Solid Mensuration 2

SOLID MENSURATION: CHA CHAPTER PTER II: II:

CUB CUBES ES;; RECT RECTA ANGUL GULAR PAR PARAL ALL LELEPIP EPIPES ES

Solid figures are three (3) dimensional figures or solids that may either be classified as either regular solids or irregular solids. In essence, esse nce, a solid is any limited portion of space bounded by surfaces. **

CUBE:

is a rectangular parallelepiped whose sixe faces are all squares. The lateral edges of a cube are all equal and the six (6) faces are congruent squares. TYPICAL PARTS OF A CUBE:

section lateral

base

diagonal altitude lateral vertex

Base bounding planes located at the top and bottom Lateral face bounding planes excluding the bases height covering the two (2) bases and is equal to the lateral Altitude

edge Lateral edge intersection of the lateral faces a straight line joining any two (2) vertices which are not on the Diagonal same plane. Vertex intersection of the edges that is usually formed by three (3) to four (4) lines. Properties of a cube:

1. The lateral lateral faces are parallelo parallelograms grams 2. The latera laterall edges edges are are equal equal and and paralle parallel. l. 3. The bases bases are equal equal and parallel parallel polyg polygons. ons. 4. All six six (6) (6) sides sides are equal equal and parallel parallel 5. Any of the the lateral lateral faces faces may also also be taken taken as the the bases. bases. 6. Any section section made made by a plane plane at right angles angles with with the the lateral lateral edge or along the sides of its base are equal polygons. 7. The section section made made by a plane plane passing passing through through any two two diagonally diagonally opposite edges is a rectangle. F OR MU LA S : TOTAL AREA:

AT

TOTAL VOLUME:

=

VT

=

where: cube’s edge

a

--

lateral edge/

SAMPLE PROBLEMS:

1. What is the weight of a block of ice 24 in. by 24 in. by 24 in., if this ice weighs 92% as much as water, and water weighs 62.5 lbs per cu.ft.?

2. Find the volume and total area of the largest cube of wood that can be cut from a log of  circular cross-section whose radius is 12.7 in..

cube the two two opposite faces. is 49 √2 cm2, find the

3. A plane is passed through two opposite edges of a forming a diagonal section bounded by opposite edges and the diagonals of  If the area of the section formed edge of the cube.

4. Pass a plane through a cube so that the section formed will be a regular hexagon. If the edge of the cube measures 2 ft, find the area of this section

5. of Lateral m.

From the figure shown, determine the area the triangle generated from the inside. edge of this cube measures 2

** RECTANGULAR PARALLELEPIPES (lateral faces) are all lateral edges are perpendicular to its

a type of polyhedron whose six faces rectangles and whose bases.

PARTS OF A RECTANGULAR

Lateral face

With a cutting plane passing

Diagonal

Altitude

Base Lateral edge vertex

Properties of a rectangular parallelepiped:

1. 2. 3. 4. 5. 6.

Opposite faces are congruent and parallel. Lateral edges are equal. Any opposite faces maybe taken as the bases. The diagonals are equal. All faces including the bases are rectangles. Every section made by a plane parallel to the base is equal in area to that of the base. 7. The section made by a plane passing through any two diagonally opposite edges is a rectangle. FORMUL AS

:

TOTAL AREA:

AT

TOTAL VOLUME:

=

2 (ab + bc where: b

SAMPLE PROBLEMS:

VT a length width

c

=

abc height