2D Shapes
2D Shapes
Key Concepts
1. What are 2D shapes?
- 2D shapes are flat shapes.
- They have:
- length
- width
- They do not have thickness.
- 2D shapes are drawn on a flat surface such as paper.
2. Sides, vertices and angles
- A side is a straight line forming part of a shape.
- A vertex (plural: vertices) is a corner point where 2 sides meet.
- An angle is the amount of turn between 2 sides.
- Shapes can be grouped by:
- number of sides
- length of sides
- size of angles
- whether sides are parallel
3. Properties of triangles
A triangle is a 2D shape with:
- 3 sides
- 3 vertices
- 3 angles
Types of triangles
Equilateral triangle
- Has 3 equal sides
- Has 3 equal angles
- Each angle is 60°
- Has 3 lines of symmetry
Isosceles triangle
- Has 2 equal sides
- The 2 angles opposite the equal sides are equal
- Has 1 line of symmetry if it is not equilateral
Right-angled triangle
- Has 1 right angle
- A right angle is 90°
- The other 2 angles are less than 90°
- It may also be isosceles if 2 sides are equal
Scalene triangle
- Has 3 unequal sides
- Has 3 unequal angles
- Has no line of symmetry
Important triangle facts
- The sum of the angles in any triangle is 180°
- A triangle can be described using more than one property:
- for example, a triangle can be right-angled and isosceles
4. Properties of quadrilaterals
A quadrilateral is a 2D shape with:
- 4 sides
- 4 vertices
- 4 angles
Square
- Has 4 equal sides
- Has 4 right angles
- Opposite sides are parallel
- Has 4 lines of symmetry
Rectangle
- Has 2 pairs of equal opposite sides
- Has 4 right angles
- Opposite sides are parallel
- Has 2 lines of symmetry
Parallelogram
- Has 2 pairs of opposite sides parallel
- Opposite sides are equal
- Opposite angles are equal
- Usually does not have right angles
- Has 0 lines of symmetry
Rhombus
- Has 4 equal sides
- Opposite sides are parallel
- Opposite angles are equal
- Usually does not have 4 right angles
- Has 2 lines of symmetry
Trapezium
- Has 1 pair of parallel sides
- The other pair of sides is not parallel
- Can have different side lengths and angle sizes
- Usually has no line of symmetry, though some special trapeziums may have 1
Important quadrilateral facts
- The sum of the angles in any quadrilateral is 360°
- A square is also:
- a rectangle
- a parallelogram
- a rhombus
- A rectangle is also:
- a parallelogram
- A rhombus is also:
- a parallelogram
5. Parallel and perpendicular lines
Parallel lines
- Parallel lines are lines that:
- stay the same distance apart
- never meet, even if extended
Perpendicular lines
- Perpendicular lines meet at a right angle
- The angle formed is 90°
6. Symmetry
Symmetry means one part of a shape matches another part exactly.
Line symmetry
- A shape has line symmetry if it can be folded into 2 matching halves.
- The fold line is called a line of symmetry.
Examples:
- Square: 4 lines of symmetry
- Rectangle: 2 lines of symmetry
- Equilateral triangle: 3 lines of symmetry
- Isosceles triangle: 1 line of symmetry
- Scalene triangle: 0 lines of symmetry
Rotational symmetry
- A shape has rotational symmetry if it matches itself when turned about a point.
- The shape must fit exactly before completing a full turn of 360°.
Examples:
- Square: rotational symmetry of order 4
- Rectangle: rotational symmetry of order 2
- Equilateral triangle: rotational symmetry of order 3
At PSLE level, students are often asked more about line symmetry, but rotational symmetry may also be useful to know.
7. Nets of 3D shapes
A net is a flat 2D pattern that can be folded to form a 3D shape.
Why nets are important
- They help us see:
- the number of faces
- the shape of each face
- how faces are connected
Common 3D shapes and their nets
Cube
- Has 6 faces
- All faces are squares
- A net of a cube has 6 squares
Cuboid
- Has 6 faces
- Faces are rectangles
- Some faces may be equal in size
- A net of a cuboid has 6 rectangles or a mix of rectangles with matching opposite faces
Triangular prism
- Has 2 triangular faces
- Has 3 rectangular faces
- A net has:
- 2 triangles
- 3 rectangles
Square-based pyramid
- Has 1 square base
- Has 4 triangular faces
- A net has:
- 1 square
- 4 triangles
What makes a correct net?
A correct net must:
- have the correct number of faces
- have faces of the correct shape
- have faces joined in the correct positions
- be able to fold without overlapping
What makes an incorrect net?
A wrong net may:
- have too many or too few faces
- have the wrong face shapes
- place faces in unsuitable positions
- cause overlapping when folded
Important Definitions
- 2D shape: a flat shape with length and width only.
- 3D shape: a solid shape with length, width and height.
- Side: a straight edge of a 2D shape.
- Vertex: the point where 2 sides meet.
- Angle: the amount of turn between 2 lines or sides.
- Right angle: an angle of 90°.
- Parallel lines: lines that stay the same distance apart and never meet.
- Perpendicular lines: lines that meet at 90°.
- Triangle: a 2D shape with 3 sides.
- Quadrilateral: a 2D shape with 4 sides.
- Equilateral triangle: a triangle with 3 equal sides and 3 equal angles.
- Isosceles triangle: a triangle with 2 equal sides.
- Right-angled triangle: a triangle with 1 right angle.
- Scalene triangle: a triangle with 3 unequal sides.
- Square: a quadrilateral with 4 equal sides and 4 right angles.
- Rectangle: a quadrilateral with opposite sides equal and parallel, and 4 right angles.
- Parallelogram: a quadrilateral with 2 pairs of opposite sides parallel.
- Rhombus: a quadrilateral with 4 equal sides and opposite sides parallel.
- Trapezium: a quadrilateral with 1 pair of parallel sides.
- Symmetry: when a shape can be divided so parts match exactly.
- Line of symmetry: a line that divides a shape into 2 matching halves.
- Rotational symmetry: when a shape matches itself after being turned.
- Net: a flat pattern that can be folded to make a 3D shape.
- Face: a flat surface of a 3D shape.
Worked Examples
Example 1: Identifying a triangle
A triangle has:
- 2 equal sides
- 2 equal angles
What type of triangle is it?
Step-by-step solution
- Look at the side lengths.
- The triangle has 2 equal sides.
- A triangle with 2 equal sides is an isosceles triangle.
- The 2 equal angles also match the property of an isosceles triangle.
Answer: It is an isosceles triangle.
Example 2: Identifying a quadrilateral
A shape has:
- 4 equal sides
- 4 right angles
What is the shape?
Step-by-step solution
- A shape with 4 sides is a quadrilateral.
- 4 equal sides suggests it might be a square or rhombus.
- 4 right angles is an important clue.
- A rhombus does not need to have 4 right angles.
- A quadrilateral with 4 equal sides and 4 right angles is a square.
Answer: The shape is a square.
Example 3: Identifying a net
A net has:
- 1 square
- 4 triangles attached around the square
What 3D shape does it form?
Step-by-step solution
- Count and identify the faces.
- There is 1 square face.
- There are 4 triangular faces.
- A 3D shape with 1 square base and 4 triangular sides is a square-based pyramid.
Answer: The net forms a square-based pyramid.
Common Mistakes to Avoid
- Confusing a square with a rectangle and saying they are completely different.
- A square is a special type of rectangle because it has 4 right angles.
- Thinking a rhombus must always look like a “diamond”.
- The important property is 4 equal sides, not the way it is tilted.
- Forgetting that a trapezium has 1 pair of parallel sides.
- Saying an isosceles triangle has 3 equal sides.
- That is an equilateral triangle.
- Mixing up parallel and perpendicular lines.
- Counting only visible shape appearance instead of checking properties carefully.
- Forgetting to count angles as well as sides when identifying shapes.
- Drawing a line of symmetry that does not divide the shape into 2 exact matching halves.
- Choosing a wrong net because the faces look correct but would overlap when folded.
- Forgetting that a cube net must have 6 square faces.
- Saying a scalene triangle has a line of symmetry.
- It has none.
Exam Tips
- When identifying a shape, always state its properties:
- “It has 4 equal sides and 4 right angles, so it is a square.”
- Use correct mathematical words such as:
- equal sides
- right angles
- parallel sides
- line of symmetry
- vertices
- For symmetry questions, imagine folding the shape.
- For net questions, check:
- number of faces
- shape of faces
- arrangement of faces
- If a question asks how you know, do not just name the shape.
- Give the property that proves it.
- Read carefully:
- “at least”
- “exactly”
- “only” These words change the meaning.
- In shape-sorting questions, remember that some shapes belong to more than one group.
- Example: a square is also a rectangle and a rhombus.
- If a diagram is slanted, do not let that confuse you.
- A rectangle tilted sideways is still a rectangle if it has 4 right angles.
Quick Summary
- A triangle has 3 sides; a quadrilateral has 4 sides.
- Equilateral triangle: 3 equal sides, 3 equal angles, each angle 60°.
- Isosceles triangle: 2 equal sides and 2 equal angles.
- Right-angled triangle: has 1 angle of 90°.
- Scalene triangle: 3 unequal sides and 3 unequal angles.
- Square: 4 equal sides, 4 right angles, opposite sides parallel.
- Rectangle: opposite sides equal and parallel, 4 right angles.
- Parallelogram: 2 pairs of opposite sides parallel.
- Rhombus: 4 equal sides, opposite sides parallel.
- Trapezium: 1 pair of parallel sides.
- Line symmetry means a shape can be folded into 2 matching halves.
- A net is a flat pattern that folds to form a 3D shape.
- A cube has 6 square faces; its net must have 6 squares.
Q1
Kenny wanted to fold the net below to form a cube. However, he realised that the net is incorrect. He has to remove one of the faces, A, B, C or D, from it to form the cube. Which of the following letters representing the face that he has to remove from the net?
Diagram: A net of a cube with 6 faces arranged in a cross-like pattern. The faces are labeled A (left), B (center-left), C (bottom-center), and D (right). One unlabeled face appears at the top of the arrangement.
2022-P6-Maths-Prelim-ACSJ 2022
Q2
Half of a symmetric figure is shown above. AB is the line of symmetry. Which of the following completes the symmetric figure?
Diagram: The top diagram shows half of a symmetric figure with vertices at A and B, where AB is the line of symmetry. The figure shown is a right triangle with a right angle marked, with the shaded region showing the portion above/along the line AB. Four answer options are provided below, each showing different possible completions of the symmetric figure by reflecting across line AB. Option 1 shows a completion with the reflected portion to the left of AB. Option 2 shows a completion with the reflected portion below and to the left of AB. Option 3 shows a completion with the reflected portion below and to the right of AB. Option 4 shows a completion with the reflected portion to the right of AB.
2022-P6-Maths-Prelim-ACSJ 2022
Q3
PQ and QR are two sides of a trapezium PQRS in which QR is parallel to PS and PS is twice the length of QR. Complete the trapezium PQRS by drawing the other two sides in the square grid below.
Diagram: The same square grid as in part (a), with points P, Q, and R already marked and connected by lines PQ and QR. Students are required to complete the trapezium by drawing sides RS and SP, where SP is parallel to QR and PS = 2 × QR.
2022-P6-Maths-Prelim-ACSJ 2022
Q4
Measure and write down the size of ∠m in the figure.
Diagram: An angle marked as m formed by two line segments meeting at a point, creating an acute angle that appears to open towards the upper right.
1 mark
2022-P6-Maths-Prelim-Catholic_High 2022
Q5
The figure above is not drawn to scale. Each of the statements below is either true, false or not possible to tell from the information given. For each statement, put a tick (✓) to indicate your answer. Statement 1: BCEF is a parallelogram. Statement 2: BE is perpendicular to EC.
Diagram: A table with three columns (True, False, Not possible to tell) and two rows for the statements about the figure.
2 marks
2022-P6-Maths-Prelim-Catholic_High 2022
Q6
Zi Xuan used identical black and white squares to form a symmetrical pattern on a large square board. The figure below shows part of the square board. Which of the following pieces will complete the pattern on the square board?
Diagram: A partially shown symmetrical checkerboard pattern with black and white squares arranged in a specific configuration. The given pattern shows an asymmetrical portion meant to be completed by one of four options, each showing different arrangements of black and white squares in a small grid pattern.
2022-P6-Maths-Prelim-Henry_Park 2022
Q7
Which of the following are isosceles triangles?
Diagram: Four triangles labeled A, B, C, and D. Triangle A has sides labeled 5 cm, 6 cm, and 7 cm. Triangle B is isosceles with two sides labeled 8 cm and base labeled 6 cm. Triangle C has two angles labeled 55° and 81°. Triangle D is a right triangle with one angle labeled 45° and a right angle marked with a square.
2022-P6-Maths-Prelim-MGS 2022
Q8
The solid below is a prism. Which of the following nets can be folded to form the solid above?
Diagram: A triangular prism is shown at the top of the question. Below are four different net options (A, B, C, D) that could potentially fold into this prism. The solid appears to be a triangular prism with two triangular bases and three rectangular faces.
2022-P6-Maths-Prelim-MGS 2022
Q9
Using the given dotted line as the line of symmetry, complete the symmetric figure by shading the correct square(s) below.
Diagram: A 4×4 grid showing part of a symmetric figure. The top half and middle sections contain shaded squares in a pattern. A vertical dotted line runs through the center of the grid indicating the line of symmetry. The figure shows: Row 1 (top): unshaded, shaded, shaded, unshaded. Row 2: shaded, shaded, unshaded, shaded. Row 3: unshaded, unshaded, unshaded, unshaded. Row 4 (bottom): unshaded, unshaded, unshaded, unshaded. The task requires completing the symmetric figure by shading the appropriate squares.
1 mark
2022-P6-Maths-Prelim-MGS 2022
Q10
Jane used a different line of symmetry that required her to shade only two squares to complete a symmetric figure. Which two squares did Jane shade? Shade in the figure below to show your answer.
Diagram: A 4×4 grid showing the same pattern with shaded squares. The grid displays: Row 1 (top): unshaded, shaded, shaded, unshaded. Row 2: shaded, shaded, unshaded, shaded. Row 3: unshaded, unshaded, unshaded, unshaded. Row 4 (bottom): unshaded, unshaded, unshaded, unshaded. The task is to identify which two additional squares need to be shaded to create a symmetric figure with a different line of symmetry than part (a).
1 mark
2022-P6-Maths-Prelim-MGS 2022
Q11
Circle the words that describe Triangle LOP correctly in the following statement: Triangle LOP ( is / is not ) an isosceles triangle because ∠LOP ( is / is not ) the same as ∠PLO.
1 mark
2022-P6-Maths-Prelim-Nan_Chiau 2022
Q12
Figure A and B are nets of solids. Circle the words that describe the figures above. Figure A is a net of a ( prism / pyramid ). Figure B is a net of a ( prism / pyramid ).
Diagram: Figure A shows a net with a triangular top and bottom connected to three rectangular sections in the middle, arranged vertically. Figure B shows a net with a square base at the bottom connected to four triangular faces meeting at a point above, arranged in a pyramid-like pattern.
2022-P6-Maths-Prelim-Nan_Hua 2022
Q13
There are 4 shaded squares in the figure. Shade 3 more squares to form a symmetric figure with AB as the line of symmetry.
Diagram: A grid of squares (4 columns × 4 rows) with a diagonal line AB drawn from bottom-left to top-right. There are 4 pre-shaded squares already marked in the grid. The student is asked to shade 3 more squares to create a figure that is symmetric with respect to the AB diagonal line.
2022-P6-Maths-Prelim-Nan_Hua 2022
Q14
A triangle ABC is drawn on a square grid.
Diagram: A square grid is shown with triangle ABC drawn on it. Point A is at the bottom left, point B is above and to the right of A, and point C is further above and to the right of B. The vertices appear to be positioned at grid intersections.
12 marks
2022-P6-Maths-Prelim-Nan_Hua 2022
Q15
Using triangle ABC, draw rhombus ABCD.
2022-P6-Maths-Prelim-Nan_Hua 2022
Q16
Which pair of lines in the square grid are parallel?
Diagram: A square grid with multiple diagonal lines drawn. Points are labeled A, B, C, D, E, F, G, H, and I. The lines form various segments connecting these points across the grid.
2022-P6-Maths-Prelim-Nanyang 2022
Q17
Which two of the following are nets of a cube?
Diagram: Four different net diagrams labeled A, B, C, and D. Each diagram shows an unfolded pattern of 6 connected squares arranged in different configurations. Net A shows squares arranged in a T-like pattern with one vertical column of 2 squares at top, connected to a row of 3 squares, with an additional square below. Net B shows a similar arrangement with squares in different positions. Net C shows another arrangement of 6 squares. Net D shows yet another configuration of 6 connected squares. Each net is shown with dotted lines indicating fold lines.
2022-P6-Maths-Prelim-Nanyang 2022
Q18
There are 5 shaded squares in the figure. Shade 5 more squares to form a symmetric figure with XY as the line of symmetry.
Diagram: A grid of squares arranged in a 3-column by 6-row structure with a horizontal dashed line XY running through the middle (between rows 3 and 4). There are 5 shaded (filled) squares shown on the top half of the grid: one in the top-left corner, one below it in the second row, and three forming an L-shape in the middle area around the line of symmetry. The remaining squares are unshaded and available for shading to complete the symmetric figure.
2022-P6-Maths-Prelim-Nanyang 2022
Q19
AB and BC form two sides of a rhombus ABCD. Complete the drawing of the rhombus ABCD.
Diagram: A square grid with points A, B, and C marked. A is located at approximately (3, 5), B is at approximately (5, 7), and C is at approximately (7, 5). Lines AB and BC are drawn connecting these points, forming two sides of the rhombus.
1 mark
2022-P6-Maths-Prelim-Nanyang 2022
Q20
AB also forms one side of a trapezium ABEF. AB is parallel to EF. The length of EF is twice the length of AB. DAF forms a straight line and AD = AF. Complete the drawing of trapezium ABEF such that it does not overlap with the rhombus.
Diagram: A square grid with the rhombus ABCD from part (a) already drawn. Points A, B, C are marked as before. The question requires drawing trapezium ABEF where EF is parallel to AB and twice its length, and DAF forms a straight line with AD = AF.
2 marks
2022-P6-Maths-Prelim-Nanyang 2022
Q21
PQRS is a trapezium and STUV is a parallelogram. Which of the following pair of angles is/are a pair of 180°?
Diagram: A geometric figure showing trapezium PQRS with parallelogram STUV, with angles marked as 180° and various angle labels
2 marks
2022-P6-Maths-Prelim-Nanyang 2022
Q22
Which of the following is not the net of a cube?
Diagram: Four different net diagrams labeled (1) through (4), each showing unfolded cube patterns made of squares arranged in different configurations. (1) shows a diagonal stair-step pattern of 6 squares. (2) shows a cross-like pattern with a vertical extension. (3) shows an L-shaped arrangement with squares. (4) shows a pattern with 3 squares in a row on top and additional squares below in an offset arrangement.
2022-P6-Maths-Prelim-PLMGS 2022
Q23
Circle the word that describes triangle BCE.
Diagram: Multiple choice completion task about triangle BCE from the figure in 12a.
1 mark
2022-P6-Maths-Prelim-PLMGS 2022
Q24
The figure below is made up of triangles and rectangles. Shade the figure so that the figure has AB as its line of symmetry with 2/3 of the figure shaded.
Diagram: A symmetric figure with vertical line of symmetry AB. The figure consists of: a triangle at the top with vertex A, a rectangle in the middle with vertical shading (already partially shaded), an inverted triangle in the middle section, and a triangle at the bottom with vertex B. Some areas are already shaded with diagonal lines. A small reference rectangle labeled with dimensions 8 and 2 is shown separately at the bottom right.
2022-P6-Maths-Prelim-Red_Swastika 2022
Q25
Name the solid.
1 mark
2022-P6-Maths-Prelim-Red_Swastika 2022
Q26
Easons drew the net of his cuboid in Figure 2 and it is incorrect. Put a cross 'X' on one face that does not fit the net of his cuboid.
Diagram: Figure 2 shows a net of a cuboid laid out flat with multiple rectangular faces. The net appears to show faces of dimensions related to 20 cm, 6 cm, and 4 cm.
1 mark
2022-P6-Maths-Prelim-Red_Swastika 2022
Q27
Which pair of lines are parallel?
Diagram: A grid with multiple labeled line segments. Point B is on the left, point A is above and to the right of B, point C is above A, point D is to the right of C, point E is to the right and below D, and point F is above E. Lines connect these points: BA, AC, CD, DE, and EF are drawn on the grid.
2022-P6-Maths-Prelim-Rosyth 2022
Q28
Which statement about the rhombus is false?
Diagram: A rhombus with vertices labeled r (top right), s (top right area), w (bottom left), and t (bottom left area). A diagonal is drawn from bottom left to top right, dividing the rhombus into two triangles.
2022-P6-Maths-Prelim-Rosyth 2022
Q29
The figure shows a pyramid. Which of the following are possible nets of the pyramid?
Diagram: A square-based pyramid is shown at the top with a square base and four triangular faces meeting at an apex. Below are three nets labeled Net A, Net B, and Net C. Net A shows a square base with four triangles attached to each side, arranged in a cross pattern with two triangles extending upward from the top. Net B shows a square base with four triangles arranged around it in a different configuration. Net C shows a square base with triangles arranged in yet another pattern, with some triangles positioned to the sides.
2022-P6-Maths-Prelim-Rosyth 2022
Q30
The figure below shows 2 parallelograms, ABCD and JKLM. Which of the following statements is true?
Diagram: Two parallelograms are shown on a coordinate grid. Parallelogram ABCD has vertices at approximately: A (1, 3), B (3, 5), C (4, 3), D (2, 1). Parallelogram JKLM has vertices at approximately: J (5, 3), K (7, 5), L (9, 3), M (7, 1). The second parallelogram appears to be rotated 45 degrees relative to the first.
2022-P6-Maths-Prelim-SCGS 2022
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