Geometry Revision Notes

Sum of Interior Angles

For any polygon with 'n' sides, the sum of its interior angles can be found using the formula: Sum = (n - 2) × 180°. This formula arises from the fact that any polygon can be divided into (n - 2) triangles by drawing diagonals from one vertex. Since each triangle has an angle sum of 180°, the total sum for the polygon is (n - 2) multiplied by 180°.

Regular Polygons: Interior and Exterior Angles

In a *regular* n-sided polygon, all interior angles are equal and all exterior angles are equal. The measure of each interior angle is given by: Angle = [(n - 2) × 180°] / n. The sum of the exterior angles of *any* convex polygon is always 360°. Therefore, for a regular n-sided polygon, each exterior angle is 360° / n. Remember that an interior angle and its corresponding exterior angle at any vertex add up to 180°.

Number of Diagonals

The number of diagonals in an n-sided polygon can be calculated using the formula: Number of diagonals = n(n - 3) / 2. Each vertex can connect to (n-3) other vertices (it cannot connect to itself or its two adjacent vertices). Since each diagonal connects two vertices, we divide by 2 to avoid counting each diagonal twice.

Problem Solving with Polygons

Higher-level problems often involve combining these concepts. You might be asked to find the number of sides given an interior or exterior angle, or to work with angles in polygons that are not regular. Sometimes, polygons are joined together, requiring you to use properties of angles on a straight line, around a point, or within other shapes like triangles. Always ensure you are clear whether you are dealing with interior or exterior angles and if the polygon is regular or irregular.

What are Polyhedra?

Polyhedra are three-dimensional (3D) solids entirely made up of flat surfaces. These flat surfaces are called faces, and each face is a polygon (a 2D shape with straight sides). Where two faces meet, they form an edge, which is a straight line. Where three or more edges (and thus faces) meet, they form a vertex (a point). Common examples include cubes, cuboids, prisms (like triangular prisms, hexagonal prisms), and pyramids (like square-based pyramids, triangular-based pyramids). Understanding these components (faces, edges, vertices) is key to describing and distinguishing different polyhedra.

Non-Polyhedral Solids

Not all 3D solids are polyhedra. Some 3D shapes have at least one curved surface. These are called non-polyhedral solids. Examples include cylinders, which have two flat circular faces and one curved surface; cones, which have one flat circular face and one curved surface meeting at an apex; and spheres, which have a single continuous curved surface and no flat faces, edges, or vertices. It's important to be able to classify a 3D shape as either a polyhedron or a non-polyhedron based on the nature of its surfaces.

Key Features: Faces, Edges, Vertices

When describing any 3D solid, particularly polyhedra, the number of faces (F), edges (E), and vertices (V) are fundamental properties. A face is a flat surface. An edge is a line segment where two faces meet. A vertex is a point where three or more edges meet. For example, a cube has 6 faces, 12 edges, and 8 vertices. You should be able to count these features accurately for common polyhedra and some non-polyhedral solids (where applicable, e.g., a cylinder has 3 faces: 2 flat, 1 curved; 2 edges, no vertices in the conventional sense for the curved part).

Euler's Formula for Polyhedra (F + V - E = 2)

For any simple polyhedron, Euler's formula states that the number of faces (F) plus the number of vertices (V) minus the number of edges (E) always equals 2. This formula (F + V - E = 2) is a valuable tool to check properties of polyhedra or to find a missing number if the other two are known. For instance, a cube has F=6, V=8, E=12. Checking with Euler's formula: 6 + 8 - 12 = 14 - 12 = 2. This formula applies only to polyhedra, not to solids with curved surfaces.

Identifying Polyhedra

Polyhedra are 3D shapes made entirely of flat faces, straight edges, and vertices (corners). Think of building blocks: cubes, cuboids, pyramids, and prisms are all polyhedra. Each 'side' of a polyhedron is a flat polygon called a face. The lines where two faces meet are called edges. The points where edges meet are called vertices.

Examples of Polyhedra

1. Cube: 6 square faces, 12 edges, 8 vertices. All faces are congruent squares. 2. Cuboid: 6 rectangular faces, 12 edges, 8 vertices. Faces are rectangles, often congruent in pairs. 3. Prism: Has two identical, parallel polygonal bases, connected by rectangular or parallelogram faces. The name of the prism comes from its base shape (e.g., triangular prism, hexagonal prism). 4. Pyramid: Has a single polygonal base and triangular faces that meet at a common point called the apex. Its name comes from the shape of its base (e.g., square-based pyramid, triangular-based pyramid).

Solids That Are NOT Polyhedra

Some 3D shapes have curved surfaces and are therefore not polyhedra. These include: 1. Cylinder: Has two circular bases that are parallel and congruent, connected by a curved surface. It has no true edges or vertices in the polyhedral sense. 2. Cone: Has one circular base and a curved surface that tapers to a single point (apex). 3. Sphere: A perfectly round 3D object where every point on its surface is the same distance from its centre. It has no faces, edges, or vertices.

Key Features to Count

You often need to count the faces, edges, and vertices of polyhedra.

• Faces (F): The flat surfaces of the shape.
• Edges (E): The lines where two faces meet.
• Vertices (V): The points where edges meet.

For any convex polyhedron, Euler's formula holds: F + V - E = 2. This can be a useful check for your counts, though not strictly required for Foundation tier problems.

3D Pythagoras and Higher Length Calculations

In a 3D solid, Pythagoras' Theorem can be applied multiple times to find lengths that are not directly on a face. For example, to find the main diagonal of a cuboid with dimensions length (l), width (w), and height (h): first find the diagonal of the base using d_base = sqrt(l^2 + w^2). Then, use this base diagonal and the height to find the space diagonal: d_space = sqrt(d_base^2 + h^2) = sqrt(l^2 + w^2 + h^2). This principle extends to other shapes where specific right-angled triangles can be identified within the solid.

Trigonometry in 3D Solids

Trigonometry (SOH CAH TOA) can be used to find angles and lengths within 3D solids. Commonly, you might need to find the angle between a line and a plane, or the angle between two planes. For an angle between a line and a plane, identify the projection of the line onto the plane. The angle is formed between the line and its projection. For example, in a cuboid, the angle between a space diagonal and the base plane. For the angle between two planes, find a line in each plane perpendicular to their common intersecting line, and the angle between these two lines is the angle between the planes. Carefully draw out the relevant 2D right-angled triangles to simplify the problem.

Visualising Cross-Sections

A cross-section is the shape formed when a 3D solid is cut by a plane. Depending on the angle and position of the cut, the cross-section will vary. For instance, a cube can have a square, rectangular, triangular, or hexagonal cross-section. For a cylinder, a cut parallel to the base gives a circle, while a cut perpendicular to the base gives a rectangle. Identifying and accurately drawing these cross-sections is crucial for calculating their area or for helping to solve further problems involving the solid. Practise visualising how a plane intersects different polyhedra and other solids.

Composite Solids and Problem Solving

Higher-level problems often involve composite solids (combinations of simpler shapes) or scenarios requiring application of multiple geometric principles. You might need to find the volume or surface area of a solid composed of a hemisphere and a cone, or determine missing dimensions given a total volume. Breaking down complex solids into simpler components is key. Always identify what information is given and what needs to be found, then plan the most efficient sequence of steps, potentially involving Pythagoras, trigonometry, and relevant area/volume formulas.