Solid Geometry

In solid geometry three dimensional shapes are studied. The solids studied are generally formed by two dimensional shapes like polygons and circles. The two main categories of solids that are studies are polyhedrons and cylinders.

Names of Geometric Shapes


The solid geometric shapes are,
 
  • Prism
  • Cube
  • Cuboid
  • Sphere
  • Hemisphere
  • Pyramid
  • Right circular cone

Polyhedrons: A polyhedron is a solid formed by flat polygonal surfaces.
  • The flat surface of polyhedron is called a face.
  • The line segments formed by the intersection of the faces are called the edges
  • The point of intersection of the edges is called a vertex.

Polyhedron

Prism

A prism is a polyhedron with two parallel bases.
  • The other faces of the prism are parallelogram and they constitute the lateral surface of the prism. 
  • In a right prism, the lateral faces are rectangles.
  • A prism whose bases are regular polygons is called a regular prism.

Constructive Solid Geometry


Triangular Prism : A triangular prism has two triangular bases and three lateral surfaces which are rectangles.

Triangular Prism

Cuboid or a rectangular prism : A rectangular prism or a cuboid is a prism with rectangular base. The lateral surfaces of the cuboid are also rectangles.

Rectangular Prism

Pentagonal Prism : A pentagonal prism has two pentagonal bases and five rectangular lateral surfaces.

Pentagonal Prism

Triangular and rectangular prisms are the common types of prisms studied in the topic of Solid Geometry. The two measures of solids which are generally discussed for solids are Surface Area and Volume.

The Surface area of a prism consists of two base areas and lateral surface area. The general formula used for the Lateral surface area of a right prism is,

Lateral surface area of a right  prism = Ph

Where P is the perimeter of the base and ‘h’ is the altitude of the prism.

Total surface area = Lateral surface area + 2 base area

Total surface area of a right prism = Ph + 2B

Where B represents the area of the base of the prism.

Volume of a prism = Bh

Where B is the area of the base and ‘h’ is the height of the prism.

Cuboid

Cuboid is another name for the rectangular prism. A cuboid has 6 rectangular faces. The 6 faces are formed by 3 pairs of congruent and parallel rectangles. Each such pair is perpendicular to the two other pairs. Any one pair of such rectangles can be considered as the bases, while the other four rectangles form the surface.
 
The formulas for surface area and volume of a cuboid are derived from the general formula as follows:


Total surface area of a cuboid = 2(lw + wh + hl) sq units

Volume of a cuboid = lwh cubic units

Where l, w, h are the lengths of the distinct edges of the cuboid.

Cube

Cube is a special cuboid, where all the six faces are congruent squares. If the length of an edge of a cube is ‘a’ units then the surface area and volume of the cube are,

Cube

Total surface area of a cube = 6a2 sq units

Volume of the cube = a3 cubic units.

Cylinders

If the bases of the solid are not polygons, but two parallel congruent closed curves, then the solid is called a cylinder. Just as the prisms, the cylinders can also be classified as oblique and right cylinders.

The commonly known cylinders are the circular cylinders where the bases are congruent parallel circles. If the line joining the centers of these circular bases is perpendicular to the bases, then the cylinder is called a right circular cylinder. The right circular cylinders is the shape we find commonly in day to day life, like cans, rollers, pencils, water tanks and barrels. In mathematical situations the word cylinder generally refers to the right circular cylinder unless otherwise stated.
 
A right circular cylinder has a curved lateral surface, which when spread out will be in the form of a rectangle whose length is equal to the circumference of the base and the width is the height of the cylinder. The formulas for the surface area and volume of a cylinder with base radius ‘r’ and height ‘h’ are,

Lateral surface area of a cylinder = 2$\pi$rh sq units

Total surface area of a cylinder = (2$\pi$rh+2$\pi r^{2}$) sq units

Volume of the cylinder = $\pi r^{2}h$ cubic units.


Cylinder

Pyramids

A pyramid has only one base as against prisms and cylinders.

Pyramid

  • Pyramid is a solid with a polygonal base and whose lateral surfaces are triangles.
  • A regular pyramid is a pyramid whose base is a regular polygon and whose altitude is perpendicular to the base at its center.
  • The lateral faces of a regular pyramid are isosceles triangles.
  • The length of the altitude of a triangular lateral face of a regular pyramid is called the slant height of the Pyramid.

All the lateral surfaces of the pyramid meet at a point which is called the vertex of the pyramid. The common type of regular pyramids are square and triangular pyramids.

The lateral surface area of a pyramid = Sum of the areas of the lateral triangles.

Total surface area of a pyramid = Lateral surface area + base area.

Volume of a pyramid = $\frac{1}{2}$ Bh ,  where B is area of the base and h is the altitude of the  pyramid.

Right Circular Cone

A cone has a circular base and a curved lateral surface which converge on vertex. The Birthday caps are examples for cones. A cone is a right circular cone if the altitude meets the base at its center. Three measures of the cone are considered for finding the formulas of surface area and volume of a cone, the base radius r, the altitude h and the slant height l.

Right Circular Cone

Base area of the cone = $\pi r^{2}$ sq units.

Lateral surface area of the cone = $\pi r l$sq units.

Total surface area of the cone = $\pi r^{2}$ + $\pi r l$ sq units.

Volume of the cone = $\frac{1}{3}$ $\pi r^{2}h$ cubic units.

Sphere

Sphere is the three dimensional equivalent of the circle. A ball is an example for a sphere.

A sphere is a set of all points equidistant from a fixed point called the center. The radius ‘r’ of a sphere is the distance between the center and any point on the surface of a sphere.

Sphere
Surface area of a sphere = 4$\pi r^{2}$ sq units

Volume of a sphere = $\frac{4}{3}$ $\pi r^{3}$ cubic units

A hemisphere is a solid got by cutting a sphere into to two congruent halves. A sphere has only a curved surface and no base surface. This against a hemisphere will have a circular base area. The radius ‘r’ of a sphere is the only dimension that determines a sphere.

Hemisphere Surface area of a hemisphere =3$\pi r^{2}$ sq units

Volume of a hemisphere = $\frac{2}{3}$$\pi r^{3}$ cubic units