Recognise and name polygons from marked properties
Name straight-sided polygons from their boundary and marked properties, including all six named quadrilateral families and irregular five-, six- and eight-sided examples.
Section 4.2 · Polygons
Exam scope and assessment notes
Foundation and Higher share all seven section4.2 requirements A–G. The Higher table says See Foundation Tier; there are no additional H4.2 rows. These seven original lessons teach polygon names, convex/concave quadrilateral sums, all six named quadrilateral families, general polygon sums, regular interior/exterior angles and congruence correspondence. F:4.2C requires both the parallelogram/rectangle/rhombus/square lesson and the trapezium/kite lesson with substantive teaching, a worked example and every Guided, Independent and Transfer stage. F and G are two distinct published rows; the same substantive congruence lesson links both without inventing a third requirement. Seven lessons and28 selected seed cases are StudyForA’s design, not a syllabus-prescribed count. Rhombus/square diagonal endpoint-angle bisection, kite symmetry-diagonal endpoint-angle bisection and isosceles-trapezium equal diagonals are conservative standard-property breadth under the broad C wording, not new verbatim syllabus bullets or an assertion that the public book chart enumerates each one. Use labelled boundary order, actual interior/reflex arcs, given or justified parallel/equal-side/right-angle marks and geometric reasons before equations. Diagrams need not be to scale; do not measure them unless instructed. Use the local exactly-one-parallel-pair trapezium convention and different-length adjacent-pair kite fixtures; the sources do not settle every inclusive naming convention. Keep rectangle equal diagonals, rhombus perpendicular/bisected diagonals, square45° endpoint halves and kite one-way diagonal bisection distinct. General simple-polygon sums include checked concave triangulation; crossed boundaries, holes and curved outlines are outside these cases. Regular examples use consistent convex exterior turns and exact integer side-count checks; unknown-n full outlines follow the deduction, while impossible marked data remain condition cards. Congruence preserves full corresponding shape and size under turns/reflections, including reflex interiors; equal area, perimeter or angles alone is insufficient. A fixed reference picture and a correct number do not establish that a learner drew or justified the configuration. Written work is not automatically marked. Attempt acknowledgements and revealed references do not award marks or establish mastery. These structural links are not exhaustive subskill approval; specialist approval and formal assessment validation remain unestablished. Section4.3 symmetry,4.4 measurement/bearings,4.5 formal instrument constructions,4.11 similarity and the full H4.7A reasoning scope remain separate. Local symmetry-diagonal use is not complete symmetry coverage; no blanket formal-proof waiver is imported from unrelated circle-theorem notes. The wider Maths A course remains incomplete. Publisher book navigation is separate evidence: Book1 Shape and Space1 has literal F4.2B/C chapter and Hub19 Geometry of Shapes2 tags, not A; map60–99 and proof start60/next100 agree. Book2 Shape and Space6 has literal D/E chapter tags, with Hub59 Angles in Polygons2 for D and Hub58 Angles in Polygons1 for E; map128–160 and proof start128/next161 agree. The map associates F/G with Shape and Space7 pages223–248, while the public proof chapter starts222 and its next chapter starts248; proof contents instead place Congruent shapes129 and Similar shapes147 in Shape and Space6. Preserve that unresolved topic/page conflict rather than moving tags silently. Hub85 Similarity1 literally tags F4.2F and F4.11A, not G. A has no literal chapter/Hub tag and G has no literal Hub tag. Only selected Book1 public uncorrected proof body pages69–71 were inspected; Book2 relevant chapters are contents-only evidence in the public sample. Full final-book and Teaching Hub body/pagination alignment remains unverified; chapter links confer no full-book or modular teaching credit.
Before you begin
- Read labelled boundary vertices and distinguish straight and curved edges.
- Recognise right angles, equal-length ticks and parallel arrows.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A simple polygon has six straight boundary sides. Which name follows from that count?
What you will learn
- Identify a simple polygon from its closed straight-sided boundary and count its sides in order.
- Recognise parallelograms, rectangles, squares, rhombuses, trapezia, kites, pentagons, hexagons and octagons from stated properties.
- Explain why a turn, reflection or inward vertex does not by itself change a polygon name, and distinguish a name from regularity.
Draw or annotate nine separate marked cards A–I. On A, opposite sides form two parallel pairs, adjacent side lengths differ and there are no right angles. On B, all four angles are right and adjacent sides are 6 cm and 3 cm. On C, all four angles are right and all four sides are equal. On D, all four sides are equal but its angles are not right. On E, exactly one pair of opposite sides is parallel. On F, two different-length pairs of adjacent sides are equal. Cards G,H,I are irregular simple polygons with 5,6,8 boundary sides respectively. Label each card’s vertices in boundary order, retain the true markings and give the most specific name from the lesson’s nine names. Which three broader families also include C?
Draw or copy each labelled boundary on paper. Keep stated side lengths, selected interior/reflex/exterior arcs, parallel arrows and equality marks. Give geometric reasons before equations and substitute back. Exact givens determine deductions; screen diagrams are illustrative and must not be measured for an exact answer.
Card A
Card B
- AB: 6 cm
- BC: 3 cm
Card C
Card D
Card E
Card F
Card G
Card H
Card I
Follow the boundary before naming the shape
A polygon has a closed boundary made from straight line segments. Each segment is a side and each meeting point of consecutive sides is a vertex. Start at a labelled vertex, follow the boundary once and count each side once. A diagonal drawn inside the shape is not another boundary side. An open path, a circle or an outline containing a curved edge does not meet this description.
The simple polygons used here have boundaries that do not cross themselves. A simple polygon can be convex or concave: in a concave one, part of the boundary turns inwards and an interior angle is reflex. That indentation does not remove a side. A pentagon has five sides, a hexagon six and an octagon eight, whether or not those sides and angles are equal. A crossed outline needs a different convention and is outside the ordinary interior-angle rules used in these lessons.
Read the quadrilateral marks
A quadrilateral has four sides. A parallelogram has two pairs of opposite parallel sides. A rectangle has four right angles, a rhombus has four equal sides, and a square has both four right angles and four equal sides. The square therefore also meets the rectangle, rhombus and parallelogram definitions. Give the most specific name asked for, while keeping those valid broader names.
For the unambiguous examples in this lesson, a trapezium has exactly one pair of opposite parallel sides. A kite has two pairs of adjacent equal sides, with the two pair lengths different. Textbooks sometimes use broader conventions for these names, so read the definition supplied with a task. Use the parallel arrows, matching side ticks and right-angle squares as givens. Do not invent an equality or parallel pair because an unscaled sketch looks nearly symmetric.
Keep properties when a diagram moves
Turning or reflecting the entire marked shape preserves its lengths, angles and side count. Keep every vertex label and mark attached to its original place in the shape. A square drawn on a corner is still a square, and a rectangle need not have a horizontal long side. Page orientation is not part of any of these definitions.
A polygon name is also not a promise of regularity. The irregular pentagon, hexagon and octagon cards have the correct numbers of sides without all their lengths or angles matching. When drawing a labelled sketch, show the properties that are given and make the boundary legible. Exact ruler-and-compass construction is a separate skill; a useful property sketch should not be presented as a measurement proof.
Sources for this lesson
Pause and explain
A marked quadrilateral has four equal sides and four right angles. Which statement is complete?
Worked example
Draw or annotate nine separate marked cards A–I. On A, opposite sides form two parallel pairs, adjacent side lengths differ and there are no right angles. On B, all four angles are right and adjacent sides are 6 cm and 3 cm. On C, all four angles are right and all four sides are equal. On D, all four sides are equal but its angles are not right. On E, exactly one pair of opposite sides is parallel. On F, two different-length pairs of adjacent sides are equal. Cards G,H,I are irregular simple polygons with 5,6,8 boundary sides respectively. Label each card’s vertices in boundary order, retain the true markings and give the most specific name from the lesson’s nine names. Which three broader families also include C?
Show the worked solution
- Follow each card’s boundary and read its marks before considering the way it sits on the page. A has the two parallel pairs of a parallelogram; B has the right angles of a rectangle without four equal sides; C has both defining square conditions.
- D is a rhombus, E a trapezium under the stated exactly-one-pair convention, and F a kite under the stated different-pair-length convention. Count G,H,I as five, six and eight sides: pentagon, hexagon and octagon. Their irregular outlines are permitted.
- C is also a rectangle because it has four right angles, a rhombus because it has four equal sides, and a parallelogram because its opposite sides are parallel. Use square as its most specific supplied name, without denying those broader memberships.
Answer A–I: parallelogram, rectangle, square, rhombus, trapezium, kite, pentagon, hexagon, octagon. C is also a rectangle, rhombus and parallelogram. Those conclusions come from the marked properties and side counts, not from page orientation.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Draw or annotate four rotated cards, labelling each boundary in order. Card J has four equal sides and four right angles. Card K has four right angles, with unequal adjacent side lengths. Card L is an irregular simple five-sided outline and M an irregular simple six-sided outline. Name all four and explain why rotating their complete marked diagrams changes none of the names.
Give me a hint
Count sides around each closed boundary, then read equal-side, right-angle and parallel marks.
Compare my reasoning
- Read J’s four equal-side marks together with its four right-angle marks: these establish a square. K has four right angles but unequal adjacent sides, so its most specific name is rectangle.
- Follow L’s five and M’s six boundary sides once. Their names are pentagon and hexagon; neither name requires the irregular card to become regular.
- A rigid turn changes the direction of each edge on the page but preserves the marks, side counts and angle measures. Thus J,K,L,M retain their names after the turn.
J is a square, K a rectangle, L a pentagon and M a hexagon. A rigid rotation preserves every defining mark and boundary side.
Look for these in your work
- Label the four boundaries and retain the givens when rotating them.
- Use the exact side counts and marked quadrilateral definitions.
- Explain that orientation does not change a name or make an irregular polygon regular.
Independently plot and close the first three labelled boundaries on grids, without a supplied name-wordbank. Use the marked fourth card Q. N has boundary vertices (0,0),(5,0),(6,2),(3,5),(0,3); O has (0,0),(4,0),(6,2),(5,5),(1,5),(−1,2); P has (1,0),(5,0),(6,1),(6,4),(5,5),(1,5),(0,4),(0,1), each joined in the stated order and back to the start. Card Q is a rotated quadrilateral marked with four equal sides and with non-right angles. Name the four polygons. Explain why the side-count names do not establish regularity and why Q need not be a square. Use coordinates to locate the vertices, then follow each boundary; no slope or distance-formula calculations are required. Explain in a sentence for each outline which boundary count or mark supports its name.
Give me a hint
Close each listed boundary and count its sides; separate a defining property from a familiar-looking outline.
Compare my reasoning
- Join each list consecutively and close it with one final side to the start. The lists for N,O,P give five, six and eight distinct boundary vertices, so their side counts are 5,6,8.
- N is a pentagon, O a hexagon and P an octagon. The supplied irregular diagrams have unequal side lengths or angles, so those names do not imply regularity.
- Q’s four equal sides establish a rhombus. A square would also need four right angles; Q is stated to have non-right angles, so it is a non-square rhombus even when rotated.
N: pentagon (5 sides); O: hexagon (6); P: octagon (8); Q: non-square rhombus (4 equal sides, non-right angles). A side-count name alone does not mean regular.
Look for these in your work
- Independently plot and close each listed boundary, then count its closing side once without a supplied name-wordbank.
- Name the 5-,6-,8-sided figures and the marked rhombus correctly.
- Explain the extra conditions needed for regularity or for a square.
Prepare seven labelled outline cards: a simple concave five-sided boundary; an open path of five straight segments; a circle; a closed outline with one curved edge; a crossed bow-tie outline; a four-sided figure with exactly one opposite parallel pair; and a kite with two different-length adjacent equal-side pairs. Make the open ends, curved edges, crossing and inward vertex visible. Which cards belong to the simple straight-sided polygon class used here? Name those that do, and explain why the crossed card is outside this lesson’s simple-polygon scope.
Give me a hint
Test closure, straight sides and whether the boundary crosses itself before choosing a polygon name.
Compare my reasoning
- Test closure and straight sides first. The concave five-sided boundary, the marked trapezium and the marked kite pass these tests and have simple non-crossing boundaries.
- The open path fails closure, the circle and curved-edge outline fail the straight-side condition. The concave card remains a pentagon because its indentation changes neither closure nor its count of five sides.
- Keep the crossed bow-tie outside this task’s simple-polygon class. It cannot be treated as an ordinary simple quadrilateral for the later interior-angle rule. The accepted names are concave pentagon, trapezium and kite.
Accept the concave pentagon, trapezium and kite. Reject the open path, circle and curved-edge outline from the polygon definition; keep the crossed outline outside the simple-polygon scope used here.
Look for these in your work
- Draw the controls so open, curved and crossed features are explicit.
- Accept the concave pentagon, trapezium and kite for correct reasons.
- Keep the ordinary simple-polygon rule separate from a crossed outline.
Common mistakes
- Counting an interior diagonal as a boundary side or omitting the closing side.
- Naming a shape from orientation or an unscaled visual guess instead of stated properties.
- Assuming every pentagon, hexagon or octagon is regular, or denying that a square also satisfies broader family definitions.
What can you explain now?
A simple closed outline has eight straight boundary sides, with no equal-side or equal-angle promise. What is its name, and must it be regular?
Compare with the explanation
It is an octagon; it need not be regular.
Octagon specifies eight sides. Regularity additionally requires all sides and all interior angles to be equal.
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Make it stick
Name all nine polygon examples in this lesson. Explain which marks establish the six quadrilateral names, why a square has broader family memberships, and why concavity or rotation alone does not change a side-count name.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.