9.2 Perimeter, Area, Surface Area & Volume of 2D and 3D Figures

9.2 Perimeter, Area, Surface Area & Volume of 2D and 3D Figures

Two-Dimensional Metric Foundations: Perimeter, Circumference & Area

Two-dimensional mensuration underpins all secondary spatial geometry on the FTCE Mathematics 6-12 exam. Perimeter measures the one-dimensional boundary length enclosing a planar region, while area quantifies the two-dimensional region enclosed.

• Circumference and Circle Area: A circle of radius $r$ has circumference $C = 2\pi r = \pi d$ and area $A = \pi r^2$.
• Triangles: Standard area is $A = \frac{1}{2}bh$, where altitude $h$ is strictly perpendicular to base $b$. When three side lengths $a, b, c$ are known without an altitude, Heron's formula applies:

$$A = \sqrt{s(s-a)(s-b)(s-c)}, \quad \text{where } s = \frac{a + b + c}{2} \text{ is the semi-perimeter}$$

• Trapezoids: With parallel bases $b_1$ and $b_2$ separated by perpendicular height $h$, area is the product of the average base and altitude:

$$A = \frac{1}{2}(b_1 + b_2)h$$

• Rhombi and Kites: Quadrilaterals with mutually perpendicular diagonals $d_1$ and $d_2$ have area:

$$A = \frac{1}{2}d_1 d_2$$

• Regular Polygons: A regular polygon with $n$ sides, side length $s$, perimeter $P = ns$, and apothem $a$ (the perpendicular distance from center to side) can be partitioned into $n$ congruent isosceles triangles, each of area $\frac{1}{2}sa$. Summing gives:

$$A = \frac{1}{2}a P$$

Trigonometry links side length $s$ and apothem $a$ via the central angle $\theta = \frac{360^\circ}{n}$. In each half-triangle with vertex angle $\frac{180^\circ}{n}$, $\tan\left(\frac{180^\circ}{n}\right) = \frac{s/2}{a} \implies a = \frac{s}{2\tan(180^\circ/n)}$.

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Three-Dimensional Polyhedra: Prisms and Pyramids

Polyhedra are 3D solids bounded by planar polygonal faces.

• Prisms: Prisms consist of two parallel, congruent polygonal bases connected by parallelogram lateral faces. In a right prism, lateral faces are rectangles perpendicular to the bases. For base area $B$, base perimeter $P$, and perpendicular height $h$:

$$V = B h$$ $$\text{Lateral Surface Area (LSA)} = P h$$ $$\text{Total Surface Area (SA)} = 2B + P h$$

• Pyramids: Pyramids possess a single polygonal base converging to a single apex. Because volume tapers uniformly from base to apex, volumetric capacity is exactly one-third of the enclosing prism:

$$V = \frac{1}{3}B h$$

For a regular pyramid (regular polygon base, apex aligned over base center), lateral faces are congruent isosceles triangles. The altitude of each lateral face is the slant height $l$. By the Pythagorean theorem, slant height $l$, vertical altitude $h$, and base apothem $a$ satisfy $l^2 = h^2 + a^2$. The surface areas evaluate to:

$$\text{LSA} = \frac{1}{2} P l$$ $$\text{Total SA} = B + \frac{1}{2} P l$$

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Curved Three-Dimensional Solids: Cylinders, Cones, and Spheres

• Right Circular Cylinders: A cylinder has circular bases of radius $r$ and perpendicular height $h$:

$$V = \pi r^2 h$$ $$\text{LSA} = 2\pi r h, \quad \text{Total SA} = 2\pi r^2 + 2\pi r h$$

• Right Circular Cones: A circular cone of radius $r$, vertical height $h$, and slant height $l = \sqrt{r^2 + h^2}$:

$$V = \frac{1}{3}\pi r^2 h$$ $$\text{LSA} = \pi r l, \quad \text{Total SA} = \pi r^2 + \pi r l$$

• Spheres and Hemispheres: A sphere of radius $r$ possesses continuous spherical curvature without planar faces:

$$V = \frac{4}{3}\pi r^3, \quad \text{Total SA} = 4\pi r^2$$

For a solid hemisphere of radius $r$, the curved dome has surface area $2\pi r^2$. Adding the circular planar base of area $\pi r^2$ gives a total closed surface area of $3\pi r^2$, and volume $V = \frac{2}{3}\pi r^3$.

• Conical and Pyramidal Frustums: Slicing off the top of a cone or pyramid with a plane parallel to the base creates a frustum of height $h$ with base areas $B_1$ and $B_2$:

$$V_{\text{frustum}} = \frac{1}{3}h\left(B_1 + B_2 + \sqrt{B_1 B_2}\right)$$

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Cavalieri's Principle: Justification of Oblique Solids and Spheres

Cavalieri's Principle is a foundational theorem in spatial measurement:

Cavalieri's Principle: If two three-dimensional solids have equal heights and have equal cross-sectional areas at every plane parallel to their bases, then the two solids have equal volumes.

This principle provides rigorous mathematical justification for several crucial spatial facts:

1. Right vs. Oblique Prisms and Cylinders: Shearing a solid changes its lateral surface area and edge lengths, but does not alter its cross-sectional slicing areas. Consequently, the volume formula $V = Bh$ for prisms and $V = \pi r^2 h$ for cylinders applies identically to both right and oblique solids, provided $h$ is strictly the perpendicular vertical distance between base planes.
2. Archimedes' Sphere Derivation: Cavalieri's Principle elegantly establishes the volume of a sphere of radius $R$. Consider a hemisphere of radius $R$ and a cylinder of radius $R$ and height $R$ from which an inverted cone of base radius $R$ and height $R$ has been hollowed out. Slice both solids at height $y$ above their bases ($0 \le y \le R$):

• In the hemisphere, the horizontal cross section is a circle of radius $r(y) = \sqrt{R^2 - y^2}$, giving area:

$$A_{\text{hemi}}(y) = \pi [r(y)]^2 = \pi (R^2 - y^2)$$

• In the hollowed cylinder, the cross section is an annulus with outer radius $R$ and inner radius $y$ (since the cone's radius equals its height $y$), giving area:

$$A_{\text{cyl-cone}}(y) = \pi R^2 - \pi y^2 = \pi (R^2 - y^2)$$

Because $A_{\text{hemi}}(y) = A_{\text{cyl-cone}}(y)$ at every height $y$, Cavalieri's Principle proves their volumes are identical:

$$V_{\text{hemi}} = V_{\text{cylinder}} - V_{\text{cone}} = \pi R^2(R) - \frac{1}{3}\pi R^2(R) = \frac{2}{3}\pi R^3$$

Multiplying by 2 confirms the full sphere volume: $V_{\text{sphere}} = \frac{4}{3}\pi R^3$.

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Composite Solids and Multi-Dimensional Scaling Laws

• Composite Solids: Practical measurement items test composite structures formed by combining or hollowing out standard geometric figures. Total volume is strictly additive: $V_{\text{total}} = \sum V_{\text{components}} - \sum V_{\text{voids}}$. Total surface area is NOT simply the sum of individual surface areas; intersecting or joined contact interfaces are internalized and must be subtracted.
• Dimensional Scaling Laws: If every linear dimension of a geometric figure is multiplied by a positive scale factor $k$:

1. All linear measurements (perimeter, circumference, slant height, radius) scale by factor $k^1 = k$.
2. All two-dimensional measurements (base area, lateral area, total surface area) scale by factor $k^2$.
3. All three-dimensional measurements (volume, capacity, mass for uniform density) scale by factor $k^3$.

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Worked Exemplar: Composite Silo Volume and Cavalieri Analysis

Problem: A grain silo consists of a right circular cylinder of radius $r = 6$ m and height $h = 20$ m, topped by a hemispherical dome of radius $r = 6$ m.

1. Calculate the exact volume of the full-scale silo in terms of $\pi$.
2. If an architect builds a scale model with a linear scale factor of $k = \frac{1}{5}$, calculate the volume of the model.

• Solution:

1. Cylinder volume: $V_{\text{cyl}} = \pi r^2 h = \pi (6^2)(20) = \pi(36)(20) = 720\pi\text{ m}^3$. Hemisphere volume: $V_{\text{hemi}} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (6^3) = \frac{2}{3}\pi(216) = 144\pi\text{ m}^3$. Total full-scale volume:

$$V_{\text{total}} = 720\pi + 144\pi = 864\pi\text{ m}^3$$

1. Since volume scales by $k^3$:

$$V_{\text{model}} = V_{\text{total}} \times k^3 = 864\pi \times \left(\frac{1}{5}\right)^3 = \frac{864\pi}{125} = 6.912\pi\text{ m}^3$$

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Comprehensive 3D Mensuration Formula Matrix