Mensuration and trigonometry Study Guide

A pyramid is a three-dimensional geometric shape that has a flat polygonal base and triangular faces that meet at a single point called the apex. The shape of the base can vary, leading to different types of pyramids, such as square-based pyramids, triangular-based pyramids (also known as tetrahedrons), or rectangular-based pyramids.

Key features of a pyramid include:

• Base: The bottom face of the pyramid. This can be any polygon.
• Apex: The single point at the very top where all the triangular faces meet.
• Lateral Faces: The triangular faces that connect the base to the apex.
• Edges: The line segments where two faces meet. These include base edges and lateral edges.
• Vertices: The corners of the pyramid, including the corners of the base and the apex.
• Height (h): The perpendicular distance from the apex to the center of the base. This is crucial for calculating volume.
• Slant Height (l): The height of one of the triangular lateral faces, measured from the midpoint of a base edge to the apex. This is used for calculating the surface area of the triangular faces.

To calculate the volume of any pyramid, the formula is always: Volume = (1/3) × Base Area × Perpendicular Height. The 'Base Area' will depend on the shape of the base (e.g., length × width for a rectangle, or (1/2) × base × height for a triangle). The 'Perpendicular Height' is the height of the pyramid from its apex straight down to the center of its base.

To calculate the surface area of a pyramid, you need to find the area of the base and add the areas of all the triangular lateral faces. The area of each triangular face is typically (1/2) × base edge length × slant height. The total surface area is the sum of the base area and the area of all the lateral faces. Remember, if the base is a regular polygon, all lateral faces will be congruent triangles. If the base is irregular, the lateral faces may have different dimensions.

Pythagoras' theorem is a fundamental principle in geometry that applies exclusively to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). If we label the lengths of the legs as 'a' and 'b', and the length of the hypotenuse as 'c', the theorem can be expressed as the equation a² + b² = c². This theorem allows us to find the length of an unknown side in a right-angled triangle if we know the lengths of the other two sides. For example, if you know the lengths of the two shorter sides, you can calculate the hypotenuse. Conversely, if you know the hypotenuse and one shorter side, you can find the length of the other shorter side. Remember, it only works for triangles with a 90-degree angle.