What it does
Degrees split a full turn into 360 parts, a number chosen thousands of years ago. A radian comes from the circle itself: it is the angle at the centre when the arc is exactly as long as the radius. Lay the radius along the circle like a bendy ruler and count: one radian, two, three … six, and a bit left over. A full turn is 2π ≈ 6.283 radians, so
π radians = 180°, 1 radian = 180/π ≈ 57.3°, 1° = π/180 ≈ 0.01745 radian.
Because the angle in radians counts “radii along the arc”, the arc length is simply s = r θ, and the sector area is ½ r² θ. That is why Class 11 maths (trigonometric functions) switches to radians, and why calculus later needs them.
For the sine, cosine and tangent of these angles, go on to the unit circle; the circle sectors tool does arcs and sectors in degrees for Class 10, and the compound angle proof proof adds two angles together.
How to use it
- Slide Angle θ from 0° to 360°, or type degrees, or type radians: a decimal such as 2.1, or with pi such as 3pi/4, π/6 or 2pi.
- Slide the Radius. The picture keeps the same size, but the numbers change.
- In the picture the arc is blue, the two radii orange, and a green mark sits every radius-length round the circle (1 to 6). The marks the arc has reached turn green, so you can read the radian measure straight off the circle.
- Wrap radii round the circle sweeps θ from 0° to 360° so you see six whole radii and the 0.28 left over.
- The box under the picture gives the angle in radians (as a multiple of π when there is one), the arc length, the number of radii along the arc and the sector area.
Worked example
The page opens at θ = 120° with r = 3 cm. The title reads 120° = 2π/3 ≈ 2.0944 radians. The box shows:
- arc length s = r θ = 3 × 2.0944 = 6.283 cm,
- radii along the arc: 2.094 (marks 1 and 2 are green, 3 is not yet reached),
- sector area ½ r² θ = ½ × 9 × 2.0944 = 9.425 cm².
Now type 3pi/4 in the radians box: the angle becomes 135°, and with r = 3 the arc is 7.069 cm.
Tips
- To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
- Learn a few by heart: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.
- When no unit is written after an angle in maths (sin 2, θ = 1.5), it is in radians.
FAQ
Why is a radian not a nice whole number of degrees? Because π is not a whole number. 180/π ≈ 57.2958° can never be written exactly as a fraction.
Does a bigger circle have bigger radians? No. The arc grows with the radius, but the ratio s ÷ r stays the same, so the angle in radians does not depend on the size of the circle.
Is anything stored? No. Everything runs in your browser. fibub is not part of NCERT; the topics follow the syllabus, but the words, pictures and examples are our own.
Where the numbers come from
- The radian, the conversion π radians = 180°, arc length s = rθ and sector area ½r²θ are standard mathematics. Wording, pictures and examples are fibub's own; no textbook text, figures or exercises are used.
- Checked against Python's math.radians and math.degrees, and sympy's nsimplify for the π forms of angles.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.