Geometry: shapes, angles and position
Bea here! Geometry is the maths of shapes, angles and where things are. Grab a ruler and some squared paper and let us explore.
1. Cheat sheet: Geometry
2-D shapes
- Square: 4 equal sides, 4 right angles.
- Rectangle: opposite sides equal, 4 right angles.
- Rhombus: 4 equal sides. Parallelogram: 2 pairs of parallel sides. Trapezium: 1 pair.
- Regular = equal sides and equal angles. Turning a shape does not change its name.
Polygon names
3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 7 heptagon, 8 octagon, 9 nonagon, 10 decagon.
3-D shapes
| Shape | F | E | V |
|---|---|---|---|
| Cube, cuboid | 6 | 12 | 8 |
| Tri. prism | 5 | 9 | 6 |
| Sq. pyramid | 5 | 8 | 5 |
A cube net has 6 squares.
Angle types
- Acute: less than 90°. Right: 90°.
- Obtuse: between 90° and 180°.
- Straight: 180°. Reflex: more than 180° and less than 360°.
- Full turn: 360°.
Angle facts
- On a line: 180°. At a point: 360°.
- Vertically opposite angles (where two straight lines cross) are equal.
- Triangle: 180°. Quadrilateral: 360°.
- Angles inside a polygon with n sides add up to (n − 2) × 180° (stretch). Regular hexagon: 120° each.
- Parallel lines: corresponding (F) and alternate (Z) angles are equal (stretch).
Circles and symmetry
- Diameter = 2 × radius.
- Lines of symmetry: square 4, rectangle 2, equilateral triangle 3, regular hexagon 6.
Coordinates
- (x, y): along the corridor, then up the stairs.
- Negative x is left, negative y is down.
- Translation: slide, same shape. Reflection: mirror image.
Patterns
- Count how much it grows each time.
- 2 more sticks each time starting at 3 (triangles in a row): 2n + 1.
- 10th pattern: 2 × 10 + 1 = 21.
2. Mini lessons
Classifying 2-D shapes, regular and irregular
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A shape is named by its properties: its sides, angles and parallel lines, not by how it is turned. A regular polygon has all sides equal and all angles equal. An irregular one does not.
Worked example: Is a square a rectangle? Is a tilted square a different shape? Is a rhombus with no right angles regular?
- A rectangle has 4 right angles and opposite sides equal. A square has all of that, so a square is a special rectangle.
- Turning a shape does not change its properties. A tilted square is still a square.
- A rhombus has equal sides but its angles are not all equal, so it is not regular.
Answer: Yes, no, no
3-D shapes and nets
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A face is a flat surface, an edge is where two faces meet, and a vertex is a corner. A net is a 3-D shape folded flat, so it shows every face. For solid shapes with flat faces, faces + vertices − edges = 2.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube / cuboid | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
| Tetrahedron | 4 | 6 | 4 |
Worked example: How many faces, edges and vertices does a triangular prism have? Check your answer.
- Faces: 2 triangles + 3 rectangles = 5.
- Vertices: 3 on each end, so 6.
- Edges: 3 on each end plus 3 joining them: 3 + 3 + 3 = 9.
- Check: 5 + 6 − 9 = 2.
Answer: 5 faces, 9 edges, 6 vertices
Angle types and turns
GEO-05
An angle is an amount of turn. A quarter turn is 90°, a half turn is 180° and a full turn is 360°. The size depends only on the turn, not on how long the drawn arms are.
Animation: an angle is an amount of turn
An angle measures how far one arm has turned from the other. The shaded part shows the turn. Less than a quarter turn is acute. A quarter turn is a right angle. More than 90° but less than 180° is obtuse. A half turn is a straight line. More than a half turn is a reflex angle: the shaded turn is bigger than a half circle. A full turn is 360°. The length of the arms never changes the angle.
Worked example: Name the angle after a three-quarter turn, and say how many degrees it is.
- A full turn is 360°. A quarter turn is 360 ÷ 4 = 90°.
- Three quarters: 3 × 90° = 270°.
- More than 180° is a reflex angle.
Answer: 270°, a reflex angle
Angles on a line, at a point and opposite
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Angles on a straight line add up to 180°. Angles round a point add up to 360°. When two lines cross, the angles opposite each other are equal.
Worked example: Two angles lie on a straight line. One is 115°. What is the other? And what is the angle opposite the 115° angle?
- On a line: 180° − 115° = 65°.
- Opposite angles are equal, so the angle opposite 115° is also 115°.
Answer: 65°, and 115°
Angles in triangles, quadrilaterals and polygons
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The angles in a triangle add up to 180° and in a quadrilateral to 360°. For any polygon with n sides, the angles add up to (n − 2) × 180°. Polygons and parallel lines are beyond KS2 but often turn up as 11+ challenges.
Worked example: A triangle has angles 52° and 68°. Find the third angle. Then find one angle of a regular pentagon.
- 52 + 68 = 120. 180 − 120 = 60. The third angle is 60°.
- Pentagon: (5 − 2) × 180 = 540° in total.
- Regular, so all 5 angles are equal: 540 ÷ 5 = 108°.
Answer: 60°, and 108°
Angles on parallel lines (stretch, beyond KS2)
GEO-08
Parallel lines never meet and are always the same distance apart. When a straight line crosses two parallel lines, some angles are equal. Corresponding angles sit in the same place at each crossing (an F shape). Alternate angles sit on opposite sides of the crossing line, between the parallel lines (a Z shape). This is stretch work, beyond KS2.
Worked example: (Stretch) A line crosses two parallel lines. The green angle in the picture is 60°. It is between the parallel lines, on the right of the crossing line. Find its corresponding angle (F shape), its alternate angle (Z shape), and the angle next to it on the lower line.
- Corresponding angle (F shape): the same place at the other crossing, which is above the upper line on the right. It is 60°.
- Alternate angle (Z shape): between the lines, on the other side of the crossing line, at the upper crossing. It is 60°.
- The angle next to the green one on the straight lower line: 180° − 60° = 120°.
Answer: 60°, 60° and 120°
Parts of a circle
GEO-09
The radius runs from the centre to the edge. The diameter goes right across through the centre, so it is twice the radius. The circumference is the distance round the edge, and a chord is any straight line joining two points on the edge.
Worked example: A circle has a radius of 6.5 cm. What is its diameter? A circular pond has a diameter of 9 m. What is its radius?
- Diameter = 2 × radius = 2 × 6.5 = 13 cm.
- Radius = diameter ÷ 2 = 9 ÷ 2 = 4.5 m.
Answer: 13 cm and 4.5 m
Lines of symmetry
GEO-10
A line of symmetry is a fold line where both halves match exactly, like a mirror. Some shapes have none, some have one and some have many.
Animation: fold along the dashed line
A kite with a dashed line down the middle. Fold the right half over the dashed line. It lands exactly on the left half, with nothing sticking out. Open it up again. Both halves match, so the dashed line is a line of symmetry. A shape with no matching fold has no line of symmetry.
Worked example: How many lines of symmetry does a rectangle (that is not a square) have?
- Fold it across the middle from top to bottom: the halves match. That is 1.
- Fold it from left to right: the halves match. That is 2.
- A diagonal fold does not match, because the corners do not land on corners.
Answer: 2
Coordinates in all four quadrants
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A coordinate pair (x, y) tells you where a point is: x says how far along and y how far up. Negative numbers go left and down. To find a missing vertex of a rectangle, use the corners you know: the new corner shares an x with one and a y with another.
Worked example: Three corners of a rectangle are A (−4, 3), B (2, 3) and C (2, −1). Find D.
- A and B have the same y, 3, so they are the top side. B and C have the same x, 2.
- D is below A, so its x is −4. D is level with C, so its y is −1.
- D = (−4, −1).
Answer: (−4, −1)
Translation and reflection
GEO-13
A translation slides a shape without turning it. Every corner moves the same distance in the same direction. A reflection flips a shape over a mirror line. Each corner ends up the same distance from the mirror, on the other side.
Animation: translate A, and reflect A in the y-axis
Triangle A (dashed outline) has corners at (−3, 1), (−1, 1) and (−3, 3). Translation: slide it 5 right and 3 down. Every corner moves the same way and the shape does not turn. Triangle B has corners (2, −2), (4, −2) and (2, 0). Now reflect the original triangle A in the y-axis. The green triangle is the mirror image of triangle A. Each corner is the same distance from the mirror, on the other side: (−3, 1) goes to (3, 1).
Worked example: Reflect the point (−3, 2) in the y-axis. Then translate it 4 right and 1 down.
- The y-axis is the mirror. (−3, 2) is 3 to the left of it, so its reflection is 3 to the right: (3, 2).
- Translate 4 right: x = 3 + 4 = 7. 1 down: y = 2 − 1 = 1.
Answer: (3, 2), then (7, 1)
Shape and number patterns
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In a shape pattern, count how much it grows from one step to the next. That tells you the rule. You can then jump to any step without drawing it.
Worked example: Matchsticks make a row of triangles. 1 triangle uses 3 sticks, 2 triangles use 5 and 3 triangles use 7. How many sticks are needed for 10 triangles?
- It grows by 2 sticks each time. Pattern 1 has 3 sticks, which is 2 × 1 + 1.
- So the rule is 2n + 1.
- For 10 triangles: 2 × 10 + 1 = 21.
- Check: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.
Answer: 21 sticks
3. Worked examples from easy to hard
Level 1 is Year 3, one step. Level 5 is a challenge with 3 to 5 steps. Try each one yourself before you read the steps.
Level 1One step
An angle measures 75°. Is it acute, right or obtuse?
- Acute angles are less than 90°.
- 75° is less than 90°, so it is acute.
Answer: acute
Level 2One or two steps
How many lines of symmetry does a regular hexagon have?
- Three lines go through pairs of opposite corners.
- Three more lines go through the middles of pairs of opposite sides.
- 3 + 3 = 6.
Answer: 6
Level 3Two steps
Three angles meet at a point. Two of them are 120° and 110°. Find the third angle.
- Angles round a point add up to 360°.
- Add the two known angles: 120 + 110 = 230.
- Subtract from 360: 360 − 230 = 130.
Answer: 130°
Level 4Exam standard
Three corners of a rectangle are at (−6, 2), (1, 2) and (1, −2). Find the fourth corner. What is the perimeter in grid units?
- The corners (−6, 2) and (1, 2) share y = 2, and (1, 2) and (1, −2) share x = 1. So the fourth corner has x = −6 and y = −2: (−6, −2).
- Length across: from x = −6 to x = 1 is 7 units.
- Height: from y = 2 to y = −2 is 4 units.
- Perimeter: 2 × (7 + 4) = 22 units.
Answer: (−6, −2) and 22 units
Level 5Challenge
A regular hexagon, a square and a third regular polygon fit together exactly around a point. How many sides does the third polygon have?
- Regular hexagon: (6 − 2) × 180 ÷ 6 = 120° at each corner.
- Square: 90°.
- Angles at a point add to 360°, so the third angle is 360 − 120 − 90 = 150°.
- Outside each corner of a regular polygon the turn is 180 − 150 = 30°. A full turn is 360°, so 360 ÷ 30 = 12 sides.
- Check: (12 − 2) × 180 ÷ 12 = 150°.
Answer: 12 sides
4. Common mistakes
Lots of children make these slips. Spot them early and you will win marks.
Longer arms, bigger angle
ANGLE_ARM_LENGTH
The mistake: Thinking an angle with longer arms is bigger.
Why it is wrong: An angle measures the turn between the arms, not how long they are drawn.
How to avoid it: Extend the short arms: the amount of turn between them stays the same, so the angle does not change.
✗ Wrong: The angle with arms 8 cm long is bigger than the one with arms 3 cm long, though both are 60°.
✓ Right: Both are 60°.
Rejecting a shape because it is tilted
SHAPE_ORIENTATION
The mistake: Saying a shape changes name when it is turned.
Why it is wrong: A shape is defined by its sides and angles, not by which way up it sits.
How to avoid it: Turn the page. Count equal sides and right angles.
✗ Wrong: A square turned on its corner is a diamond, not a square.
✓ Right: It still has 4 equal sides and 4 right angles, so it is a square.
Answering the angle that is already given
TECH_MISREAD_QUESTION
The mistake: Writing down a number from the question, or the total, instead of the missing angle.
Why it is wrong: The question asks for the unknown angle, not the total or a given angle.
How to avoid it: Mark the angle you need with a ? and say what you are finding.
✗ Wrong: Angles on a line: 115° and x. Answer: 180°.
✓ Right: x = 180° − 115° = 65°.
5. Quick check
Five questions, easy to hard. Have a go on paper, then open the answer.
How many edges does a hexagonal prism have?
Show answer
18
6 on each hexagonal end, plus 6 joining the ends: 6 + 6 + 6 = 18.
Two angles lie on a straight line. One is 133°. What is the other?
Show answer
47°
180° − 133° = 47°.
An isosceles triangle has a 36° angle at the top, between the two equal sides. What size is each of the other two angles?
Show answer
72°
180 − 36 = 144. The other two angles are equal: 144 ÷ 2 = 72.
The point (−6, 3) is reflected in the y-axis. What are the new coordinates?
Show answer
(6, 3)
The y-axis is the mirror. The point is 6 left of it, so the image is 6 right of it. The y value does not change.
What is the size of each angle in a regular nonagon (9 sides)? (Stretch)
Show answer
140°
The angles add up to (9 − 2) × 180 = 1,260°. Divide by 9: 140°.
6. For grown-ups
- Use the real world: ask your child to find right angles in the room, count faces and edges on a cereal box, or spot lines of symmetry in letters of the alphabet.
- Make nets from card. Cutting out and folding a cube net is worth ten pictures. Try different nets and see which fold up.
- For angles, a paper "protractor": fold a sheet to make 90° and 45° and compare other angles with those before using a real protractor.