Geometry Revision Notes

What are 2D Shapes?

2D shapes are flat figures that can only be measured in two dimensions: length and width. They do not have any depth. Key terms to remember for 2D shapes include:

• Sides: The lines that form the boundary of the shape.
• Vertices (or corners): Points where two or more sides meet.
• Angles: The space between two intersecting sides.

Common 2D shapes you should recognise are squares, rectangles, triangles, circles, parallelograms, rhombuses, trapeziums, and various polygons (shapes with many straight sides).

What are 3D Shapes?

3D shapes, also called solid shapes, occupy space and have three dimensions: length, width, and depth/height. They are made up of 2D faces. Key terms for 3D shapes are:

• Faces: The flat surfaces of the 3D shape (these are 2D shapes themselves).
• Edges: Where two faces meet.
• Vertices (or corners): Points where three or more edges meet.

Examples of 3D shapes include cubes, cuboids, spheres, cylinders, cones, pyramids, and prisms.

Key Properties to Identify Shapes

When identifying or describing shapes, focus on their defining properties:

• For 2D shapes: Count the number of sides, the number of vertices, and look at the types of angles (right, acute, obtuse). Consider if sides are parallel or perpendicular, and if all sides are equal in length.
• For 3D shapes: Count the number of faces, edges, and vertices. Describe the shape(s) of its faces. For example, a cube has 6 square faces, 12 edges, and 8 vertices. A cylinder has 2 circular faces and 1 curved face. This systematic approach will help you categorise and understand different shapes.

Identifying and Describing 2D Shapes

2D shapes are flat shapes that only have length and width. Key examples include squares (4 equal sides, 4 right angles), rectangles (4 sides, opposite sides equal, 4 right angles), triangles (3 sides, sum of angles is 180 degrees), circles (all points equidistant from a centre), and various polygons (shapes with three or more straight sides). You need to be able to identify these shapes and describe their properties, such as the number of sides, angles, and lines of symmetry. For instance, a regular polygon has all sides and all angles equal.

Identifying and Describing 3D Shapes

3D shapes are solid objects that have length, width, and height (or depth). Common 3D shapes include cubes (6 square faces, 12 edges, 8 vertices), cuboids (6 rectangular faces, 12 edges, 8 vertices), cylinders (2 circular bases, 1 curved surface), spheres (perfectly round, all points equidistant from centre), cones (1 circular base, 1 curved surface, 1 vertex), and pyramids (a polygon base and triangular faces meeting at an apex). You should be able to identify these and state the number of faces, edges, and vertices they possess. For example, a triangular prism has 5 faces, 9 edges, and 6 vertices.

Developing Spatial Reasoning

Developing spatial reasoning involves understanding how shapes move and relate to each other in space. This includes visualising nets (2D patterns that can be folded to form 3D shapes), understanding plans and elevations (different views of a 3D object from above, front, or side), and recognising symmetry. Being able to sketch 2D representations of 3D solids, such as an isometric drawing of a cuboid, is also a valuable skill. This helps in understanding the hidden parts of shapes and how they are constructed.

Basic Calculations for 2D and 3D Shapes

For 2D shapes, foundational calculations involve perimeter (the total distance around the outside of the shape) and area (the amount of surface a shape covers). For example, the perimeter of a rectangle is '2 × (length + width)' and its area is 'length × width'. For 3D shapes, key calculations are volume (the amount of space a shape occupies) and surface area (the total area of all faces). For a cuboid, volume is 'length × width × height', and surface area is '2lw + 2lh + 2wh'. Remember to use consistent units for all measurements.

Advanced 2D Geometry Techniques

For 2D shapes, higher methods often involve non-right angled trigonometry. The sine rule is used to find unknown sides or angles when you have an angle and its opposite side, plus one other piece of information (angle or side): a/sin A = b/sin B = c/sin C. The cosine rule is used when you have two sides and the included angle (to find the third side), or all three sides (to find an angle): a² = b² + c² - 2bc cos A. The area of a non-right angled triangle can be found using Area = (1/2)ab sin C. Remember to apply these rules strategically, sometimes combining them with Pythagoras' theorem or properties of specific polygons like parallelograms or trapeziums. Vector geometry can also be used for working with positions and transformations of 2D shapes.

3D Shapes: Advanced Calculations and Visualisation

In 3D geometry, higher methods involve calculating volumes and surface areas of composite solids, and finding angles or distances within 3D figures. Visualising slices or cross-sections of 3D shapes is crucial. Often, problems require you to identify right-angled triangles within 3D space to apply Pythagoras' theorem or basic trigonometry (SOH CAH TOA). For example, finding the space diagonal of a cuboid involves applying Pythagoras' theorem twice. Angles between a line and a plane, or between two planes, can be determined by constructing appropriate right-angled triangles. Always draw clear diagrams or mark points on given diagrams to aid visualisation and problem-solving. Pay attention to units and ensure consistency.

Angles in 3D Shapes

Calculating angles in 3D shapes often involves finding a specific cross-section that forms a 2D triangle containing the angle. There are two main types: angles between lines and angles between lines and planes. To find the angle between a line and a plane:

1. Identify the line and the plane.
2. Find the projection of the line onto the plane (imagine a light shining directly down onto the plane).
3. The angle is between the original line and its projection. This typically forms a right-angled triangle where the 'height' is perpendicular to the plane and the 'base' is the projection. Use trigonometry (SOH CAH TOA) to find the angle.

Solving Complex Geometric Problems

Higher-level problems often combine multiple geometric concepts. This might involve finding the volume of a frustum (a cone with the top cut off), where you need to use similar triangles to find the height of the removed cone. Or it could be calculating the surface area of a complex solid made of cylinders and hemispheres. Break down complex problems into smaller, manageable steps. Identify knowns and unknowns, then choose the appropriate formula or theorem for each step. Often, a crucial step is to construct an imaginary line or plane to create a simpler 2D problem within the 3D context. Always check your units and the reasonability of your answer.

Types of Angles

Angles are measured in degrees (°). Knowing the different types of angles is fundamental. An acute angle is between 0° and 90°. A right angle is exactly 90°, often marked with a square. An obtuse angle is between 90° and 180°. A straight angle is exactly 180°, forming a straight line. A reflex angle is between 180° and 360°. A full turn (or complete angle) is exactly 360°.

Angles on a Straight Line

Angles that lie on a straight line always add up to 180°. This property is very useful for finding unknown angles. For example, if you have one angle of 70° on a straight line, the other angle adjacent to it on that line must be 180° - 70° = 110°. Remember to state 'angles on a straight line add to 180°' as your reason.

Angles Around a Point

Angles around a central point always add up to 360°. This is equivalent to a full revolution. If you have several angles sharing a common vertex and covering the entire space around that vertex, their sum will be 360°. For instance, if three angles around a point are 90°, 120°, and x, then 90 + 120 + x = 360, meaning x = 150°. The reason is 'angles around a point add to 360°'.

Vertically Opposite Angles

When two straight lines intersect, they form two pairs of vertically opposite angles. These pairs of angles are always equal. For example, if two lines cross to form angles A, B, C, and D, where A and C are opposite, and B and D are opposite, then A = C and B = D. The reason for this equality is 'vertically opposite angles are equal'.