Congruence, similarity and 3D solids
Congruence criteria (SSS, SAS, ASA, RHS), similar shapes and enlargement, classifying triangles and quadrilaterals, and 3D solid properties
About four lessons of 45 to 60 minutes
Same shape and size, same shape but a different size, or a genuinely different shape
2 shapes can relate to each other in several precise ways. CONGRUENT shapes are identical, same size and same shape, just possibly moved, turned or flipped. SIMILAR shapes have exactly the same shape but a DIFFERENT size, one is an ENLARGEMENT of the other. And every shape, 2D or 3D, has its own set of defining PROPERTIES, side lengths, angles, or (in 3D) faces, edges and vertices, that let you classify and reason about it precisely instead of just by eye.
Knowing exactly how much information is enough to GUARANTEE 2 triangles are congruent (without measuring every single side and angle) is one of the most useful shortcuts in geometry, and it extends naturally into similar shapes (where lengths scale by a constant factor) and into describing 3D solids by counting their faces, edges and vertices.
- 2 jigsaw pieces cut from the same templatecongruent: identical shape and size
- A photo and its enlarged poster printsimilar: same shape, scaled up by a constant factor
- A flat-pack shelf bracket, checked against its technical drawingcongruence, confirmed with just 2 sides and the angle between them (SAS)
- A dice: 6 faces, 12 edges, 8 verticesthe same face/edge/vertex counts as any cube
What students will be able to do
Students will use the 4 triangle congruence criteria (SSS, SAS, ASA, RHS) to identify congruent triangles and find missing sides or angles, find missing lengths and scale factors for similar shapes and enlargements, classify triangles and quadrilaterals by their properties, and state the faces, edges and vertices of common 3D solids, applying Euler's formula where it holds.
- I can name the 4 triangle congruence criteria (SSS, SAS, ASA, RHS) and use them to find a missing side or angle.
- I can find a missing length or the scale factor for a pair of similar shapes.
- I can classify a triangle by its sides (equilateral, isosceles, scalene) and by its angles (acute, right, obtuse).
- I can name a quadrilateral (square, rectangle, rhombus, parallelogram, trapezium, kite) from its properties.
- I can state the faces, edges and vertices of a 3D solid, and verify Euler's formula (F + V - E = 2) for a polyhedron.
Standards this unit teaches
- KS3 Maths: Geometry and measuresUK National Curriculum (England)Geometry and measures
Statutory requirement (Department for Education, "National curriculum in England: mathematics programmes of study", updated 28 September 2021, Key stage 3, "Geometry and measures" strand, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study): pupils should be taught to "use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles"; "derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies"; "identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids"; "use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 6 area and volume teaching unit2D and 3D shape fluency this unit builds on
- Congruent in the glossarythe student-facing definition of congruence
- Similar shapes in the glossarythe student-facing definition of similarity and scale factor
- Geometry learn guideshape vocabulary and properties this unit assumes
Words to teach and display
- Congruent
- identical in shape and size; 1 shape is an exact copy of the other, possibly moved, rotated or reflected
- Similar
- the same shape but a different size; 1 shape is an enlargement of the other by a constant scale factor
- SSS / SAS / ASA / RHS
- the 4 minimum sets of matching information that GUARANTEE 2 triangles are congruent: side-side-side, side-angle-side, angle-side-angle, right angle-hypotenuse-side
- Scale factor
- the number every length is multiplied by when a shape is enlarged (or divided by, when it is shrunk back)
- Polyhedron
- a 3D solid with only flat faces (no curved surfaces), such as a cube, prism or pyramid
- Euler's formula
- the rule F + V - E = 2 (faces plus vertices minus edges equals 2), true for every convex polyhedron
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Triangle congruence criteria: SSS, SAS, ASA, RHS
Concrete2 triangles are congruent if 1 can be moved (translated, rotated or reflected) exactly onto the other. Rather than checking all 3 sides and all 3 angles every time, only 1 of 4 specific combinations of matching information is needed to GUARANTEE congruence: SSS (all 3 sides), SAS (2 sides and the angle between them), ASA (2 angles and the side between them), or RHS (a right angle, the hypotenuse, and 1 other side).
Triangle ABC has AB = 5 cm, BC = 8 cm and angle B = 40 degrees. Triangle DEF has DE = 5 cm, EF = 8 cm and angle E = 40 degrees. Are the triangles congruent, and by which criterion?
- 2 sides (AB/DE and BC/EF) and the INCLUDED angle between them (angle B / angle E) all match.
- That is exactly the SAS criterion.
Answer: Yes, congruent by SAS.
- Why is 'SSA' (2 sides and a non-included angle) NOT one of the 4 valid congruence criteria?
- What criterion applies when you know all 3 angles of 2 triangles match, but no sides?
2. Similar shapes and enlargement
PictorialWhen a shape is enlarged, EVERY length is multiplied by the same scale factor, but every ANGLE stays exactly the same (angles never change under enlargement). If the scale factor is greater than 1, the shape gets bigger; if it is between 0 and 1, the shape gets smaller.
Area does not scale the same way as length: since area is a length TIMES a length, it scales by the scale factor SQUARED. A shape enlarged by scale factor 3 has 3 times the perimeter, but 9 times the area.
Shape A has a side of 6 cm. Similar shape B has the corresponding side 18 cm. Find the scale factor, and find the length on shape B of a 2nd side that measures 4 cm on shape A.
- Scale factor = 18 / 6 = 3.
- A 4 cm side on shape A scales to 4 x 3 = 12 cm on shape B.
Answer: Scale factor = 3. The 4 cm side becomes 12 cm.
- If a shape is enlarged by scale factor 5, by what factor does its AREA scale?
3. Classifying triangles and quadrilaterals
PictorialTriangles are classified 2 different ways at once: by their SIDES (equilateral: all 3 equal; isosceles: exactly 2 equal; scalene: none equal) and by their ANGLES (acute: all under 90 degrees; right: exactly 1 angle of 90 degrees; obtuse: 1 angle over 90 degrees). Quadrilaterals are classified by a combination of side, angle and parallel-line properties: squares, rectangles, rhombuses, parallelograms, trapeziums and kites each have their own defining set.
Classify a triangle with sides 5 cm, 5 cm, 8 cm by its sides. Name a quadrilateral with 4 equal sides and no right angles.
- 2 of the 3 sides (5 cm and 5 cm) are equal, and the 3rd is different, so it is isosceles.
- 4 equal sides with no right angles is the definition of a rhombus (a square would also have 4 equal sides, but WITH right angles).
Answer: Isosceles. A rhombus.
- What is the key difference between a rhombus and a square?
- What is the key difference between a rectangle and a parallelogram?
4. 3D solid properties and Euler's formula
AbstractEvery 3D solid can be described by its faces (flat surfaces), edges (where 2 faces meet) and vertices (corners, where edges meet). For any convex POLYHEDRON (a solid built entirely from flat faces, like a cube or a prism), Euler's formula always holds: faces plus vertices minus edges equals 2.
Solids with a CURVED surface (a cylinder, cone or sphere) are not polyhedra, so Euler's formula does not apply to them; they are instead described using 'curved surfaces' alongside any flat faces they have.
A triangular prism has 5 faces, 9 edges and 6 vertices. Verify Euler's formula.
- F + V - E = 5 + 6 - 9 = 2.
- This matches Euler's formula exactly.
Answer: 5 + 6 - 9 = 2. Euler's formula holds.
- Why does Euler's formula NOT apply to a cylinder?
- A square-based pyramid has 5 faces and 5 vertices. How many edges must it have, using Euler's formula?
Common misconceptions and how to address them
MisconceptionIf 2 triangles have 2 matching sides and any matching angle (not necessarily the angle between them), they must be congruent.
Why it happens: Students remember 'SAS is a valid criterion' but forget that the angle must specifically be the one INCLUDED between the 2 given sides.
How to address it: Show a genuine counterexample: 2 triangles with the same 2 sides and the same NON-included angle that are NOT congruent (this is the classic 'ambiguous case', sometimes called the 'SSA trap'), proving the included-angle condition really matters.
MisconceptionEnlarging a shape by scale factor 2 doubles its area as well as its side lengths.
Why it happens: Students apply the length scale factor directly to area without realising area is 2-dimensional and so scales by the factor SQUARED.
How to address it: Count squares directly on a small enlarged example (e.g. a 2x2 square enlarged by scale factor 2 becomes 4x4): the side length doubled (2 to 4), but the area went from 4 to 16 squares, 4 TIMES as much, not 2.
MisconceptionA rhombus is just a 'tilted square', so it is basically the same shape.
Why it happens: Both have 4 equal sides, so the visual similarity overrides the actual defining difference (right angles).
How to address it: Draw both side by side and measure the angles directly: a square's angles are all exactly 90 degrees; a rhombus's usually are not (though a square IS technically a special rhombus with right angles).
MisconceptionEuler's formula (F + V - E = 2) works for every 3D solid, including a cylinder or a sphere.
Why it happens: The formula is taught as a general-sounding rule without enough emphasis on the polyhedron (flat-faces-only) restriction.
How to address it: Test it directly on a cylinder (2 flat faces + 1 curved surface, 2 edges, 0 vertices): F + V - E = 2 + 0 - 2 = 0, NOT 2, proving the formula genuinely fails for solids with a curved surface.
Guided practice (with answers)
1. Triangle ABC has all 3 sides equal to triangle DEF's 3 sides. Name the congruence criterion.
Answer: SSS (side-side-side).
2. Right-angled triangles ABC and DEF share the same hypotenuse and 1 other matching side. Name the congruence criterion.
Answer: RHS (right angle-hypotenuse-side).
3. Shape A has a side of 4 cm. Similar shape B's corresponding side is 20 cm. Find the scale factor.
Answer: 5, because 20 / 4 = 5.
4. Classify a triangle with angles 90, 40 and 50 degrees.
Answer: Right-angled (it has exactly 1 angle of 90 degrees).
5. Name a quadrilateral with 2 pairs of equal opposite sides and 4 right angles, but not all 4 sides equal.
Answer: A rectangle.
6. A cube has 6 faces and 8 vertices. Use Euler's formula to find its number of edges.
Answer: 12, because F + V - E = 2 gives 6 + 8 - E = 2, so E = 12.
Independent practice worksheets
Practise congruence, similarity, shape classification and 3D solid properties with computed, never-wrong answer keys.
Differentiation
- Keep a printed reference card of the 4 congruence criteria with a small labelled diagram for each, until they can be named from memory.
- For similarity, always write 'scale factor = new length / original length' before substituting, every time.
- For classifying quadrilaterals, build a properties table (equal sides? right angles? parallel sides?) and tick columns rather than guessing from a mental image.
- For 3D solids, physically hold or point to a real object (a dice, a can) while counting faces, edges and vertices for the first few examples.
- Prove that 2 given triangles are NOT necessarily congruent from an 'SSA' style description, by constructing a genuine counterexample.
- Given a shape's AREA before and after enlargement, find the scale factor (working backward through the square-of-the-scale-factor relationship).
- Classify a shape given only its coordinates (calculate side lengths first using the distance between points).
- Investigate why Euler's formula fails for a non-convex (dented) polyhedron, and research the corrected general formula.
Assessment: exit ticket
A three-question exit ticket sampling congruence, similarity and 3D solids.
1. Triangle ABC has angle A = 30 degrees, angle B = 70 degrees and side AB = 6 cm, matching triangle DEF's angle D = 30 degrees and side DE = 6 cm. Name the congruence criterion.
Answer: ASA (angle-side-angle).
2. Shape A's perimeter is 12 cm. Similar shape B has scale factor 4. Find shape B's perimeter.
Answer: 48 cm, because 12 x 4 = 48.
3. A hexagonal prism has 8 faces and 18 edges. Use Euler's formula to find its number of vertices.
Answer: 12, because F + V - E = 2 gives 8 + V - 18 = 2, so V = 12.
Teacher notes and timings
- Rough timing across 4 lessons: Lesson 1 congruence criteria (section 1), Lesson 2 similarity/enlargement (section 2), Lesson 3 classifying shapes (section 3), Lesson 4 3D solids (section 4) plus the exit ticket.
- Congruence and 3D solid properties are combined into 1 unit deliberately (following the same 'group closely related bullets' judgment call earlier batches used, e.g. batch 6 combining 3 angle-relation types into 1 unit): both sit under the same 'derive/illustrate shape properties' family of KS3 Geometry bullets, and combining them keeps the unit count for this closing batch manageable.
- Every SSS triangle in the matching worksheet is regenerated if it fails the triangle inequality (a + b > c for all 3 pairs), so every triangle described is genuinely constructible, never an impossible combination of side lengths.
- Similarity/enlargement and shape classification ship without a diagram for those specific sub-skills: the site's 'transformation' figure kind only supports reflection (fixed, over the y-axis), rotation (fixed, 90 degrees about the origin) and translation, with no 'enlargement' (scale-factor) variant, and no figure kind draws a generic labelled polygon for classification. Section 1's congruence figure reuses the existing translation transformation (exact, no new figure kind added); every worked example and worksheet item elsewhere in this unit is still fully computed and correct, just described in words and numbers.