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Ch 12 - Surface Areas and Volumes10th Mathematics

Core Formula: Properties of a Cuboid

MEMORY Formula

\begin{aligned} V &= lbh \\ CSA &= 2h(l + b) \\ TSA &= 2(lb + bh + hl) \end{aligned}

Fundamental equations defining the capacity and surface areas of a standard cuboid.

Derivation

Volume is derived fundamentally by multiplying the base area ( $l \\times b$ ) by the perpendicular height ( $h$ ).
CSA (Lateral Area) is the perimeter of the base ( $2l + 2b$ ) multiplied by the height $h$ .
TSA is the sum of the areas of the 3 distinct opposite pairs of rectangles forming the cuboid.

Component Anatomy

$l, b, h$ : Length, breadth, and height of the cuboid.
$V$ : Volume representing total 3D capacity.
$CSA$ : Lateral/Curved Surface Area (area of the 4 vertical walls).
$TSA$ : Total Surface Area (area of all 6 faces).

Boundary Conditions / Domain

All spatial dimensions must be strictly positive: $l > 0$ , $b > 0$ , $h > 0$ .

Prism Memory Hook

TSA wraps it all (2 pairs of each rectangular face).

Core Formula: Properties of a Cube

MEMORY Formula

\begin{aligned} V &= a^3 \\ CSA &= 4a^2 \\ TSA &= 6a^2 \end{aligned}

Fundamental equations defining the capacity and surface areas of a perfect cube.

Derivation

Derived directly by substituting $l = a$ , $b = a$ , and $h = a$ into the standard cuboid formulas.

Component Anatomy

$a$ : Uniform edge length of the cube.
$V$ : Total internal volume.
$CSA$ : Lateral surface area (4 square faces).
$TSA$ : Total surface area (6 square faces).

Boundary Conditions / Domain

Edge length must be strictly positive: $a > 0$ .

Prism Memory Hook

A cube is just a cuboid where strictly l = b = h = a.

Core Formula: Properties of a Right Circular Cylinder

MEMORY Formula

\begin{aligned} V &= \pi r^2 h \\ CSA &= 2\pi rh \\ TSA &= 2\pi r(h + r) \end{aligned}

Formulas defining the Volume, Curved Surface Area, and Total Surface Area of a solid cylinder.

Derivation

Volume: Area of the circular base ( $\\pi r^2$ ) extruded along the height ( $h$ ).
TSA Algorithmic SOP: Add the completely unrolled curved rectangular surface ( $2\\pi rh$ ) to the areas of the two enclosed circular bases ( $2 \\times \\pi r^2$ ).
Algebraically factor out $2\\pi r$ to yield the condensed formula $2\\pi r(h + r)$ .

Component Anatomy

$r$ : Radius of the circular base.
$h$ : Perpendicular height spanning strictly between the two bases.

Boundary Conditions / Domain

$r > 0$ and $h > 0$ strictly. Base must be a perfect circle and the axis must be perfectly perpendicular to the base.

Prism Memory Hook

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Also in this chapter

35 more cards to study in Prism

4Core Formula: Properties of a Right Circular Cone
5Geometric Relationship: Slant Height of a Cone
6Core Formula: Properties of a Sphere
7Terminology Trap: Sphere Surface Areas
8Core Formula: Properties of a Hemisphere
9Core Principle: TSA of Combined Solids
10Core Principle: Volume of Combined Solids
11Trap Application: Stacked Cylinders TSA
12Application: Real-World Deconstruction (Set 1)
13Application: Real-World Deconstruction (Set 2)
14Algorithmic SOP: TSA of Conjoined Cubes
15Geometric Constraint: Maximum Surmounted Diameter
16Algorithmic SOP: Solid with a Hemispherical Depression
17Core Principle: Conservation of Recasting
18Algorithmic SOP: Recasting into Multiple Solids
19Algorithmic SOP: Volume of a Hollow Shell
20Algorithmic SOP: Maximum Sphere Inscribed in a Cube
21Shortcut: Ratio Conversion (Volume to Surface Area)
22Algorithmic SOP: Flowing Water Volume (Part 1 - Kinematic Length)
23Algorithmic SOP: Flowing Water Volume (Part 2 - Volume Mapping)
24Algorithmic SOP: Area Irrigated by Discharged Water
25Algorithmic SOP: Embankment Height (Part 1 - Geometric Setup)
26Algorithmic SOP: Embankment Height (Part 2 - Volume Equating)
27Algorithmic SOP: Interstitial Voids (Part 1 - Available Volume)
28Algorithmic SOP: Interstitial Voids (Part 2 - Discrete Object Count)
29Algorithmic SOP: Porous Displacement (Part 1 - Gross Solid Volume)
30Algorithmic SOP: Porous Displacement (Part 2 - Net Effective Volume)
31Mathematical Definition: Frustum of a Cone
32Application: Real-World Frustum (Funnel)
33Core Formula: Volume of a Frustum
34Geometric Relationship: Slant Height of a Frustum
35Core Formula: Curved Surface Area (CSA) of a Frustum
36Core Formula: Total Surface Area (TSA) of a Frustum
37Algorithmic SOP: Bisected Cone (Part 1 - Radius Derivation)
38Algorithmic SOP: Bisected Cone (Part 2 - Volume Ratio)

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