Curved Solids ☰

Classification of Curved Solids
Curved solids can be classified into different categories based on their properties and the shapes of their surfaces:

1. Convex Solids: These are solids in which every pair of points within the solid is connected by a line segment that lies entirely within the solid. Spheres, cones, and cylinders are examples of convex solids.

2. Non-Convex Solids: These are solids that do not meet the convexity criteria. A non-convex solid has at least one pair of points within the solid that are connected by a line segment that does not lie entirely within the solid. Examples include tori (doughnut-shaped solids) and some polyhedra with concave faces.

Theorems Related to Curved Solids

• Pappus's Centroid Theorem: This theorem states that the volume of a solid of revolution generated by revolving a plane figure around an axis is equal to the product of the area of the figure and the distance traveled by the figure's centroid during the rotation.
• Cavalieri's Principle: This principle states that if two solids have the same height and their cross-sectional areas are equal at every height, then the volumes of the two solids are equal. This principle can be used to derive the formulas for the volumes of cones and pyramids.

Spherical Geometry
Spherical geometry is a non-Euclidean geometry that studies figures on the surface of a sphere. This geometry differs from the traditional Euclidean geometry, as the shortest distance between two points on a sphere is not a straight line but a great circle arc.

Properties of Spherical Geometry

• There are no parallel lines in spherical geometry.
• The angles of a spherical triangle always add up to more than 180 degrees.
• The area of a spherical triangle is proportional to its excess angle (the amount by which its angle sum exceeds 180 degrees).

Solids of Revolution
Many curved solids, including spheres, cones, and cylinders, can be generated by rotating a two-dimensional shape around an axis. These shapes are called "solids of revolution."

• A sphere can be generated by revolving a semicircle around its diameter.
• A cone can be generated by revolving a right triangle around one of its legs (adjacent to the right angle).
• A cylinder can be generated by revolving a rectangle around one of its sides.

Understanding the properties and applications of curved solids is essential in various fields such as mathematics, engineering, architecture, and physics. These shapes often provide the basis for more complex structures and systems.

Sphere ☰

A sphere is a perfectly symmetrical solid with all points on its surface equidistant from a fixed point called the center. The distance between the center and any point on the sphere is called the radius. Spheres have infinite lines of symmetry and the largest volume-to-surface-area ratio of any solid, making them ideal for minimizing heat loss or evaporation.

Formulas:
Surface Area (SA): \( SA = 4 \pi r^2 \)

Volume (V): \( V = \frac{4}{3} \pi r^3 \)

Diameter (D): \( D=2r \)

Properties

• All points on the surface of a sphere are equidistant from the center.
• A sphere has the smallest surface area among all solids with a given volume.
• The great circle of a sphere is the largest circle that can be drawn on its surface, with the plane of the circle passing through the center of the sphere.
• A sphere is perfectly symmetrical and has infinite lines of symmetry.
• A sphere has the largest volume-to-surface-area ratio of any solid, making it an ideal shape for minimizing heat loss or evaporation.

Real-World Applications

• Planets and celestial bodies are often approximately spherical due to gravity pulling matter inward, forming a shape with the least amount of potential energy.
• Soap bubbles form spheres as the surface tension minimizes the surface area for a given volume of air.
• Spherical tanks are used to store pressurized gases such as propane, as they can withstand high pressure with minimal stress concentration.

Cone ☰

A cone is a solid formed by connecting a flat, usually circular, base to a single point called the vertex or apex. The curved surface of the cone is created by the points on the base connected to the vertex. Cones have only one plane of symmetry, passing through the vertex and the center of the base.

Formulas
The lateral surface area (LSA): \( S_{\text{LSA}} = \pi rl \)

Surface Area (SA): \( S_{\text{SA}} = \pi r(r+l) \) Where \(r\) is the radius of the base, and \(l\) is the slant height of the cone.

Volume (V): \( \frac{1}{3} \pi r^2 h \) Where \(h\) is the height of the cone.

Slant Height (l): \( l = \sqrt{r^2 +h^2 } \)

A frustum is a section of a cone obtained by cutting the top portion with a plane parallel to the base. The volume \((V)\) of a frustum of a cone is given by the formula:
\( V= \frac{1}{3} \pi h(R^2+r^2+Rr) \) where \(R\) is the radius of the larger base, \(r\) is the radius of the smaller base, and \(h\) is the height of the frustum.

Properties

• A cone has only one plane of symmetry, which passes through its vertex and the center of its base.
• The lateral surface of a cone forms a right triangle when unrolled, with the slant height as the hypotenuse, and the base circumference and height as the other two sides.
• A cone has a unique curved surface called the "nappe."
• The volume of a cone is one-third the volume of a cylinder with the same base and height.

Applications

• Cones are used in traffic cones, rocket nose cones, and conical roofs.
• Reflective cones, or parabolic reflectors, are used in antennas, microphones, and telescopes to focus incoming waves onto a single point.

Cylinder ☰

A cylinder is a solid that consists of two parallel, congruent, flat bases connected by a curved surface. The axis of the cylinder is the line segment connecting the centers of the two bases and is perpendicular to both bases. Cylinders have two planes of symmetry and are a type of prism, with the same cross-sectional area at every height.

Formulas
Surface Area (SA): \( SA = 2 \pi r^2 +2 \pi rh \)

The lateral surface area (LSA): \( LSA= 2 \pi rh \)

Volume (V): \( V = \pi r^2 h \)

Right Circular Cylinder
A right circular cylinder has a circular base, and its axis is perpendicular to the base. In this case, the height and the lateral edge of the cylinder are the same.

Elliptical Cylinder
An elliptical cylinder has an elliptical base, with major axis \(a\) and minor axis \(b\). The volume \((V)\) of an elliptical cylinder is given by the formula:
\(V = \pi abh \) where \(a\) and \(b\) are the semi-major and semi-minor axes of the ellipse, respectively, and \(h\) is the height of the cylinder.

Properties

• A cylinder has two planes of symmetry: one passing through the axis and parallel to the bases, and the other perpendicular to the axis and bisecting the height.
• The lateral surface area of a cylinder is equal to the base circumference multiplied by the height.
• A cylinder has the same cross-sectional area at every height, making it a type of prism.
• If the bases of a cylinder are not parallel, the cylinder is called an "oblique cylinder."

Real-World Applications

• Cylindrical shapes are used in various engineering and architectural applications, such as pipes, columns, and storage tanks, due to their strength and ease of manufacturing.
• Cylindrical lenses can be used to correct astigmatism in eyeglasses by focusing light onto a single focal line.