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Curved Solids (area , volume)

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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

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

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."

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.


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.

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

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

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


Real-World Applications


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.

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.




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.

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.


Real-World Applications