Grade 10 · Measure central-angle proportions · 546 of 750
Measure central-angle proportions · practice 59
A circle has radius 7. Find the area and arc length of its 75° sector. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A circle has radius 7. Find the area and arc length of its 75° sector.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
The sector uses the same angle fraction of the full area and circumference.
Work through the reasoning
Step 1
Translate the givens
The sector fraction is 75/360. The full disk area is π × 7² = 49π.
Step 2
Calculate with the model
Sector area = (75/360) × 49π ≈ 32.07043 square units.
Step 3
Check and interpret
Full circumference is 2π × 7 = 14π; its selected fraction gives arc length (75/360) × 14π ≈ 9.16298. Area and arc have different units.
The answer
Area = (75/360) × 7²π ≈ 32.07043 square units; arc = (75/360) × 2π × 7 ≈ 9.16298 length units.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
A sector takes the same fraction of a circle's area as its central angle takes of a full turn. Its curved boundary similarly takes that fraction of the circumference. Radius is a length; area uses its square, while arc length uses its first power. Starting quantities: Radius = 7; Final central angle = 75.
sector area = (θ/360)πr²; arc = (θ/360)2πr
03 · Reflect and transfer
Explain what changes and why.
If the radius doubles while the angle stays fixed, how do sector area and arc length change?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.