# Solid Mensuration

**Topics:**Cube, Prism, Rectangle

**Pages:**10 (1850 words)

**Published:**March 5, 2013

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. As the volume of a cube is the third power of its sides , third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length+width+height).

* Parts:

Side/ lateral Face

Height (s)

Bases

* formulae: Surface Area= 6s2

Volume: s3

Face Diagonal:

Space Diagonal:

Radius of Circumscribed Sphere:

Radius of sphere tangent to edges

Radius of inscribed sphere

Angles between faces (in radians)

Questions:

1. What is the volume of a cube with sides of length of 6 cm?

2. How many dices with sides of 1.8 cm in length will fit in a cube box with side of 30 cm in length?

* Cuboid/ Rectangular prism

In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While some mathematical literature refers to any such polyhedron as a cuboid, other sources use "cuboid" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. In a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. It is also a right rectangular prism.

* Parts:Rectangular cuboid

length

width

side face

height front/ back face

bases (B)

the number of faces ('F'), vertices (V), and edges (E) of any convex polyhedron are related by the formula "F + V - E" = 2 . In the case of a cuboid this gives 6 + 8 - 12 = 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges.

* Formulae:

If the dimensions of a cuboid are a, b and c, then its volume is abc and its surface area is 2ab + 2ac + 2bc. Surface Area= Lateral area + area of bases

= LA+2B

Volume = Bh ; where B= lw

Questions:

1. Find the volume of a rectangular prism with base length 6 cm, base width 5 cm, and height 4 cm.

2. What is the volume of a rectangular solid with base length 11 dm, base with 5 dm, and height 6 dm ?

* Square Pyramid

In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4vsymmetry. Its faces are consists of 4 triangles and 1 square as the base. It has 6 edges and 5 vertices A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base. Square frustum is a square pyramid with the apex truncated.

* Parts:

height (h) slant height (s)

base (b)

* Formulae:

Surface Area= Lateral area + area of bases ; where LA= n(1/2 bs) ; n= number of lateral faces

b= base of the triangular face

s= slant height

SA= LA+B

Volume = 1/3 Bh; B= area of base

Questions:

1. Find the surface area of a square pyramid whose base is 6 cm on a side and whose slant height...

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