Comprehensive Geometric Formulas and Solid Geometry Notes

Geometric Foundations and Calculations for Prisms (برزم)

The study of prisms involves understanding both the linear components, such as length and width, and the two-dimensional area of the base. For a rectangular prism, the perimeter of the base, denoted by the symbol $P$, is calculated by adding the length and the width and multiplying the sum by two. This is expressed by the formula $P = 2 \times (\text{Length} + \text{Width})$. The base area, denoted as $B$, is the product of these two dimensions, represented as $B = \text{Length} \times \text{Width}$.

The surface area of a prism is divided into two parts: the lateral area and the total surface area. The lateral area ($LA$) is the sum of the areas of all the side faces and is determined by multiplying the perimeter of the base by the height of the prism, using the formula $LA = P \times h$. To find the total surface area ($SA$), one must add the areas of the two bases to the lateral area. This is calculated using the formula $SA = LA + 2B$. Finally, the volume ($V$) of a prism is the total three-dimensional space it occupies, which is found by multiplying the base area by the height, expressed as $V = B \times h$.

Geometric Properties and Formulas for Cylinders (سلندر)

A cylinder (سلندر) is a solid with two parallel circular bases. The lateral area ($LA$) of a cylinder represents the area of the curved surface that connects these two circles. This area is calculated by multiplying the circumference of the circle ($2 \pi r$) by the height ($h$) of the cylinder. The formula provided in the transcript is $LA = 2 \pi r h$. It is important to note that the transcript uses the shorthand $P$ or similar notations to represent the circular perimeter or radius-related values.

The total surface area ($SA$) of a cylinder is the sum of the lateral area and the areas of the two circular bases. Since the area of a single circle is $\pi r^2$, the formula for the surface area is $SA = LA + 2 \pi r^2$. For volume calculations, a cylinder follows the same logic as a prism, where the volume is the base area times the height. Given that the base is a circle, the volume ($V$) is expressed by the specific formula $V = \pi r^2 h$.

Geometry of Pyramids (الهرم) and Cones (المخروط)

Pyramids (الهرم) and cones (المخروط) differ from prisms and cylinders because they taper to a single point called the apex. A critical measurement for these shapes is the slant height (الارتفاع المائل), which is the distance from the apex down to a point on the perimeter of the base. In calculations of surface area, the lateral area ($LA$) of a pyramid or cone is defined as half the product of the base perimeter ($P$) and the slant height ($l$), recorded as $LA = \frac{1}{2} P l$.

The total surface area ($SA$) for these figures is calculated by adding the lateral area to only a single base area ($B$), as they do not have a top base. This is represented by the formula $SA = LA + B$. For a cone specifically, the base area is circular, so the transcript notes the application of $\pi$ in these calculations. The volume ($V$) of a pyramid or a cone is exactly one-third the volume of a prism or cylinder with the same base and height, which leads to the formula $V = \frac{1}{3} B h$.

The Geometry of Spheres (الكرة) and Hemispheres (نصف الكرة)

The sphere (الكره) is a perfectly symmetrical three-dimensional shape where every point on its surface is equidistant from its center. The surface area ($SA$) of a sphere is a primary calculation. Additionally, the transcript documents the volume of the sphere, which is calculated using the formula $V = \frac{4}{3} \pi r^3$. While the transcript uses shorthand like "ter³", this refers to the cubic radius which is standard for spherical volume calculations.

A hemisphere (نصف الكره) represents exactly half of a sphere. When calculating the properties of a hemisphere, it is essential to account for the flat circular base that is created when a sphere is cut in half, though the volume is simply half of the sphere's total volume. The transcript lists both the sphere and the hemisphere as key objects for calculation within this geometric framework.

Spatial Relations: Sphere Within a Cube (سفير داخل كيوب)

The transcript explores a specific spatial relationship: a sphere located inside a cube (سفير داخل كيوب يعني كره داخل صندوق). In this scenario, the sphere is perfectly inscribed within the box. This implies that the diameter of the sphere is equal to the side length of the cube. If the side of the cube is denoted as $s$, the radius of the sphere ($r$) would be $\frac{s}{2}$. This relationship is used to solve complex problems where the volume or surface area of one shape must be determined relative to the other. The transcript notes associated values for such calculations, such as the area of a square face ($s^2$) and its relation to the sphere's surface area ($SA$).