Solid Geometry: Formulas, Dimensional Ratios, and Composite Solids

Fundamentals of Three-Dimensional Solids

  • Definition of a Solid:

    • A solid is a three-dimensional geometric figure bounded by surfaces or plane figures.

  • Surface Area and Volume Definitions:

    • Lateral Surface Area (LSA\text{LSA}) / Surface Area:
    • The area of all curved or vertical lateral surfaces of a solid figure, excluding the area of its base(s).
    • Always expressed in square units (units2\text{units}^2).
    • Total Surface Area (TSA\text{TSA}):
    • The entire boundary area of the solid figure, defined as the sum of the areas of all the bases and the lateral surface area (LSA\text{LSA}).
    • Always expressed in square units (units2\text{units}^2).
    • Volume (VV):
    • The measure of three-dimensional space enclosed or occupied by the solid.
    • Always expressed in cubic units (units3\text{units}^3).

  • Primary Classifications and Structural Components of Solids:

    • Cylinder:
    • Bounded by two parallel circular bases (Base 1\text{Base } 1 and Base 2\text{Base } 2) of radius rr, and a curved lateral side.
    • The unrolled lateral side forms a rectangle whose length equals the base circumference (Cb=2×\pi×rC_b = 2\times \backslash pi \times r) and whose height is the altitude (hh).
    • Cone:
    • Characterized by a single circular flat base with circumference Cb=2πrC_b = 2\pi r, a vertex, a vertical altitude (hh), and a slant height (ll).
    • Sphere:
    • A symmetrical round solid whose surface points are equidistant from the center by radius rr.
    • Prism:
    • A polyhedron composed of two congruent, parallel polygonal bases connected by lateral faces that are parallelograms.
    • Common types include the rectangular prism (parallelepiped) with length (ll), width (ww), and height (hh), and the triangular prism with triangular bases.
    • Cube:
    • A regular hexahedron bounded by six congruent square faces, with all edges equal in length (ss).

Whiteboard with definitions, diagrams, and formulas for geometric solids

Geometric Formulas for Standard Solids

  • General Volume Principles:

    • Uniform Cross-Section Solids (Prisms and Cylinders):
    • V=BhV = Bh
    • In this formula, B=area of baseB = \text{area of base} and h=height of solidh = \text{height of solid}.
    • Tapered Solids with a Single Base (Cones and Pyramids):
    • V=13BhV = \frac{1}{3}Bh
    • In this formula, B=area of baseB = \text{area of base} and h=height of solidh = \text{height of solid}.

  • Specific Solid Formulas:

    • Cube:
    • Variable definition: s=measure of its edges = \text{measure of its edge}
    • Lateral Surface Area:
      • LSA=4s2\text{LSA} = 4s^2
    • Total Surface Area:
      • TSA=6s2\text{TSA} = 6s^2
    • Volume:
      • V=s3V = s^3
    • Sphere:
    • Variable definition: r=radius of spherer = \text{radius of sphere}
    • Total Surface Area:
      • TSA=4πr2\text{TSA} = 4\pi r^2
    • Volume:
      • V=43πr3V = \frac{4}{3}\pi r^3
    • Rectangular Prism (Parallelepiped):
    • Variable definitions: l=lengthl = \text{length}, w=widthw = \text{width}, h=heighth = \text{height}
    • Lateral Surface Area:
      • LSA=2lh+2wh\text{LSA} = 2lh + 2wh
    • Right Circular Cone:
    • Variable definitions: r=base radiusr = \text{base radius}, h=heighth = \text{height}, l=slant heightl = \text{slant height}
    • Slant height relation:
      • l=h2+r2l = \sqrt{h^2 + r^2}
    • Lateral Surface Area derivation:
      • LSA=12(l)(2πr)=πrl\text{LSA} = \frac{1}{2}(l)(2\pi r) = \pi r l
    • Volume:
      • V=13πr2hV = \frac{1}{3}\pi r^2 h
    • Right Circular Cylinder:
    • Variable definitions: r=base radiusr = \text{base radius}, h=heighth = \text{height}
    • Lateral Surface Area:
      • LSA=2πrh\text{LSA} = 2\pi r h
    • Total Surface Area:
      • TSA=2πrh+2πr2\text{TSA} = 2\pi r h + 2\pi r^2
    • Volume:
      • V=πr2hV = \pi r^2 h

Ratio Analysis of Cubes Based on Space Diagonal

  • Problem Description:

    • A cube has an edge equal to the space diagonal of another cube. Determine the ratio of their volumes and the ratio of their total surface areas.

  • Variable Definitions and Given Relationships:

    • Let Cube 2 be the base cube with edge length:
    • s2=xs_2 = x
    • The space diagonal (dd) of Cube 2 across its three dimensions is:
    • d2=x2+x2+x2=3x2=x3d_2 = \sqrt{x^2 + x^2 + x^2} = \sqrt{3x^2} = x\sqrt{3}
    • Cube 1 has an edge (s1s_1) equal to the diagonal of Cube 2:
    • s1=e1=d2=x3s_1 = e_1 = d_2 = x\sqrt{3}

  • Derivation of Volume Ratio:

    • Volume of Cube 1:
    • V1=s13=(x3)3=33 x3V_1 = s_1^3 = (x\sqrt{3})^3 = 3\sqrt{3}\,x^3
    • Volume of Cube 2:
    • V2=s23=x3V_2 = s_2^3 = x^3
    • Ratio of Volumes (V1V2\frac{V_1}{V_2}):
    • V1V2=s13s23=33 x3x3=33\frac{V_1}{V_2} = \frac{s_1^3}{s_2^3} = \frac{3\sqrt{3}\,x^3}{x^3} = 3\sqrt{3}
    • Expressed as a ratio: 33:13\sqrt{3} : 1

  • Derivation of Total Surface Area Ratio:

    • Total Surface Area of Cube 1:
    • TSA1=6s12=6(x3)2=6(3x2)=18x2\text{TSA}_1 = 6s_1^2 = 6(x\sqrt{3})^2 = 6(3x^2) = 18x^2
    • Total Surface Area of Cube 2:
    • TSA2=6s22=6x2\text{TSA}_2 = 6s_2^2 = 6x^2
    • Ratio of Total Surface Areas (TSA1TSA2\frac{\text{TSA}_1}{\text{TSA}_2}):
    • TSA1TSA2=6s126s22=18x26x2=3\frac{\text{TSA}_1}{\text{TSA}_2} = \frac{6s_1^2}{6s_2^2} = \frac{18x^2}{6x^2} = 3
    • Expressed as a ratio: 3:13 : 1

Altitude and Dimensions of a Right Circular Tank

  • Problem Statement:

    • The lateral area of a right circular tank is 48π m248\pi\,\text{m}^2 and its volume is 72π m372\pi\,\text{m}^3. Find the altitude (hh) of the tank.

  • Given Parameters:

    • Lateral Surface Area: LSA=48π m2\text{LSA} = 48\pi\,\text{m}^2
    • Volume: V=72π m3V = 72\pi\,\text{m}^3
    • Target Variable: Altitude hh

  • System of Equations Formulation:

    • Using the lateral surface area formula for a cylinder:
    • LSA=2πrh\text{LSA} = 2\pi r h
    • 48π=2πrh48\pi = 2\pi r h
    • rh=48π2π=24rh = \frac{48\pi}{2\pi} = 24
    • h=24rh = \frac{24}{r} (Equation 1)
    • Using the volume formula for a cylinder:
    • V=πr2hV = \pi r^2 h
    • 72π=πr2h72\pi = \pi r^2 h
    • r2h=72r^2 h = 72
    • h=72r2h = \frac{72}{r^2} (Equation 2)

  • Step-by-Step Algebraic Solution:

    • Equating Equation 1 and Equation 2:
    • 24r=72r2\frac{24}{r} = \frac{72}{r^2}
    • 24r2=72r24r^2 = 72r
    • 24r=7224r = 72
    • r=7224=3 mr = \frac{72}{24} = 3\,\text{m}
    • Alternative method via ratio of Volume to Lateral Surface Area:
    • VLSA=πr2h2πrh=r2\frac{V}{\text{LSA}} = \frac{\pi r^2 h}{2\pi r h} = \frac{r}{2}
    • 72π48π=32\frac{72\pi}{48\pi} = \frac{3}{2}
    • r2=32  ⟹  r=3 m\frac{r}{2} = \frac{3}{2} \implies r = 3\,\text{m}
    • Substituting r=3 mr = 3\,\text{m} into Equation 1 to find the altitude hh:
    • h=24r=243=8 mh = \frac{24}{r} = \frac{24}{3} = 8\,\text{m}

  • Final Result:

    • Base radius: r=3 mr = 3\,\text{m}
    • Altitude of the tank: h=8 mh = 8\,\text{m}

Composite Solids: Analysis of a Cone-Cylinder Structure

  • System Description:

    • A composite solid consisting of a right circular cone mounted directly on top of a right circular cylinder with a common circular base interface.
    • Base radius of both cone and cylinder: r=10 cmr = 10\,\text{cm}
    • Height of the conical section: hcone=25 cmh_{\text{cone}} = 25\,\text{cm}
    • Height of the cylindrical section: hcylinder=30 cmh_{\text{cylinder}} = 30\,\text{cm}

  • Step 1: Slant Height of the Conical Component:

    • Calculated using the Pythagorean theorem with radius and cone height:
    • l=hcone2+r2l = \sqrt{h_{\text{cone}}^2 + r^2}
    • l=252+102=625+100=725l = \sqrt{25^2 + 10^2} = \sqrt{625 + 100} = \sqrt{725}
    • l≈26.93 cml \approx 26.93\,\text{cm}

  • Step 2: Lateral Surface Area of the Conical Component:

    • Formula for conical lateral area:
    • LSAcone=πrl\text{LSA}_{\text{cone}} = \pi r l
    • LSAcone=π(10 cm)(26.93 cm)\text{LSA}_{\text{cone}} = \pi (10\,\text{cm})(26.93\,\text{cm})
    • LSAcone=846 cm2\text{LSA}_{\text{cone}} = 846\,\text{cm}^2

  • Step 3: Volume of the Conical Component:

    • Formula for conical volume:
    • Vcone=13πr2hconeV_{\text{cone}} = \frac{1}{3}\pi r^2 h_{\text{cone}}
    • Vcone=13π(10 cm)2(25 cm)V_{\text{cone}} = \frac{1}{3}\pi (10\,\text{cm})^2 (25\,\text{cm})
    • Vcone=2500π3≈2,618 cm3V_{\text{cone}} = \frac{2500\pi}{3} \approx 2{,}618\,\text{cm}^3

  • Step 4: Volume of the Cylindrical Component:

    • Formula for cylindrical volume:
    • Vcylinder=πr2hcylinderV_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}}
    • Vcylinder=π(10 cm)2(30 cm)V_{\text{cylinder}} = \pi (10\,\text{cm})^2 (30\,\text{cm})
    • Vcylinder=3000π≈9,425 cm3V_{\text{cylinder}} = 3000\pi \approx 9{,}425\,\text{cm}^3

  • Step 5: Total Composite Volume:

    • The sum of the volumes of the two component solids:
    • Vcomposite=Vcone+VcylinderV_{\text{composite}} = V_{\text{cone}} + V_{\text{cylinder}}
    • Vcomposite=2,618 cm3+9,425 cm3V_{\text{composite}} = 2{,}618\,\text{cm}^3 + 9{,}425\,\text{cm}^3
    • Vcomposite=12,073 cm3V_{\text{composite}} = 12{,}073\,\text{cm}^3

  • Step 6: Total Surface Area of the Composite Solid:

    • Consists of the lateral area of the top cone, the lateral area of the cylinder, and the bottom circular base of the cylinder (excluding the joined circular boundary between the cone and cylinder):
    • TSAcomposite=LSAcone+LSAcylinder+Basecylinder\text{TSA}_{\text{composite}} = \text{LSA}_{\text{cone}} + \text{LSA}_{\text{cylinder}} + \text{Base}_{\text{cylinder}}
    • TSAcomposite=πrl+2πrhcylinder+πr2\text{TSA}_{\text{composite}} = \pi r l + 2\pi r h_{\text{cylinder}} + \pi r^2
    • TSAcomposite=846 cm2+2π(10 cm)(30 cm)+π(10 cm)2\text{TSA}_{\text{composite}} = 846\,\text{cm}^2 + 2\pi (10\,\text{cm})(30\,\text{cm}) + \pi (10\,\text{cm})^2
    • TSAcomposite=846+600π+100π\text{TSA}_{\text{composite}} = 846 + 600\pi + 100\pi
    • TSAcomposite=846+700π\text{TSA}_{\text{composite}} = 846 + 700\pi
    • 700π≈2199.11 cm2700\pi \approx 2199.11\,\text{cm}^2
    • TSAcomposite=846+2199.11=3045.11≈3045 cm2\text{TSA}_{\text{composite}} = 846 + 2199.11 = 3045.11 \approx 3045\,\text{cm}^2