Circles, angle facts & constructions
Circles: radius, diameter, circumference, chord, disc as a set of points; angle facts (at a point, on a line, vertically opposite, parallel-line angles); angle sum of a triangle/polygon; perpendicular bisector & angle bisector; circumscribed circle; congruence & similarity; (KS3: Pythagoras, ruler-and-compass constructions).
A circle is one of the simplest shapes to draw and one of the richest to reason about. This concept gathers the language of circles, the small set of angle facts that let you work out unknown angles without measuring, and the two classic ruler-and-compass constructions.
What it means
A circle is the set of all points that sit exactly the same distance from one fixed centre point. That fixed distance is the radius. A straight line from one side of the circle to the other passing through the centre is the diameter — always twice the radius. The distance all the way around the edge is the circumference, and it is always about 3.14 × diameter; that number is π (pi). A chord is any straight line joining two points on the circle (a diameter is the longest chord). The filled-in circle — the circle plus everything inside it — is the disc.
Angle facts let you find angles by reasoning instead of measuring. An angle is the amount of turn between two lines meeting at a point, measured in degrees, where a full turn is 360°.
- Angles around a point add to
360°. - Angles on a straight line add to
180°. - When two lines cross, the angles opposite each other are equal (vertically opposite).
- When a straight line (a transversal) crosses two parallel lines: corresponding angles (same position at each crossing, an “F” shape) are equal, alternate angles (opposite sides of the transversal, a “Z” shape) are equal, and co-interior angles (same side, between the parallels, a “C” shape) add to
180°. - The three angles inside any triangle add to
180°; inside any four-sided shape they add to360°. In general, the interior angles of a polygon withnsides add to(n − 2) × 180°, because the polygon can be cut inton − 2triangles.
The distance between two points is simply the length of the straight segment joining them, measured with a ruler (or, on a coordinate grid, read off from the axes). The midpoint of a segment is the point exactly halfway along it — the same distance from each end. A perpendicular bisector of a line segment is the line that cuts it exactly in half at a right angle (90°), so it passes through that midpoint, and every point on it is equidistant from the two ends. An angle bisector is the line that splits an angle into two equal halves. Both are built with only a straightedge and a pair of compasses — no protractor: for a perpendicular bisector, open the compasses wider than half the segment and draw an arc from each end, then join the two crossing points. Draw the perpendicular bisector of each side of a triangle and the three lines meet at one point, which is the centre of the circumscribed circle that passes through all three corners.
Two shapes are congruent when they are identical in size and shape — one can be laid exactly on top of the other by sliding, turning or flipping. They are similar when they have the same shape but possibly different size — all matching angles are equal and all sides scale by the same factor (a photocopy enlargement is similar to the original). Finally, in any right-angled triangle the sides obey Pythagoras’ theorem: the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two, a² + b² = c².
Worked examples
Find the missing angle on a line. Two angles sit on a straight line, one is 115°.
missing = 180° − 115° = 65°Angles in a triangle. A triangle has angles 40° and 75°.
third = 180° − (40° + 75°) = 65°Vertically opposite. Two lines cross; one angle is 50°. The angle directly across is also 50°, and each neighbouring angle is 180° − 50° = 130°.
Angle sum of a polygon. A pentagon has 5 sides.
sum = (5 − 2) × 180° = 540°Pythagoras. A right-angled triangle has short sides 3 and 4.
hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5Circumference. A wheel has radius 10 cm, so diameter 20 cm.
circumference ≈ 3.14 × 20 = 62.8 cmThe generative-art connection
Circles are where numbers become pictures. Mark n equally spaced points around a circle, then for each point k draw a chord to point 2k (its times-table partner, wrapping around). Do this for every point and the envelope of chords traces a cardioid — a heart-shaped curve — purely from the two-times table. Change the multiplier to 3, 4 or 5 and new petal patterns bloom. The Coding Train’s Times-Tables Cardioid challenge builds exactly this. Because every chord is just “join point A to point B on a circle”, the mathematics of the circle literally is the drawing. Rotating one motif by a fixed angle, over and over, is the same move that turns a single shape into a symmetric mandala — see the dot-multiplier tool, where one dot fans into a radially symmetric starburst.
Common misconceptions
- Radius vs diameter. The diameter is twice the radius, not the same thing — halve the diameter before using a radius formula.
- π is not exactly 3.14. It is a never-ending decimal;
3.14is a handy rounding, not the true value. - Angle facts need the right setup. “Angles add to 180°” holds for angles on a straight line or inside a triangle — check the configuration matches before you subtract.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
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A circle has centre `O`. Name each of these and give the fact linking them: the distance from `O` to the edge, a straight line from edge to edge through `O`, and the distance all the way round the edge. If the radius is `6 cm`, what is the diameter, and roughly what is the circumference (use `π ≈ 3.14`)?
Answer
Radius = distance centre→edge; diameter = edge-to-edge through the centre = `2 × radius`; circumference = distance around = `π × diameter`. Diameter `= 2 × 6 = 12 cm`. Circumference `≈ 3.14 × 12 = 37.68 cm ≈ 37.7 cm`.
Art hook Draw a circle, then a radius, a diameter and a chord in three different hues. Add a slider for the radius: as it grows, live-print diameter (2r) and circumference (2πr) beside the shape so the child sees the numbers scale together with the drawing.
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Two straight lines cross at a point. One of the four angles made is `130°`. Work out the other three angles and say which fact you used for each.
Answer
Vertically opposite to `130°` is `130°`. The two neighbouring angles are on a straight line with it: `180° − 130° = 50°` each. So the four angles are `130°, 50°, 130°, 50°` (opposite angles equal; angles on a line sum to `180°`).
Art hook Draw an X of two crossing lines on Canvas. Colour opposite wedges the same hue (so two hue pairs). Let the user drag one line to change the crossing angle; the four printed angle values update live, always showing the two equal pairs and each adjacent pair summing to 180.
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A straight line crosses two parallel lines. At the top crossing, one angle measures `72°`. Find (a) the corresponding angle at the bottom crossing and (b) the co-interior angle (same side, between the parallels). Justify each.
Answer
(a) Corresponding angles are equal, so `72°`. (b) Co-interior angles add to `180°`, so `180° − 72° = 108°`.
Art hook Render two horizontal parallel lines and a slanted transversal. Highlight the F-shape (corresponding, equal hue) and the C-shape (co-interior, complementary hues). Drag the transversal's slope and watch the corresponding pair stay equal while co-interior pair always totals 180.
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Find the sum of the interior angles of a regular hexagon (`6` sides), then the size of one interior angle. Explain the triangle-cutting idea behind the sum.
Answer
A hexagon splits into `6 − 2 = 4` triangles, so the interior angles sum to `4 × 180° = 720°`. Regular means all `6` are equal: `720° ÷ 6 = 120°` each.
Art hook Draw a regular n-gon and fan lines from one vertex to split it into (n−2) triangles, each shaded a rotating hue. A slider changes n from 3 to 12; print the running sum (n−2)×180 and, when regular, the single interior angle.
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Describe, step by step, how to construct the perpendicular bisector of a line segment `AB` using only a straightedge and compasses (no protractor). What is special about every point on the line you draw?
Answer
Open the compasses wider than half of `AB`. With the point on `A`, draw an arc above and below the segment; keeping the same width, repeat from `B`. The arcs cross at two points; join them with the straightedge. This line cuts `AB` in half at `90°`, passing through the midpoint. Every point on it is the same distance from `A` as from `B` (equidistant from the two ends).
Art hook Animate the construction: two compass arcs sweep from A and B, their intersection points pop, then a line snaps through them. Let the user place A and B anywhere; the bisector redraws, and a moving dot on it shows equal distances to A and B in real time.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Match each word to its meaning: **radius**, **diameter**, **chord**, **circumference**. Meanings: (1) the distance all the way around the edge; (2) centre to edge; (3) any straight line joining two points on the circle; (4) edge to edge through the centre.
Answer
radius = (2); diameter = (4); chord = (3); circumference = (1). A diameter is also the longest possible chord.
Art hook On a single circle, let the user click two edge points to draw a chord; if the chord passes through the centre it snaps to a special colour and is labelled 'diameter'. Builds intuition that a diameter is the longest chord.
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A circle has diameter `18 cm`. What is its radius? A different circle has radius `7 mm`; what is its diameter?
Answer
Radius `= 18 ÷ 2 = 9 cm`. Diameter `= 2 × 7 = 14 mm`. (Diameter is always twice the radius.)
Art hook Two concentric circles: outer has a diameter line, inner shows the radius as exactly half of it, animated by folding the diameter onto itself so the halving is visible.
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Two angles sit together on a straight line. One is `43°`. What is the other?
Answer
`180° − 43° = 137°`. (Angles on a straight line add to `180°`.)
Art hook A horizontal line with a ray you can drag like a clock hand from 0° to 180°. The two angles it makes are shaded in two hues and their labels always add to 180.
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Three angles meet around a single point with no gaps. Two of them are `150°` and `90°`. Find the third.
Answer
`360° − (150° + 90°) = 360° − 240° = 120°`. (Angles around a point add to `360°`.)
Art hook A pie-slice fan around a centre point: drag the dividers to set the first two slice angles, and the third slice fills the remaining wedge, its size and angle label updating so all three always sum to 360.
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A triangle has two angles of `55°` and `55°`. Find the third angle. What kind of triangle is this?
Answer
Third `= 180° − (55° + 55°) = 70°`. Two equal angles mean two equal sides, so it is an **isosceles** triangle.
Art hook Draw a triangle whose three vertices you can drag. Colour equal angles the same hue; when two angles match, the two equal sides light up to reveal the isosceles symmetry, with an automatic mirror line.
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Find the sum of the interior angles of a quadrilateral (`4` sides), then of an octagon (`8` sides). Use the rule `(n − 2) × 180°`.
Answer
Quadrilateral: `(4 − 2) × 180° = 360°`. Octagon: `(8 − 2) × 180° = 6 × 180° = 1080°`.
Art hook Any-shape polygon builder: click to drop vertices, and as each new corner is added the app cuts the shape into triangles and tallies (n−2)×180 with a growing colour-coded triangle fan.
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A straight line crosses two parallel lines. One alternate angle (the 'Z' shape) is `65°`. What is the other alternate angle, and why?
Answer
`65°`. Alternate angles between parallel lines are equal (the 'Z' rule).
Art hook Highlight the Z-shape between two parallels; drag the transversal and watch the two Z-angles stay locked as equal, printed in matching colour, so the equality feels inevitable.
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**Spot the mistake.** Sam says: 'A triangle has angles `90°`, `60°` and `40°`, because a right angle plus these fits.' Is Sam right? Explain and give a correct third angle if the first two are `90°` and `60°`.
Answer
Sam is wrong: `90 + 60 + 40 = 190°`, but a triangle's angles must total `180°`. With `90°` and `60°`, the third is `180° − 150° = 30°`.
Art hook A 'does it close?' checker: the three angle rays try to join into a triangle; if they sum to over 180 the triangle can't close and a gap flashes red, if under 180 the sides overshoot — only exactly 180 snaps shut.
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Are these two shapes **congruent**, **similar**, or neither? Shape A is a square of side `4 cm`. Shape B is a square of side `8 cm`. Shape C is a square of side `4 cm` that has been rotated `45°`.
Answer
A and C are **congruent** (same size and shape; rotation doesn't change size). A and B are **similar** (same shape, all sides scaled by `2`, angles equal) but not congruent.
Art hook Show one motif, then two copies: one rotated (congruent, same size) and one enlarged ×2 (similar). A toggle overlays them so congruent pair matches exactly and the similar pair nests as a scaled echo — a self-similar zoom animation.
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Describe how to construct the **angle bisector** of an angle using only straightedge and compasses, and say what the bisector does to the angle.
Answer
Put the compass point on the vertex and draw an arc crossing both arms of the angle. From each of those two crossing points draw an arc of equal radius so they meet inside the angle. Join the vertex to that meeting point: this ray splits the angle into two equal halves.
Art hook Animated bisector: arcs sweep from the vertex and arms, the meeting point pops, and a ray splits the angle. The two half-angles are shaded mirror-image hues; drag an arm and the bisector re-centres to keep both halves equal.
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In a right-angled triangle the two shorter sides are `6 cm` and `8 cm`. Use Pythagoras `a² + b² = c²` to find the hypotenuse.
Answer
`c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm`.
Art hook Draw the triangle and literal squares on each side (6×6, 8×8, 10×10). Fill the two small squares' area of dots and animate them pouring into the big square to show 36 + 64 = 100 dots exactly.
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**Reason it out.** You draw the perpendicular bisector of each of the three sides of a triangle. What is special about where the three lines meet, and what circle can you draw from that point?
Answer
The three perpendicular bisectors all cross at a single point. That point is equidistant from all three corners, so you can draw one circle centred there passing through all three vertices — the **circumscribed circle** (circumcircle).
Art hook Draggable triangle: draw all three perpendicular bisectors live; they always meet at one point, and a circle snaps through the three corners. Moving a vertex slides the centre and re-fits the circumcircle in real time.