CELE Engineering Mathematics — Plane, Solid Geometry and MensurationConcept Map

Concept

Plane Figures

Sub Concepts

• Triangles (Heron's formula, base-height method)
• Circles and circular segments
• Polygons (regular and irregular)
• Trapezoids and quadrilaterals
• Sectors and arcs

Relationship To Central

Foundation for calculating areas of 2D shapes used in building footprints, land parcels, and cross-sectional calculations

Concept

Solid Figures

Sub Concepts

• Prisms and cylinders
• Pyramids and cones
• Spheres and hemispheres
• Frustums (truncated pyramids and cones)
• Composite solids

Relationship To Central

Core for calculating volumes and surface areas of 3D structures (buildings, tanks, embankments, concrete members)

Concept

Pappus Theorems

Sub Concepts

• First Theorem (Surface Area of Revolution)
• Second Theorem (Volume of Revolution)
• Centroid location and axis distance

Relationship To Central

Advanced technique for solids of revolution; applies to irregular shapes generated by rotation about an axis

Concept

Prismatoid Formula

Sub Concepts

• End areas (A₁ and A₂)
• Mid-section area (Aₘ)
• Height between sections

Relationship To Central

Critical for earthwork volume calculations; used to estimate volumes of irregular solids by sectional areas

Concept

Practical Engineering Applications

Sub Concepts

• Earthwork volume and cut-fill calculations
• Concrete quantity estimation (ACI 318 standards)
• Water tank and reservoir capacity design
• Foundation and footing area calculations
• Bridge deck and structural member dimensions
• Stormwater and wastewater facility sizing

Relationship To Central

Real-world use cases demonstrating why mensuration is essential in civil engineering practice

To

Solid Figures

From

Plane Figures

Strength

strong

Relationship

Plane figures form the bases and cross-sections of solid figures; calculating solid volume requires knowing plane figure areas

To

Prismatoid Formula

From

Solid Figures

Strength

strong

Relationship

Prismatoid formula extends solid geometry to irregular solids by using area measurements at multiple cross-sections

To

Solid Figures

From

Pappus Theorems

Strength

moderate

Relationship

Pappus theorems provide alternative methods to calculate volumes and surface areas of solids of revolution, complementing direct formulas

To

Pappus Theorems

From

Plane Figures

Strength

strong

Relationship

Pappus theorems require precise calculation of plane figure areas and centroids as input data

To

Practical Engineering Applications

From

Prismatoid Formula

Strength

strong

Relationship

Prismatoid formula is the primary tool for earthwork volume calculations in highway and dam engineering

To

Practical Engineering Applications

From

Solid Figures

Strength

strong

Relationship

Direct solid geometry formulas are applied to calculate concrete volumes, tank capacities, and structural member dimensions per ACI 318

To

Cylinder

From

Circle

Strength

strong

Relationship

Cylinder volume is calculated using circular base area: V = πr²h

To

Pyramid

From

Triangle

Strength

strong

Relationship

Pyramid volume uses triangular base area: V = 1/3 × Base Area × h

To

Cone

From

Circle

Strength

strong

Relationship

Cone volume uses circular base area: V = 1/3 × πr²h

To

Pyramid and Cone

From

Frustum

Strength

strong

Relationship

Frustum is a truncated pyramid or cone; requires both end areas (A₁, A₂) to calculate volume

To

Pappus Theorems

From

Centroid Location

Strength

strong

Relationship

Pappus theorems depend critically on accurate centroid location and distance to rotation axis

To

Prismatoid Formula

From

Trapezoid

Strength

moderate

Relationship

Trapezoidal cross-sections are common in earthwork; prismatoid formula combines multiple trapezoidal areas for volume

To

Solid Figures

From

Regular Polygon

Strength

moderate

Relationship

Regular polygonal bases form prisms and pyramids; area formula used in volume calculations

To

Earthwork

From

Practical Engineering Applications

Strength

strong

Relationship

Earthwork volume calculations use prismatoid formula and average end area method for cut-fill design

To

Concrete Design

From

Practical Engineering Applications

Strength

strong

Relationship

ACI 318 structural design requires accurate volume calculations for beam, column, footing, and slab dimensions