Geometry Arc Length and Sector Area Test Solutions and Scoring (PDF)

Geometry math test solutions: Arc Length and Sector Area – Maths PDF

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This arcs sectors test answers for Geometry covers arc length and sector area (Common Core G-C.B.5), with a free PDF to print.

Use this answer key and scoring guide for the arc length and sectors test. Each solution shows the full work, and the scoring note tells how the points are split between method and final answer. Partial credit is listed so that grading stays consistent from one paper to the next.

Geometry · Common Core State Standards: G-C.B.5

Arcs sectors test answers: worked solutions

Question 1 – Arc length [2 pts]

Arc = 6π.

\(s = \frac{72}{360} \cdot 2\pi \cdot 15 = 6\pi\) = 18.85.

6π, about 18.85

Scoring: 1 point for the setup, 1 point for the answer.

Question 2 – Sector area [2 pts]

\(\frac{150}{360} \cdot \pi \cdot 12^{2} = 60\pi\)

60π

Scoring: 1 point for the fraction, 1 point for the area.

Question 3 – Radians [3 pts]

\(225 \cdot \frac{\pi}{180} = \frac{5\pi}{4}\)
\(\frac{7\pi}{6} \cdot \frac{180}{\pi} = 210^{\circ}\)
\(s = 9 \cdot \frac{2\pi}{3} = 6\pi\)

(a) 5π/4; (b) 210°; (c) 6π

Scoring: 1 point per part.

Question 4 – Find the radius [2 pts]

\(\frac{45}{360}\pi r^{2} = 8\pi\)

\(r^{2} = 64\), so \(r = 8\).

r = 8

Scoring: 1 point for the equation, 1 point for r.

Question 5 – Central angle [2 pts]

\(\text{theta} = \frac{s}{r} = \frac{5\pi}{12}\) radians.

\(\frac{5\pi}{12} \cdot \frac{180}{\pi} = 75^{\circ}\).

5π/12 rad = 75°

Scoring: 1 point each.

Question 6 – Sector perimeter [2 pts]

Arc: \(\frac{60}{360} \cdot 2\pi \cdot 9 = 3\pi\) = 9.4.

Perimeter: \(9 + 9 + 3\pi\) = 27.4.

27.4 units

Scoring: 1 point for the arc, 1 point for the perimeter.

Question 7 – Pizza slices [2 pts]

Radius 9; each slice is \(\frac{1}{6}\) of the circle.

\(\frac{1}{6} \cdot \pi \cdot 9^{2} = 13.5\pi\) = 42.4.

42.4 square inches

Scoring: 1 point for the fraction, 1 point for the area.

Question 8 – Wheel turn [2 pts]

\(\text{theta} = \frac{s}{r} = \frac{70}{14} = 5\) radians.

\(\frac{5}{2\pi} = 0.8\) turns.

5 radians; 0.8 turns

Scoring: 1 point each.

Question 9 – Lawn [3 pts]

Garden bed.

Area: \(\frac{135}{360} \cdot \pi \cdot 20^{2} = 150\pi\) = 471.2 sq ft.

Arc: \(\frac{135}{360} \cdot 2\pi \cdot 20 = 15\pi\) = 47.1 ft.

471.2 sq ft; 47.1 ft

Scoring: 1 point for each setup, 1 point for both decimals.

Total score

Total

20 points

Key facts behind this arcs sectors test answers

Key facts

Arc length = (angle/360) x 2 pi r.
Sector area = (angle/360) x pi r squared.
180 degrees = pi radians.
In radians: arc length s = r times theta, area A = (1/2) r squared theta.
A radian is the central angle whose arc length equals the radius.

How to use this arcs sectors test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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