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Grade 10 · Circles

Design challenge: Measure central-angle proportions

Design challenge investigation: measure central-angle proportions. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Design challenge: Measure central-angle proportions. Current sector angle: 0°. Current sector area: 0. Current arc length: 0. Whole circumference: 34.558Sector fraction = angle / 360°0°r = 5.5Sector area: 0 · arc: 0
Angles are measured in degrees from 0 to 360. The circle uses one length unit and is scaled to fit; numeric radius changes area and circumference even when the displayed outline is fitted to the panel.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Current sector angle
0°
Current sector area
0
Current arc length
0
Whole circumference
34.558

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

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. This investigation begins with radius = 5.5; final central angle = 195. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

sector area = (θ/360)πr²; arc = (θ/360)2πr

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Identify the radius and distinguish it from the full diameter across the circle.

  2. STEP 2

    Follow the changing model

    Watch the central angle grow to the selected final value. The highlighted sector grows with the angle fraction.

  3. STEP 3

    Check and explain the relationship

    Apply the same fraction θ/360 to πr² for area and to 2πr for arc length, keeping the units different.

Your turn to explain

Make a prediction. Test your reasoning.

Design challenge: Find the area of a 195° sector of radius 5.5.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

16.385π ≈ 51.476 square units. The matching arc is 18.719 length units.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 10 · Geometry

Use the same turn fraction for arc and sector

Find arc length and sector area from a central angle in degrees.