What is the unit circle used for?

The unit circle is a circle of radius 1 centered at the origin. For any angle θ, the point where its ray meets the circle is (cos θ, sin θ), so the circle lets you read exact sine and cosine values, understand their signs by quadrant, and connect degrees to radians. It is the foundation for trigonometry, graphing sine and cosine, and much of calculus.

How do you convert between degrees and radians?

Use the fact that 180° equals π radians. To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. For example, 30° × π/180 = π/6, 45° becomes π/4, 60° becomes π/3, and 90° becomes π/2. This tool shows both measures at once so you can compare them directly.

What are the coordinates on the unit circle?

Each point is (cos θ, sin θ). The only magnitudes that appear are 0, 1/2, √2/2, √3/2, and 1. For instance 30° is (√3/2, 1/2), 45° is (√2/2, √2/2), and 60° is (1/2, √3/2). The signs depend on the quadrant: cosine is negative in Quadrants II and III, and sine is negative in Quadrants III and IV.

What are the 16 standard angles on the unit circle?

The standard reference angles are the multiples of 30° and 45°: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°. In radians these are 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4, and 11π/6.

Can I make a blank unit circle to practice on?

Yes. Turn off degrees, radians, and coordinates and you get a blank circle with just the angle marks — a printable worksheet students can fill in from memory. You can also show only degrees, only radians, or only coordinates to quiz one skill at a time.

Can I download or print the unit circle?

Yes. The Precise circle mode exports a clean SVG or a high-resolution PNG (rendered at 2x) that drops neatly into worksheets, slides, and study guides. Precise mode runs entirely in your browser and does not use any image-generation credits.

Math Tool

Unit Circle Generator with Radians, Degrees & Coordinates

Create a clean, printable unit circle chart with the standard angles labeled in degrees and radians and their exact (cos, sin) coordinates. Toggle degrees, radians, and coordinates, highlight an angle with its reference triangle, shade a quadrant, and export SVG or PNG. Deterministic precise mode plus an AI sketch mode.

All 16 standard angles labeledDegrees, radians & (cos, sin) togglesHighlight an angle’s reference triangleSVG & PNG export

LabelsShow degrees (30°, 45°…)Show radians (π/6, π/4…)Show (cos, sin) coordinates

Angle set

Highlight an angle (reference triangle)

Shade a quadrant

Unit Circle Examples

Common unit-circle reference layouts for degrees, radians, and coordinates

View:

Complete Unit Circle

The full reference chart: degrees, radians, and (cos, sin) at every standard angle.

completereference

Unit Circle in Radians

Every standard angle labeled as an exact fraction of π around the circle.

radians

Unit Circle in Degrees

The standard angles marked in degrees for a simple, uncluttered reference.

degrees

Unit Circle Coordinates

The exact (cos θ, sin θ) point where each terminal ray meets the circle.

coordinatescos-sin

Unit Circle Quadrants

The four quadrants highlighted to teach the sign pattern of cos and sin.

quadrants

Blank Unit Circle Worksheet

A printable blank circle for students to fill in the angles and coordinates.

blankworksheet

What is the unit circle?

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. It is one of the most useful pictures in trigonometry because it turns angles into points: for any angle θ measured counter-clockwise from the positive x-axis, the point where the terminal ray meets the circle has coordinates exactly (cos θ, sin θ). Because the radius is 1, the horizontal distance to that point is the cosine of the angle and the vertical distance is the sine, so the circle lets you read off sine and cosine values directly. The standard reference chart marks sixteen common angles — the multiples of 30° and 45° — and labels each one with its degree measure, its radian measure, and its exact coordinates.

Degrees vs radians

Angles on the unit circle can be measured two ways. Degrees split a full turn into 360 equal parts, so a right angle is 90° and a full circle is 360°. Radians instead measure the arc length along a radius-1 circle, so a full turn is 2π radians and a right angle is π/2.
The two systems are linked by 180° = π radians. To convert degrees to radians, multiply by π/180; to go the other way, multiply by 180/π. That is why 30° becomes π/6, 45° becomes π/4, and 60° becomes π/3.
In the tool you can show degrees, radians, or both at once. Seeing them side by side is the fastest way to build the mental map between the two systems that trigonometry and calculus both rely on.

The (cos, sin) coordinates

At each angle, the point on the unit circle is (cos θ, sin θ). Reading the coordinates off the chart gives you exact trig values without a calculator: at 30° the point is (√3/2, 1/2), so cos 30° = √3/2 and sin 30° = 1/2.
The signs of the coordinates follow the quadrant. In Quadrant I both are positive; in Quadrant II the x-coordinate (cosine) turns negative; in Quadrant III both are negative; and in Quadrant IV the y-coordinate (sine) is negative while cosine is positive again.
Only three magnitudes ever appear — 1/2, √2/2, and √3/2 — plus 0 and 1 on the axes. Once you know those values and the sign rule, you can reconstruct every coordinate on the circle.

Memorizing the standard angles

A common trick for the first quadrant is to write sin as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. That simplifies to 0, 1/2, √2/2, √3/2, 1 — the sine values in order. Cosine is the same list reversed.
For the other quadrants, find the reference angle (the acute angle to the nearest x-axis), read its coordinates, then apply the quadrant sign rule for cosine and sine.
Use the highlight-angle option in the tool to draw one angle’s reference triangle. Seeing the right triangle drop from the point to the x-axis makes the connection between the circle and the familiar 30-60-90 and 45-45-90 triangles obvious.

Tips for a clean, printable chart

Turn off the labels you do not need. A degrees-only or radians-only circle is far easier to read for a quiz or a first lesson than the fully loaded chart.
Switch to the quadrantal set (0°, 90°, 180°, 270°) when you only want to teach the axis values and the sign pattern, without the diagonal angles crowding the figure.
Export SVG for crisp printing at any size, or PNG (rendered at 2x) for slides and handouts. Precise mode runs entirely in your browser and uses no image-generation credits, so you can print a whole class set of blank worksheets for free.

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Unit Circle Generator with Radians, Degrees & Coordinates

All 16 standard angles labeledDegrees, radians & (cos, sin) togglesHighlight an angle’s reference triangleSVG & PNG export

What is the unit circle?

Degrees vs radians

Angles on the unit circle can be measured two ways. Degrees split a full turn into 360 equal parts, so a right angle is 90° and a full circle is 360°. Radians instead measure the arc length along a radius-1 circle, so a full turn is 2π radians and a right angle is π/2.

The two systems are linked by 180° = π radians. To convert degrees to radians, multiply by π/180; to go the other way, multiply by 180/π. That is why 30° becomes π/6, 45° becomes π/4, and 60° becomes π/3.

In the tool you can show degrees, radians, or both at once. Seeing them side by side is the fastest way to build the mental map between the two systems that trigonometry and calculus both rely on.

The (cos, sin) coordinates

At each angle, the point on the unit circle is (cos θ, sin θ). Reading the coordinates off the chart gives you exact trig values without a calculator: at 30° the point is (√3/2, 1/2), so cos 30° = √3/2 and sin 30° = 1/2.

The signs of the coordinates follow the quadrant. In Quadrant I both are positive; in Quadrant II the x-coordinate (cosine) turns negative; in Quadrant III both are negative; and in Quadrant IV the y-coordinate (sine) is negative while cosine is positive again.

Only three magnitudes ever appear — 1/2, √2/2, and √3/2 — plus 0 and 1 on the axes. Once you know those values and the sign rule, you can reconstruct every coordinate on the circle.

Memorizing the standard angles

A common trick for the first quadrant is to write sin as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. That simplifies to 0, 1/2, √2/2, √3/2, 1 — the sine values in order. Cosine is the same list reversed.

For the other quadrants, find the reference angle (the acute angle to the nearest x-axis), read its coordinates, then apply the quadrant sign rule for cosine and sine.

Use the highlight-angle option in the tool to draw one angle’s reference triangle. Seeing the right triangle drop from the point to the x-axis makes the connection between the circle and the familiar 30-60-90 and 45-45-90 triangles obvious.

Tips for a clean, printable chart

Turn off the labels you do not need. A degrees-only or radians-only circle is far easier to read for a quiz or a first lesson than the fully loaded chart.

Switch to the quadrantal set (0°, 90°, 180°, 270°) when you only want to teach the axis values and the sign pattern, without the diagonal angles crowding the figure.

Export SVG for crisp printing at any size, or PNG (rendered at 2x) for slides and handouts. Precise mode runs entirely in your browser and uses no image-generation credits, so you can print a whole class set of blank worksheets for free.