Priya's formula is correct — she just made an arithmetic error

F

Challenge / Extension

1.

A circle has radius $r = 9$ cm and central angle $\theta = \dfrac{4\pi}{3}$ radians. Find the arc length $s$ . Express your answer in terms of $\pi$ .

2.