Plane, Solid Geometry and Mensuration - Cheat Sheet
Your 30-minute rapid-fire reference for plane figures, solids, and mensuration formulas. Covers all exam-critical material for areas, volumes, surface areas, and Pappus theorems used in earthwork, concrete calculations, and tank/reservoir design.
PLANE FIGURES — Areas
Formulas
A = ½bh
Meaning
Triangle area: b = base, h = perpendicular height
Watch out
Height MUST be perpendicular to base, not slant side; confusing height with side length is the #1 error
When to use
When base and vertical height are given; fastest method
A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2
Meaning
Heron's formula: a, b, c = triangle sides, s = semi-perimeter
Watch out
Must compute semi-perimeter s FIRST; students forget s is NOT the perimeter
When to use
When all three sides are known but height is not; three-side triangles
A = πr²
Meaning
Circle area: r = radius
Watch out
Students use diameter instead of radius; HALVE diameter first if given diameter
When to use
For any circle; fundamental formula
A = ½r²θ (θ in radians)
Meaning
Circular sector area: r = radius, θ = central angle
Watch out
If angle given in degrees, CONVERT to radians by multiplying by π/180°; this causes >50% of errors
When to use
Sector of a circle; angle MUST be in radians
A_segment = A_sector − A_triangle = ½r²(θ − sin θ)
Meaning
Circular segment: area between chord and arc
Watch out
Angle must be in radians; segment area is SMALLER than sector area
When to use
When chord cuts off part of circle; found by subtracting triangle from sector
A = ½(b₁ + b₂)h
Meaning
Trapezoid area: b₁, b₂ = parallel bases, h = perpendicular distance
Watch out
h is perpendicular distance between parallel sides, not slant height; sum bases BEFORE multiplying by ½h
When to use
Trapezoid with two parallel sides
A = ¼ns² cot(180°/n)
Meaning
Regular polygon: n = number of sides, s = side length
Watch out
Angle in degrees (180°/n); use cot, not tan; common for hexagon (n=6) in earthwork
When to use
Regular hexagon, octagon, pentagon, etc. with side length
A = ½ap
Meaning
Regular polygon alternate: a = apothem (center to side), p = perimeter
Watch out
Apothem is perpendicular from center, not radius; use radius r = a/cos(180°/n)
When to use
When apothem is given instead of side length
A = ½d₁d₂
Meaning
Rhombus or kite area: d₁, d₂ = diagonals
Watch out
Diagonals must BISECT each other (rhombus/square); for kite, one diagonal may not bisect other
When to use
Rhombus, square, or kite when diagonals are known
Common values
Value
3.14159...
Symbol
π
Quantity
π (pi)
Value
1.41421...
Symbol
√2
Quantity
√2 (diagonal of unit square)
Value
1.73205...
Symbol
√3
Quantity
√3 (height ratio in equilateral triangle)
Value
√3 ≈ 1.732
Symbol
cot(30°)
Quantity
cot(30°) for regular polygon (n=6)
Important facts
- Circle circumference C = 2πr = πd (d = diameter)
- Arc length L = rθ (θ in radians); L = πrθ/180° (θ in degrees)
- Segment area = ½r²(θ − sin θ) with θ in radians
- Regular hexagon = 6 equilateral triangles; area = (3√3/2)s² where s = side
- Regular polygon interior angle = (n−2)180°/n; central angle = 360°/n
- Inscribed circle radius (inradius) a = A/s where A = area, s = semi-perimeter for triangle
Key definitions
Radius (r)
Example
Circle with r = 5 m has area = π(5)² = 78.54 m²
Distance from center of circle to circumference; half the diameter.
Sector
Example
90° sector of 4 m radius circle: A = ½(4)²(π/2) = 4π ≈ 12.57 m²
Pie-slice portion of circle bounded by two radii and an arc.
Segment
Example
Segment area = sector area minus triangle area
Region between a chord and the arc it subtends.
Apothem
Example
Regular hexagon side 6 m, apothem = 6/2tan(30°) ≈ 5.196 m
Perpendicular distance from center of regular polygon to a side.
Diagrams to know
- Circle with radius, diameter, sector, segment, and chord labeled
- Triangle with base, height (perpendicular), and semi-perimeter notation
- Regular polygon with side, apothem, and central angle marked
- Trapezoid with parallel bases b₁, b₂ and height h perpendicular to bases
- Circular segment showing chord and arc
SOLIDS — Volumes & Surface Areas
Formulas
V = A_base × h
Meaning
Prism or cylinder volume: A_base = base area, h = height
Watch out
Height h must be PERPENDICULAR to base, not slant; common in concrete slab and tank calculations per ACI 318
When to use
Any prism (rectangular, triangular) or cylinder with perpendicular height
V = πr²h
Meaning
Cylinder volume: r = radius, h = height
Watch out
Use radius, not diameter; halve diameter first; height must be perpendicular to base
When to use
Cylindrical tanks, pipes, silos; standard in Philippine water tank design
S = 2πrh + 2πr²
Meaning
Cylinder surface area: lateral 2πrh + two circular bases 2πr²
Watch out
Lateral area only is 2πrh (no bases); don't double-count circular area
When to use
Total surface area including top and bottom
V = ⅓A_base × h
Meaning
Pyramid or cone volume: A_base = base area, h = perpendicular height
Watch out
Height h is perpendicular from apex to base plane; NOT the slant height; pyramid volume ⅓ not ½
When to use
Pyramid (triangular/square base) or cone; volume is ⅓ of equivalent prism
V = ⅓πr²h
Meaning
Cone volume: r = base radius, h = perpendicular height
Watch out
h is perpendicular height from apex to base, NOT slant height L; confused with ½ factor
When to use
Conical spoil piles in earthwork; standard in fill volume calculations
S_lateral = πrL
Meaning
Cone lateral surface area: r = base radius, L = slant height
Watch out
Slant height L ≠ perpendicular height h; L = √(r² + h²) is Pythagorean, not slant height given directly
When to use
Lateral (side) surface only; find L = √(r² + h²)
V = ⁴⁄₃πr³
Meaning
Sphere volume: r = radius
Watch out
Coefficient is 4/3 not 1/3; use radius r, not diameter d
When to use
Spherical tanks, pressure vessels, ball-shaped structures
S = 4πr²
Meaning
Sphere surface area: r = radius
Watch out
Surface area is 4πr² (not ⅘πr³); this is derived from rotating semicircle via Pappus
When to use
Coating, painting, or surface treatment of spherical tank
V = (h/3)(A₁ + A₂ + √(A₁A₂))
Meaning
Frustum volume (truncated cone/pyramid): A₁, A₂ = end areas, h = height between parallel faces
Watch out
Third term is √(A₁A₂), NOT A₁A₂; students forget square root; formula different from prismatoid
When to use
Cone/pyramid cut parallel to base; common in embankment and dam slope calculations per NSCP
V = (h/6)(A₁ + 4A_m + A₂)
Meaning
Prismatoid volume (general formula): A₁, A₂ = end areas, A_m = middle cross-section, h = length
Watch out
Coefficient 1/6 with middle section weighted 4×; NOT same as frustum; A_m is at h/2, not average area
When to use
Earthwork volumes (cuts/fills); basis of prismoidal rule in surveying per NSCP 2015
Common values
Value
3.14159...
Symbol
π
Quantity
π (pi)
Value
1.33333...
Symbol
4/3
Quantity
4/3 (sphere volume coefficient)
Value
1.41421...
Symbol
√2
Quantity
√2 (Pythagorean in diagonal)
Important facts
- Cone volume = ⅓ cylinder volume with same base and height
- Pyramid volume = ⅓ prism volume with same base and height
- Sphere surface area S = 4πr² can be derived from Pappus by rotating semicircle
- Frustum formula V = (h/3)(A₁ + A₂ + √(A₁A₂)) applies to both cone and pyramid frustums
- Prismoidal rule V = (h/6)(A₁ + 4A_m + A₂) is exact for all polyhedra and approximation for curved solids
- For right circular cone, lateral area = πrL where L = √(h² + r²)
- Hemisphere volume = ½ sphere = ⅔πr³; hemisphere surface (curved only) = 2πr²
Key definitions
Slant height (L)
Example
Cone h = 12 m, r = 5 m: L = √(144 + 25) = √169 = 13 m
Distance along slant surface of cone/pyramid from apex to base edge; L = √(h² + r²) for cone.
Apothem (pyramid)
Example
Square pyramid slant apothem different from perpendicular height h
Perpendicular distance from apex to midpoint of a base edge; used for lateral surface area of pyramid.
Frustum
Example
Frustum with A₁ = 25 m², A₂ = 4 m², h = 6 m: V = (6/3)(25 + 4 + √100) = 2(39) = 78 m³
Solid formed by cutting a cone or pyramid with a plane parallel to the base.
Prismatoid
Example
Earthwork volumes calculated using prismoidal rule V = (h/6)(A₁ + 4A_m + A₂)
Polyhedron with two parallel polygonal bases and trapezoidal or triangular lateral faces.
Diagrams to know
- Cylinder with radius r and height h; lateral surface unwraps to rectangle 2πr × h
- Cone showing apex, perpendicular height h, radius r, and slant height L forming right triangle
- Pyramid (square base) with apex height h and slant apothem on face
- Frustum with two parallel bases A₁ and A₂ separated by height h
- Sphere with diameter and radius labeled
- Right triangle for Pythagorean relationship: L² = h² + r²
PAPPUS THEOREMS — Solids & Surfaces of Revolution
Formulas
S = 2π d̄ L
Meaning
First Pappus Theorem (surface): S = surface area of revolution, d̄ = distance from centroid of curve to axis of rotation, L = arc length of curve
Watch out
d̄ is distance to axis, NOT to a point; L is arc length, not straight-line distance; curve must NOT cross axis
When to use
Surface of revolution generated by rotating a curve (line, arc, or polygon edge) around an axis; used for cone lateral area, torus surface
V = 2π d̄ A
Meaning
Second Pappus Theorem (volume): V = volume of revolution, d̄ = distance from centroid of area to axis of rotation, A = area of region revolved
Watch out
d̄ is centroid distance, NOT average position; area A must be planar region, not 3D; axis must be external to area
When to use
Volume generated by rotating a planar area around an external axis; used for torus, composite tank volumes
Common values
Value
2r/π
Symbol
d̄
Quantity
Semicircle arc centroid distance from center
Value
4r/(3π)
Symbol
d̄
Quantity
Semicircular area centroid distance from diameter
Important facts
- Cone lateral surface S = πrL can be derived: semicircle arc L = πr rotated about perpendicular axis at distance d̄ = r (from Pappus 1)
- Sphere surface S = 4πr²: semicircle (length πr, centroid at 2r/π from diameter) rotated about diameter → S = 2π(2r/π)(πr) = 4πr²
- Torus volume V = (2πd̄)(πr²) where d̄ = distance from circle center to axis, r = circle radius
- Centroid of semicircle arc from center: d̄ = 2r/π; for semicircular area: d̄ = 4r/3π from diameter
- Centroid of triangle from base: h/3 (one-third the height)
- Pappus theorems apply ONLY when curve/area does NOT cross the axis of rotation
Key definitions
Centroid (first moment)
Example
Circle of radius r rotated about axis at distance d̄ from center: volume = 2π(d̄)(πr²) = 2πd̄πr² (torus)
Geometric center of a line, area, or volume; for Pappus, the distance from this center to the axis of rotation.
Solid of Revolution
Example
Cone generated by rotating right triangle about one leg; sphere generated by rotating semicircle about diameter
3D solid generated by rotating a 2D shape (area or curve) about a fixed axis.
Torus
Example
Circle r = 2 m, centroid distance d̄ = 5 m: V = 2π(5)(π·4) = 40π² ≈ 394.78 m³
Doughnut-shaped solid generated by rotating a circle about an external axis in its plane.
Diagrams to know
- Semicircle rotated about its diameter to form sphere; mark centroid of arc and area
- Right triangle rotated about vertical leg to form cone; show axis of rotation
- Circle rotated about external axis to form torus; mark circle center and distance d̄ from axis
- General planar area with centroid marked and axis of rotation shown
- Arc with centroid d̄ labeled perpendicular to axis of rotation
COMPOSITE FIGURES & COMPOUND SOLIDS
Formulas
A_total = A₁ + A₂ + ... + Aₙ
Meaning
Composite area: sum of non-overlapping component areas
Watch out
Sketch and divide carefully; subtract cutout areas if present; ensure no overlap or gap
When to use
Irregular polygons or shapes divided into simpler figures (triangles, rectangles, circles)
V_total = V₁ + V₂ + ... + Vₙ
Meaning
Composite volume: sum of component volumes
Watch out
Subtract void volumes if solid is hollow; ensure components don't overlap; use consistent units
When to use
Compound solids (e.g., cylinder + cone for silo, rectangular prism + pyramid for roof tank)
x̄ = (A₁x₁ + A₂x₂ + ... + Aₙxₙ) / (A₁ + A₂ + ... + Aₙ)
Meaning
Centroid (first moment) of composite area: Aᵢ = component area, xᵢ = distance from reference axis
Watch out
Centroid of each component must be known; cutout areas are NEGATIVE; reference axis position critical
When to use
Finding centroid of irregular shape for structural design per NSCP 2015; used with Pappus theorem
Important facts
- Centroid of rectangle: center (bh/2 at b/2 and h/2 from corner)
- Centroid of triangle: (b/3, h/3) from corner opposite base
- Centroid of semicircle: 4r/(3π) from diameter (along perpendicular bisector)
- Centroid of quarter circle: 4r/(3π) from both axes (by symmetry in first quadrant)
- For composite areas with cutouts, subtract the cutout area as negative value
Key definitions
Composite Figure
Example
L-shaped area = large rectangle minus cutout rectangle
Complex shape formed by combining or subtracting simpler geometric figures.
Moment of Area (First Moment)
Example
Rectangle 3 m × 4 m, distance from axis 5 m: Q = 12 × 5 = 60 m³
Product of area and perpendicular distance from reference axis; Q = A × d̄; used to find centroid.
Diagrams to know
- L-shaped polygon divided into two rectangles with areas and centroids marked
- T-shaped section with horizontal and vertical components; centroid of each labeled
- Irregular profile (e.g., channel cross-section) decomposed into rectangles
- Circle with sector cutout; show positive and negative area components
CIVIL ENGINEERING APPLICATIONS — Earthwork & Mensuration
Formulas
V = (h/6)(A₁ + 4A_m + A₂)
Meaning
Prismoidal rule for earthwork: A₁ = area of first cross-section, A_m = area at midpoint h/2, A₂ = area of second cross-section
Watch out
A_m MUST be at exact midpoint (h/2); coefficient 1/6 with 4× weight on middle section; NOT average (A₁ + A₂)/2
When to use
Computing volumes of cuts and fills in road/bridge construction per NSCP 2015; most accurate for irregular terrain
V_cone = (1/3)πr²h
Meaning
Volume of conical spoil pile: r = base radius, h = height
Watch out
Often measured height (slant) ≠ perpendicular height h; h must be vertical from base center to apex
When to use
Estimating volume of excavated soil piled in conical form; common in grading calculations
V = πr²h (cylinder)
Meaning
Volume of cylindrical tank or stockpile: r = radius, h = height
Watch out
Height h perpendicular to base; diameter often given, so r = d/2; measure inside radius for capacity
When to use
Water tanks, fuel storage, bulk material (cement, sand) stockpiles per ACI 318 and material handling specs
V = (1/3)(A₁ + A₂ + √(A₁A₂))h
Meaning
Frustum volume for tapered embankments or fills: A₁, A₂ = cross-sectional areas at two elevations, h = vertical distance
Watch out
√(A₁A₂) is geometric mean, NOT arithmetic average; h is vertical (perpendicular), not slant height
When to use
Dam embankments, fills on sloping terrain, tapered columns (per NSCP 2015 structural design)
Important facts
- NSCP 2015 recommends prismoidal rule (h/6 formula) for engineering calculations; simpler average-end-area rule gives ±5% for most terrain
- Conical stockpile volume often measured indirectly: height h from surveyed elevation difference
- Trapezoidal cross-section (road cut on sloping terrain) is most common in Philippines infrastructure projects
- Centroid of trapezoid used in earthwork calculations for compound fill geometry
- For irregular terrain, cross-sections at 50 m or 100 m intervals standard for major works
Key definitions
Cross-Section (A)
Example
Trapezoidal cut: bases 10 m and 8 m, height 3 m: A = (10+8)/2 × 3 = 27 m²
Area of cut or fill perpendicular to the longitudinal axis of a construction project (road, canal).
Prismoidal Rule
Example
Road cut: A₁ = 50 m², A_m = 55 m² (at 100 m), A₂ = 52 m²; V = (200/6)(50 + 4·55 + 52) = 10,400 m³
NSCP 2015 standard method for computing volumes of cuts/fills using end areas and mid-section area.
Embankment
Example
Dam embankment with frustum geometry over length requiring volume estimation
Raised structure of earth/fill material; volume calculated from cross-sections perpendicular to centerline.
Diagrams to know
- Longitudinal profile of road showing two cross-sections A₁ and A₂ separated by distance h
- Trapezoidal cross-section with top width, bottom width, and height labeled
- Conical pile with height and radius marked for volume calculation
- Frustum of pyramid showing two parallel bases A₁, A₂ and height h
Must remember
ALWAYS use perpendicular height h, NOT slant height L, in pyramid V = ⅓A_base·h and cone V = ⅓πr²h. Slant height L is for lateral surface area only. This is the #1 exam error.
ALWAYS convert angle θ from degrees to radians when using sector area A = ½r²θ. Multiplying by π/180: θ_rad = θ_deg × (π/180). Forgetting this causes 50%+ student errors.
Frustum volume uses GEOMETRIC MEAN √(A₁A₂), NOT product A₁A₂. Formula: V = (h/3)(A₁ + A₂ + √(A₁A₂)). Prismatoid is different: V = (h/6)(A₁ + 4A_m + A₂) with A_m at mid-point.
Pappus Theorem 1 (surface) uses arc length L and applies to curves; Pappus Theorem 2 (volume) uses planar area A. Centroid distance d̄ in both must be measured perpendicular to axis of rotation.
For circle/circular shapes: Use RADIUS r, not diameter d. If diameter is given, HALVE it first: r = d/2. This applies to area (πr²), volume (πr²h, ⅓πr²h), and surface (4πr²).
Regular polygon area A = ¼ns²cot(180°/n) where n = number of sides, s = side length. Angle 180°/n is in DEGREES. For hexagon (n=6): A = (3√3/2)s² ≈ 2.598s².
Triangle area by Heron's formula requires SEMI-PERIMETER s = (a+b+c)/2 FIRST. Then A = √[s(s−a)(s−b)(s−c)]. Students forget s is half the perimeter, not the perimeter itself.
Prismoidal rule V = (h/6)(A₁ + 4A_m + A₂) is standard per NSCP 2015 for earthwork volumes. The middle area A_m MUST be at exactly h/2 (midpoint). Coefficient is 1/6, not 1/3 or 1/2.
Centroid of composite area: x̄ = (ΣAᵢxᵢ)/(ΣAᵢ). For areas with cutouts, subtract the cutout as NEGATIVE area. Identify centroid of each component first, then apply formula.
Cone lateral surface area S = πrL where L = √(h² + r²) is slant height. Perpendicular height h and radius r form a right triangle with L. Common error: confusing h with L or using πrh instead of πrL.
Last minute tips
BEFORE computing, SKETCH the figure. Label all given dimensions (height, radius, base, angles). Identify what you're solving for (area, volume, lateral surface). Wrong sketches lead to wrong formula selection.
HEIGHT vs SLANT HEIGHT: In cones and pyramids, h (perpendicular) is for VOLUME (⅓A·h) and base calculations. Slant height L = √(h² + r²) is ONLY for lateral surface area (πrL). This distinction appears in >60% of exam problems.
ANGLES in RADIANS for circular formulas: Sector area ½r²θ REQUIRES θ in radians. If angle is in degrees, MULTIPLY by π/180 to convert. This is the #1 source of wrong answers; check your angle units before computing.
COMPOSITE FIGURES: Use positive area for the main shape and SUBTRACT (negative) any cutouts. Write out A_total = A_main − A_cutout step-by-step to avoid sign errors.
EARTHWORK VOLUMES (NSCP 2015): Use prismoidal rule V = (h/6)(A₁ + 4A_m + A₂) for official calculations. Verify that A_m is the cross-sectional area at the EXACT midpoint between A₁ and A₂. Improper A_m placement causes large errors in fill volume estimates.
Comparison tables
Triangle vs Trapezoid Area Formulas
| Shape | Formula | Variables | When to Use | Common Mistake |
|---|---|---|---|---|
| Triangle (base-height) | A = ½bh | b = base, h = perpendicular height | Base and perpendicular height known | Forgetting ½ factor; using slant height instead of perpendicular |
| Triangle (Heron's formula) | A = √[s(s−a)(s−b)(s−c)] | a,b,c = sides, s = (a+b+c)/2 | Three sides known; no height available | Forgetting to compute s first; using perimeter instead of semi-perimeter |
| Trapezoid | A = ½(b₁ + b₂)h | b₁, b₂ = parallel bases, h = perpendicular distance | Two parallel sides and perpendicular height | h is perpendicular distance between parallel sides, NOT a slant; summing bases before ½ |
Pyramid vs Cone vs Prism — Volume Comparison
| Solid | Base Shape | Volume Formula | Lateral Surface | Key Distinction |
|---|---|---|---|---|
| Prism | Any polygon | V = A_base × h | Rectangle faces; no apex | Uniform cross-section perpendicular to height |
| Pyramid | Any polygon | V = ⅓A_base × h | Triangular faces meeting at apex | ⅓ factor; apex reduces volume vs prism |
| Cylinder | Circle | V = πr²h | Curved lateral surface 2πrh | Prism with circular base |
| Cone | Circle | V = ⅓πr²h | Lateral area πrL (L = slant height) | ⅓ factor; apex point; cone = ⅓ cylinder |
Sector vs Segment vs Whole Circle
| Region | Definition | Area Formula | Boundary | Exam Frequency |
|---|---|---|---|---|
| Sector | Pie-slice from center to circumference | A = ½r²θ (θ in radians) | Two radii + arc | Very common |
| Segment | Region between chord and arc | A = ½r²(θ − sin θ) | Chord + arc (no radii) | Common |
| Whole Circle | Complete circle | A = πr² | Circumference only | Fundamental |
Frustum vs Prismatoid Volume Formulas
| Formula | Usage | Variables Required | Accuracy | Common Error |
|---|---|---|---|---|
| Frustum: V = (h/3)(A₁ + A₂ + √(A₁A₂)) | Cone or pyramid truncated by plane parallel to base | A₁, A₂ = parallel end areas, h = perpendicular distance | Exact for cones, pyramids, and tapered solids | Using A₁A₂ instead of √(A₁A₂); forgetting geometric mean |
| Prismatoid: V = (h/6)(A₁ + 4A_m + A₂) | General prismatoid or earthwork cuts/fills (NSCP 2015) | A₁, A₂ = end areas, A_m = mid-section area at h/2 | Exact for polyhedra; excellent approximation for curved surfaces | A_m position must be at exact midpoint h/2; coefficient 1/6 with 4× weight on A_m |
Pappus Theorem 1 (Surface) vs Pappus Theorem 2 (Volume)
| Pappus Theorem | What Rotates | Formula | Variables | Example |
|---|---|---|---|---|
| First Theorem (Surface) | A curve (arc or line) | S = 2π d̄ L | S = surface area, d̄ = centroid distance to axis, L = arc length | Rotating semicircle (arc L = πr, d̄ = 2r/π) about diameter generates sphere S = 4πr² |
| Second Theorem (Volume) | A planar area | V = 2π d̄ A | V = volume, d̄ = centroid distance to axis, A = area | Rotating circle (A = πr², d̄ = external distance) about axis generates torus V = 2π d̄ πr² |
Diagrams
Plane Figures & Area Formulas Hierarchy
Hierarchical mind map showing area formulas for all major plane figures, organized by shape category. Useful for quickly identifying the correct formula based on given information.
Volume Formula Decision Tree for Solids
Decision flowchart for selecting the correct volume formula based on solid geometry characteristics. Includes branching logic for prismatoid vs frustum selection and emphasizes perpendicular height requirement.
Circular Region Identification & Area Selection
Diagnostic flowchart for identifying whether a circular region is a sector, segment, or full circle based on boundary description. Includes reminder to convert angles to radians.
Height Type Selection for Cones and Pyramids
State diagram showing decision logic between perpendicular height h (for volume) and slant height L (for lateral surface area) in cones and pyramids. Includes Pythagorean relationships for computing missing height.
Pappus Theorem Application Decision Tree
Decision tree for applying Pappus theorems based on whether a curve (surface) or area (volume) is being revolved. Includes centroid distance and arc/area relationships.