GELE Mathematics — Plane, Solid Geometry and MensurationStudy Notes

Plane geometry deals with two-dimensional shapes. Understanding how to calculate areas and perimeters is fundamental to civil engineering, as these measurements directly feed into three-dimensional volume and surface area problems. **Triangle** The most versatile plane figure in engineering. Three methods exist for computing area: 1. **Base-Height Method:** A = (1/2)bh Where b is the base and h is the perpendicular height. This is the simplest and most commonly used. 2. **Heron's Formula:** A = √[s(s-a)(s-b)(s-c)] Where s = (a+b+c)/2 is the semi-perimeter, and a, b, c are the side lengths. Use this when heights are unknown. 3. **Coordinate Method:** For a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): A = (1/2)|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| **Circle** Circles appear frequently in civil engineering—circular columns, pipe cross-sections, roundabouts: - Area: A = πr² - Circumference: C = 2πr = πd - Sector area (θ in radians): A_sector = (1/2)r²θ - Sector area (θ in degrees): A_sector = (θ/360°)πr² - Segment area: A_segment = A_sector − A_triangle = (1/2)r²(θ − sin θ) **Regular Polygon** (n sides, each of length s) Used in architectural elements and structural design: A = (1/4)ns² cot(180°/n) Alternatively, for a regular polygon inscribed in a circle of radius r: A = (1/2)nr² sin(360°/n) **Trapezoid** (parallel sides b₁ and b₂, height h) Common in cross-sectional earthwork calculations: A = (1/2)(b₁ + b₂)h **Rectangle and Square** - Rectangle: A = length × width - Square (side a): A = a² **Parallelogram** A = base × height (perpendicular height, not slant height) **Key Distinction for Engineers:** When calculating earthwork cross-sections or trapezoidal channel sections per NSCP 2015, always use perpendicular height, never slant distance.

Solid geometry extends plane figures into three dimensions. Volume calculations are essential for: - Concrete pouring (structural members) - Earthwork excavation and fill - Tank and reservoir capacity - Material cost estimation **Prism and Cylinder** Both have constant cross-sectional area along their height. - Volume: V = A_base × h - Cylinder (circular base): V = πr²h - Lateral (curved) surface area of cylinder: S_lateral = 2πrh - Total surface area of cylinder: S_total = 2πrh + 2πr² = 2πr(h+r) **Pyramid and Cone** Both have an apex and taper from base to point. Volume is 1/3 of corresponding prism/cylinder: - Volume: V = (1/3)A_base × h - Cone: V = (1/3)πr²h - Cone lateral surface area: S_lateral = πrL (where L is slant height) - Cone total surface area: S_total = πrL + πr² = πr(L+r) - Relationship between slant height L, radius r, and vertical height h: L = √(r²+h²) **Sphere** A perfect three-dimensional round object: - Volume: V = (4/3)πr³ - Surface area: S = 4πr² **Frustum of Cone or Pyramid** A truncated cone/pyramid with two parallel bases (radii r₁ and r₂, or areas A₁ and A₂): - Volume: V = (h/3)(A₁ + A₂ + √(A₁A₂)) - For cone frustum with radii r₁ (top) and r₂ (bottom): A₁ = πr₁², A₂ = πr₂² - Slant height (cone frustum): L = √[h² + (r₂−r₁)²] - Lateral surface area (cone frustum): S_lateral = π(r₁+r₂)L **Prismatoid Formula** (Fundamental for Earthwork) For any solid with two parallel bases A₁ and A₂ and height h, with A_m being the area of the mid-section: V = (h/6)(A₁ + 4A_m + A₂) This is the basis for earthwork volume calculations in NSCP 2015, where cross-sections at different stations are measured and volumes computed using this formula. It provides high accuracy compared to simpler methods. **Critical Distinction:** - Use the simple formula V = (h/3)(A₁ + A₂ + √(A₁A₂)) ONLY for frustra - Use prismatoid formula for general solids where the middle section area differs from the arithmetic mean of end areas

The Pappus theorems provide elegant shortcuts for calculating surface areas and volumes of shapes generated by revolving a plane figure around an external axis. These theorems are powerful tools in civil engineering for analyzing tanks, pipes, and structural elements with radial symmetry. **First Pappus Theorem (Surface Area of Revolution)** When a plane curve of length L is revolved around an external axis: S = 2π d̄ L Where: - S = surface area of the solid of revolution - d̄ = perpendicular distance from the curve's centroid to the axis of revolution - L = length (arc length) of the curve being revolved - The 2π d̄ represents the circumference traced by the centroid **Second Pappus Theorem (Volume of Revolution)** When a plane area A is revolved around an external axis: V = 2π d̄ A Where: - V = volume of the solid of revolution - d̄ = perpendicular distance from the area's centroid to the axis of revolution - A = area of the plane figure being revolved - Again, 2π d̄ is the path traced by the centroid **Key Insight:** Both theorems reduce the problem to finding the centroid location and multiplying by the circumference path (2π d̄). This avoids integration and is particularly useful for composite or irregular shapes. **Requirements for Pappus Theorems:** 1. The axis of revolution must NOT intersect the plane figure (external axis only) 2. The figure must not cross the axis 3. The plane figure and axis must be coplanar **Common Applications:** - Torus (doughnut shape): revolving a circle - Hollow pipes and tanks: revolving rectangular sections - Lamp shades and conical sections: revolving triangles - Architectural domes: revolving curve profiles **How to Find Centroids of Common Shapes:** For a triangle: centroid is at (1/3) of the height from the base, or at the intersection of medians. For a rectangle: centroid is at the geometric center (L/2, W/2). For a semicircle: centroid is at distance 4r/(3π) from the diameter. For a composite area: use weighted average: d̄ = Σ(A_i × d_i) / Σ(A_i)

Real-world engineering problems rarely involve simple, isolated geometric shapes. Engineers must decompose complex structures into manageable geometric components, calculate volumes and areas for each, then sum or subtract to find the overall result. **Composite Solids Strategy** When faced with an irregular solid: 1. **Decompose** the solid into recognizable geometric shapes (prisms, pyramids, cylinders, cones, spheres, frustums) 2. **Orient** each component so its volume formula is straightforward 3. **Calculate** the volume (or surface area) of each component 4. **Combine** by addition (for solid components) or subtraction (for voids/holes) 5. **Verify** the answer makes physical sense (positive volume, reasonable magnitude) **Hollow Cylinders and Pipes** Common in water supply, sewage, and structural columns: V = π(R² − r²)h Where R is outer radius, r is inner radius, h is height. Alternatively: V = V_outer − V_inner = πR²h − πr²h **Cylindrical Tanks with Domed or Conical Bottoms** A practical water tank often combines: - Cylindrical main body: V_cyl = πr²h_cyl - Hemispherical bottom (half-sphere): V_hemi = (2/3)πr³ - Total: V_total = πr²h_cyl + (2/3)πr³ **Truncated Cones in Stepped Excavations** Earthwork often requires stepped slopes. A frustum between two levels calculates the volume of cut or fill. **Partial Solids** When a solid is partially filled or cut: - Partially filled cylinder: requires segment area calculation first - Wedge (portion of prism): calculate proportionally based on height ratio - Spherical cap (portion of sphere): V_cap = (πh²/3)(3r − h), where h is cap height and r is sphere radius **Integration Method for Irregular Solids** For complex geometries not fitting standard formulas: V = ∫ A(x) dx from x₁ to x₂ Where A(x) is the cross-sectional area as a function of position x. For prismatoid (most practical earthwork): V = (h/6)(A₁ + 4A_m + A₂) **Symmetry and Simplification** Always look for symmetry: - Central axis symmetry: calculate one section, multiply by 2 - Radial symmetry: use Pappus theorems - Similar sections: use proportionality **Common Errors in Composite Solid Problems** 1. **Unit inconsistency**: mixing meters, centimeters, etc. (always use SI—meters) 2. **Double counting**: including the same volume twice 3. **Sign error**: forgetting to subtract voids 4. **Formula confusion**: using cone volume (1/3) instead of cylinder (1×) 5. **Centroid misplacement**: Pappus theorems require correct centroid location 6. **Slant vs. vertical height**: using L instead of h in volume formulas

Mensuration directly supports daily engineering decisions in practice and licensure exams. **Concrete and Material Estimation** When designing reinforced concrete beams, columns, slabs, and foundations: - Column volume: V = A_cross-section × height (if uniform) or sum of segments if tapered - Footing volume: typically a frustum (larger top, smaller bottom) or trapezoidal prism - Concrete mix design requires accurate volume to calculate cement, sand, aggregate quantities - Example: A square footing 2.0 m × 2.0 m × 0.5 m deep requires V = 2.0 × 2.0 × 0.5 = 2.0 m³ concrete, translating to ~300 bags of cement (at ~10 kg/bag) **Water Tank and Reservoir Design** Capacity calculations ensure adequate supply per RA 544 (Philippine Water Code): - Minimum capacity = daily demand (for small systems) or 25–50% of daily demand (for larger systems) - Tank shape optimization: cylindrical with hemispherical ends minimizes surface area (cost) for a given volume - Partially filled tanks require segment area calculations to determine water level vs. volume - Example: A cylindrical tank 5 m diameter × 8 m high = π(2.5)²(8) ≈ 157.08 m³ serves ~5,000 people at 30 L/capita/day **Earthwork Calculations (Cut-and-Fill)** Dam, embankment, and foundation construction requires precise volume estimates: - Uses prismatoid formula: V = (h/6)(A₁ + 4A_m + A₂) per NSCP 2015 § 207.3 - Cross-sectional areas measured at regular stations (often 50 m or 100 m spacing) - Cut volume (excavated) must match fill volume (compacted) for balance - Cost = volume × unit rate per m³ (excavation: ₱100–200/m³; fill: ₱50–100/m³ depending on material) **Structural Columns and Piles** - Circular column: V = π(d/2)²h; stress = load/A = load/[π(d/2)²] - Rectangular column: V = b × d × h; stress = load/(b×d) - Pile group: V_total = n × V_single pile (where n is number of piles) - Example: A 0.4 m diameter circular column, 6 m high: V = π(0.2)²(6) ≈ 0.754 m³ concrete **Pipe and Conduit Networks** - Hollow cylinder: V = π(R² − r²)L (for pipe material) - Water flow capacity: Q = A × v, where A = π(d_inner/2)² (cross-sectional area) - Example: 1.0 m diameter pipeline over 2 km: Material volume (assuming 100 mm thick walls) = π[(0.5)² − (0.4)²] × 2000 ≈ 565.5 m³ **Excavation and Slope Stability** - Trapezoidal embankment: cross-section area = (1/2)(top + bottom) × height - Volume between stations: use prismatoid for accuracy - Slope angle α affects embankment area: flatter slopes (smaller α) increase area - Example: A road embankment 4 m high with 1:1.5 slope (H:V) has width = 4 × 1.5 × 2 = 12 m **Spherical and Hemispherical Structures** - Storage domes, water tanks, architectural features - Volume per unit height relates to load distribution analysis - Surface area determines weather protection, insulation, and painting requirements **Landscape and Site Grading** - Terracing: calculate cut/fill volumes for each terrace - Pond and retention basin sizing: V = (1/2)(A_top + A_bottom) × depth for trapezoidal sections - Topographic surveying: use contour maps and prismatoid formula to determine excavated or filled volumes **Quality Control in Manufacturing** Concrete blocks, precast elements, and manufactured products: - Verify product dimensions: if nominal block is 200×100×200 mm, actual volume ≈ 0.004 m³ - Batch production: 1000 blocks = 4 m³ material (minus allowable tolerances) **Cost and Resource Management** - Earthwork: Volume × rate = project cost - Material: Volume × density = mass → material cost - Labor: Volume ÷ productivity rate = person-hours → labor cost - Example: Excavation 5000 m³ at ₱150/m³ = ₱750,000 budget **Exam Contexts** PRC Licensure Examination frequently tests: - Frustum volume for foundation/pile design - Prismatoid formula for earthwork - Sphere/hemisphere for tank design - Composite solids (cylinder + cone) for complex structures - Pappus theorems for special cases (rarely, but important for distinction)