Solid Mensuration

Overview

• Also known as Solid Geometry
• Study of various solids, measuring volume, area, height, length, etc.

Key Concepts

Point

• No dimensions
• No length, width, or thickness
• Represents an exact location in space

Line

• No width or thickness, only length
• Has one dimension

Plane

• No thickness
• Contains infinite points and lines
• Extends indefinitely in all directions

Space

• Set of all points

Intersections

• Common point, line, or plane shared by multiple geometric objects

Types of Lines

Parallel Lines

• Lines on a plane that do not intersect

Intersecting Lines

• Lines on a plane that meet at one point

Coinciding Lines

• Lines that contain the same set of points

Collinear Points

• Points that lie on the same line

Coplanar Points

• Points that lie in the same plane

Concurrent

• Three or more distinct lines that pass through a common point

Polygons

• Plane closed figures with sides that are line segments

Parts of a Polygon

• Sides or Edge: Line segments that make up the polygon
• Adjacent sides: Pairs of sides sharing a common endpoint
• Vertices: Endpoints of each side
• Adjacent Vertices: Endpoints of a side
• Diagonal: Line segment joining two non-adjacent vertices
• Interior Angle: An angle formed by two adjacent sides inside the polygon
• Exterior Angle: An angle adjacent to and supplementary to an interior angle

Types of Polygons

• Equiangular Polygon: All angles are congruent
• Equilateral Polygon: All sides are equal
• Regular Polygon: Both equiangular and equilateral
• Irregular Polygon: Neither equiangular nor equilateral
• Convex Polygon: Every interior angle less than 180 degrees; a line intersects two sides at most
• Concave Polygon: At least one interior angle more than 180 degrees; a line may intersect more than two sides

Specific Polygons

Triangle

• Simplest polygon with the least number of sides

Quadrilateral

• Polygon with four sides

Types of Quadrilaterals

• Parallelogram: Two pairs of opposite sides parallel
  • Rectangle: Four right angles and congruent diagonals
  • Rhombus: Four congruent sides, diagonals perpendicular
  • Square: Rectangle with four congruent sides
• Trapezoid: One pair of opposite sides parallel
• Trapezium: General quadrilateral with no parallel opposite sides

Additional Terms

Radius

• Distance from the center of a regular polygon to any vertex

Apothem

• Altitude of the isosceles triangle formed
• Line from the center of the polygon perpendicular to its side
• Bisects the central angle and its opposite side

Perimeter

• Sum of the length of the sides of a polygon

Circle

• Set of all points equidistant from a fixed center point

Types of Circles

• Inscribed Circle: Tangent to all polygon sides
• Circumscribed Circle: Contains all the vertices of the polygon
• Sector: Region bounded by two radii and their intercepted arc
• Ratio: Fractional part of the circle in the sector
• Chord: Straight line segment connecting two circumference points
• Segment: Region bounded by an arc and a chord
• Concentric Circles: Circles with different radii and a common center
• Annulus: Area bounded by any concentric circles
• Annular Region: Shaded or annulus region
• Tangent Circles: Circles tangent to the same line at the same point