# SOLID MENSURATION

## PLANE AND SOLID MENSURATION

SOLID MENSURATION - also known as solid geometry, it is the study of various solids (Volume Area, Height, etc.)

SOLID - any limited portion of space bounded by surfaces of plane figures

PLANE FIGURE - two dimensional

SOLID FIGURE - three dimensional

### PARTS OF SOLIDS

EDGES - the intersection of the bounding plane

FACES - the portions of the bounding planes included by the edges

VERTICES - the intersections of the edge

DIAGONALS - are any straight lines joining any two vertices that are not in the same face

SECTION - plane figure formed by the intersection of a plane and a solid

RIGHT SECTION - a section of solid that is perpendicular to one of its lateral edges

### TYPES OF SOLIDS

• CUBE

• RECTANGULAR PRISM

• TRIANGULAR PRISM

• CYLINDER

• SPHERE

• CONE

• PYRAMID

VOLUME OF A SOLID - a volume of a solid is the amount of space it occupies. It is denoted in units of cubic length

AREA OF A PLANE - the space enclosed by the boundary of a plane figure. It is denoted in units of square length

SURFACE AREA OF A SOLID - also known as Total Surface area, it is the area of a 3-dimensional surface or solid

LATERAL AREA OF A SOLID - areas of the lateral or side surfaces except for the base and top

### SPECIAL TYPES OF SOLIDS

POLYHEDRON - is a solid which is bounded by polygons joined at their edges

- REGULAR - made of the same, regular polygons

- IRREGULAR - made of the same or different irregular polygons

• PLATONIC SOLIDS OR REGULAR POLYHEDRA - solids that have polygonal faces that are similar in form, height, angles, and edges. It is also a 3-dimensional convex and regular solid

- TETRAHEDRON

- CUBE

- OCTAHEDRON

- DODECAHEDRON

- ICOSAHEDRON

• IRREGULAR POLYHEDRA -

- PRISM - a solid shape that is bound on all its sides by plane faces

- PYRAMID - is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face

SIMILAR SOLIDS - two or more solids that have the same shape but not the same size

## BASIC MATHEMATICAL CONCEPTS

AXIOM METHOD - used to arrive at a theory based on axioms

AXIOMS/POSTULATES - self-evident statements that do not require proof

THEOREMS - statements that need to be proved

PROPOSITION - same definition, but of lesser importance than a theorem

LEMMA - a proven statement used to prove a theorem

COROLLARY - statements consequent to the theorem

POINT - a fixed location in space represented by a dot

LINES - is contained within two points and extends indefinitely in two opposite directions

LINE SEGMENT - line with two endpoints

RAY - a portion of a line that consists of an endpoint and a set of all points on one side of the line

COLLINEAR POINTS - points that lie on the same line

NONCOLLINEAR POINTS - no line on which all of the points lie

PLANE - a flat surface that extends indefinitely. Three noncollinear points determine a plane

COPLANAR POINTS - are points that lie on the same plane

PARALLEL LINES - are lines in the same plane that have no points in common

SKEW LINES - non-coplanar lines that do not parallel nor intersect

PARALLEL PLANES - planes that do not intersect

## ANGLES AND POLYGONS

ANGLES - set of all points that are the union of rays having the same endpoint. It may be named using capital letters, Greek letters, or three capital letters

TYPES OF ANGLES

NAME

SIZE OF ANGLE

ACUTE ANGLE

0-90 degrees

RIGHT ANGLE

exactly 90 degrees

OBTUSE ANGLE

90-180 degrees

STRAIGHT ANGLE

exactly 180 degrees

REFLEX ANGLE

180-360 degrees

360 ANGLE

one complete rotation or revolution

VERTICAL ANGLES - angles lying on opposite sides or a pair of non-adjacent angles formed by two lines that are intersecting

CONGRUENT ANGLES - have the same angle measure

ALTERNATE EXTERIOR ANGLES - pair of angles that are formed on the other side of the parallel lines but on the opposite side of the transversal

CORRESPONDING ANGLES - angles formed in matching corners or corresponding corners with the transversal

DIHEDRAL ANGLE - an angle formed by two intersecting planes

ADJACENT DIHEDRAL ANGLE - two dihedral angles having a common edge and a common face

PERPENDICULAR PLANES - are two intersecting planes that form a right dihedral angle

POLYHEDRAL ANGLE - an angle formed by three or more intersecting planes at a common point or vertex

TYPES OF POLYHEDRAL ANGLES

• tetrahedral

• pentahedral

• hexahedral

• heptahedral

POLYGON - closed plane figure formed by three or more line segments

SIDES - line segments

VERTEX - the intersection of two sides in a polygon

CONSECUTIVE SIDES - are sides that share a common vertex

CONSECUTIVE VERTICES - two endpoints of any side of a polygon

DIAGONAL - a line segment whose endpoints are nonconsecutive vertices of a polygon

CONVEX POLYGON - where diagonals lie inside of a polygon

CONCAVE POLYGON - where diagonals lie outside of a polygon

EQUILATERAL - congruent sides

EQUIANGULAR - congruent angles

REGULAR POLYGON - a polygon that is both equilateral and equiangular