Mensuration and trigonometry Study Guide

A polyhedron is a three-dimensional solid made up of flat polygonal faces, straight edges, and sharp corners or vertices. The word 'polyhedra' is the plural form of polyhedron. Common examples of polyhedra include cubes, cuboids, pyramids, and prisms. Spheres and cylinders are not polyhedra because they have curved surfaces.

To describe a polyhedron, we often count its key features: Faces (F), Edges (E), and Vertices (V).

• Faces (F): These are the flat surfaces of the polyhedron. Each face is a polygon (e.g., triangle, square, pentagon).
• Edges (E): These are the line segments where two faces meet. An edge is a straight line.
• Vertices (V): These are the points where three or more edges meet. A vertex is a corner.

For any convex polyhedron (a polyhedron where a line segment connecting any two points on its surface lies entirely within or on the surface of the polyhedron), Euler's Formula provides a fundamental relationship between these features: F - E + V = 2. This formula is incredibly useful for checking calculations or finding a missing value if the other two are known. For example, a cube has 6 faces, 12 edges, and 8 vertices. Using Euler's formula: 6 - 12 + 8 = 2. This relationship holds true for all convex polyhedra. Understanding these basic components is crucial for visualising and analysing 3D shapes in geometry.

Cones and spheres are three-dimensional geometric shapes that you will encounter in GCSE Mathematics. A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a perfectly round ball. It is defined by its radius, which is the distance from its centre to any point on its surface. No matter which direction you measure, the radius will always be the same. The key formulas for a sphere are its surface area and its volume.

A cone is a three-dimensional geometric shape that smoothly tapers from a flat base (usually circular) to a point called the apex or vertex. It has a circular base, a perpendicular height (h) from the centre of the base to the apex, and a slant height (l) which is the distance from the apex to any point on the circumference of the base. For a right cone, the height, radius, and slant height form a right-angled triangle, meaning you can use Pythagoras' theorem ($r^2 + h^2 = l^2$) to find any of these values if the other two are known. The formulas for a cone include its volume and its curved surface area. The total surface area of a cone also includes the area of its circular base.

It is important to understand which dimensions are needed for each formula. For spheres, only the radius is required. For cones, the radius of the base, the perpendicular height, and/or the slant height are needed, depending on whether you're calculating volume, curved surface area, or total surface area. You should practice applying these formulas to various problems, ensuring you use the correct units for your answers (e.g., cm3 for volume, cm2 for area) and rounding appropriately if specified.