Circle theorems
Six theorems, each drawn on its own circle rather than written as a sentence. Find the picture that matches the question and the rule is underneath it.
Reference chart
Circle theorems
Angle at the center
Twice the angle at the edge
both stand on the same arc
Same segment
Same arc, same angle
move the point — it does not change
Angle in a semicircle
A diameter gives 90°
wherever the point sits
Cyclic quadrilateral
Opposite angles add to 180°
all four corners on the edge
Tangent and radius
A tangent meets it at 90°
touches once, never crosses
Perpendicular to a chord
It cuts the chord in half
the two ticks are equal
An angle is fixed by the arc it stands on, not by where its point sits. That is why the same arc gives the same angle from anywhere on the circle, and why a diameter — half the circle — always gives 90°.
What's on this chart
Parts of a circle names the pieces; this is what happens once two of them are in the same drawing. The angle at the center is twice the angle at the edge standing on the same arc, two points on the edge looking at the same arc give the same angle, and a diameter always makes a right angle at the circumference. Those three are one idea seen from three sides.
Why this chart helps
The other three cover the cases a question is most likely to be built from: a four-sided shape with every corner on the circle has opposite angles adding to 180°, a tangent meets the radius at 90° where it touches, and a line from the center that meets a chord square cuts it exactly in half — which is the usual way into finding a chord's length.
More printable charts
- Naming 2D shapes
Print Count the straight sides on a shape, then read its name below it. - Quadrilateral family
Print Follow the branches to see which names a shape inherits. - Lines of symmetry
Print Fold each shape in your mind and count the matching halves. - Types of angles
Print Find the picture that matches yours, and the name is under it. - Angles in parallel lines
Print Match each numbered pair to its position around the transversal. - Perimeter formulas
Print Follow the traced edge on the figure, then read the formula under it. - Area formulas
Print Match your shape to a figure and use the letters drawn on it. - Volume formulas
Print Find your solid, then multiply the base area by how tall it stands.