Geometry

Parts of a circle

Explore radii, chords, arcs and regions.

Predict, test and explain

Step 1 of 3

Predict

Keep radius 2 fixed. If a sector angle grows from 90° to 180°, what happens to its area?

r=4 cmr=4\,\text{cm}
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CircumferenceC=2πr≈25.13 cmC=2\pi r\approx 25.13\,\text{cm}AreaA=πr2≈50.27 cm2A=\pi r^2\approx 50.27\,\text{cm}^2

InvestigateWhat happens to arc length when you double the angle?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

A sector has radius 6 cm and central angle 60 degrees. Find its area exactly.

Hint

The sector occupies one sixth of the whole circle.

Reveal answer

Take one sixth of the circle's area.

A=60360π(62)=6π cm2A=\frac{60}{360}\pi(6^2)=6\pi\,\text{cm}^2

Build an explanation

Hint 1

A radius runs from the centre to the circle; a diameter is twice as long.

Hint 2

For an arc or sector, compare its angle with the full 360° turn.

Hint 3

Multiply circumference by θ/360 for arc length, or circle area by θ/360 for sector area.

Worked example

This example uses fixed values, separate from the diagram controls.

Find the area of a 90° sector in a circle of radius 4 cm.

  1. The sector is 90/360 = 1/4 of the circle.
  2. The full area is π × 4² = 16π cm².
  3. Take one quarter.

Answer: 4π cm², approximately 12.57 cm².

Connect this idea

Watch out: A sector uses two radii and an arc; a segment uses a chord and an arc.