Comprehensive Study Notes on Solid Mensuration and Plane Figures

Fundamental Principles of Solid Mensuration

• Solid mensuration is defined as the study of the measurement and calculation of the lengths, areas, and volumes of geometric figures, specifically focusing on three-dimensional shapes and the plane figures that comprise their surfaces.

Properties and Formulas for Plane Figures

Triangles

A triangle is a polygon consisting of three sides.

• General Area Formulas:
  • Using base and height: $A = \frac{1}{2} b \times h$
  • Using two sides ($a$ and $b$) and the included angle ($\theta$): $A = \frac{1}{2} a \times b \times \sin(\theta)$
  • Heron's Formula (when three sides $a$, $b$, and $c$ are given):
  • Semi-perimeter: $s = \frac{1}{2}(a + b + c)$
  • Area: $A = \sqrt{s(s-a)(s-b)(s-c)}$
  • Given one side and three angles:
  • $A = \frac{a^2 \times \sin(B) \times \sin(C)}{2 \times \sin(A)}$
  • $A = \frac{b^2 \times \sin(A) \times \sin(C)}{2 \times \sin(B)}$
  • $A = \frac{c^2 \times \sin(A) \times \sin(B)}{2 \times \sin(C)}$
• Specific Triangle Types:
  • Isosceles Triangle:
  • Area: $A = \frac{1}{2} b \times h$
  • Perimeter: $P = 2a + b$
  • Equilateral Triangle:
  • Area: $A = \frac{\sqrt{3}}{4} a^2$
  • Perimeter: $P = 3a$
  • Right-Angled Triangle:
  • Area: $A = \frac{1}{2} b \times h$
  • Relationship: $b \times h = \text{hypotenuse} \times h$

Quadrilaterals

There are five primary types of quadrilaterals detailed for study:

• Square:
  • Area: $A = s^2$
  • Perimeter: $P = 4s$
• Rectangle:
  • Area: $A = L \times W$
  • Perimeter: $P = 2(L + W)$
• Parallelogram:
  • Area: $A = b \times h$
  • Perimeter: $P = 2(h + b)$
• Rhombus:
  • Area: $A = \frac{1}{2} d_1 \times d_2$
  • Perimeter: $P = 4 \times \text{side}$
• Trapezoid:
  • Area: $A = \frac{1}{2}(a + b)h$
  • Perimeter: $P = a + b + c + d$
• Trapezium:
  • Area: $A = \frac{1}{2}(a+b)h$
  • Perimeter: $P = a + b + c + d$
• General Quadrilateral Area (based on diagonals):
  • $A = \frac{1}{2} d_1 d_2 \sin(\theta)$

Circles and Related Figures

• Circle:
  • Area: $A = \pi r^2$ or $A = \frac{\pi d^2}{4}$
  • Circumference/Perimeter: $C = 2\pi r$ or $C = \pi d$
• Semicircle:
  • Area: $A = \frac{1}{2} \pi r^2$ or $A = \frac{\pi d^2}{8}$
• Sector of a Circle:
  • Area: $A = \frac{1}{2} \theta r^2$ (where $\theta$ is in radians). If in degrees, convert using $\frac{\pi}{180}$.
• Ellipse:
  • Area: $A = \pi ab$
  • Outer Circumference (Appx): $C = 2\pi \sqrt{\frac{a^2+b^2}{2}}$
• Parabolic Segment:
  • Area: $A = \frac{2}{3} L \times D$

General Polygons

• Sum of Interior Angles: $S = (n - 2) \times 180^{\circ}$
• Number of Diagonals: $\frac{n(n - 3)}{2}$

Formulas for Solid Shapes (3D Figures)

Exponent Rules in Calculations

• Power of a Product Rule: $(ab)^n = a^n b^n$
  • Example: $(ab^2)^3 = a^3 b^6$
  • Example: $(e^{-2} f^2)^7 = e^{-14} f^{14} = \frac{f^{14}}{e^{14}}$
• Power of a Quotient Rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
  • Example: $(\frac{z^3}{4})^3 = \frac{z^9}{4^3} = \frac{z^9}{64}$

Practical Applications and Activity Solutions

Exercise 1: Square Measurements

• Problem: If the perimeter of a square is $100\,m$, find its area.
• Calculation:
  • Perimeter $P = 4s = 100\,m$
  • Side $s = 25\,m$
  • Area $A = s^2 = (25\,m)^2 = 625\,m^2$

Exercise 2: Rectangle Dimensions

• Problem: A rectangle has a length of $6\,m$ and an area of $28\,m^2$. Find the perimeter in cm.
• Calculation:
  • Area $A = L \times W$
  • $28 = 6 \times W \rightarrow W = 4.67\,m$
  • Perimeter $P = 2(L + W) = 2(6 + 4.67) = 21.34\,m$
  • Conversion: $21.34\,m \times 100\,cm/m = 2,134\,cm$

Exercise 3: Circle Geometry

• Problem: The diameter of a circle is twice the value of $\pi$. Find the circumference and area in cm.
• Calculation:
  • Diameter $D = 2\pi \approx 6.28\,cm$
  • Radius $r = \pi \approx 3.14\,cm$
  • Circumference $C = 2\pi r = 2(3.14)(3.14) = 19.72\,cm$
  • Area $A = \pi r^2 = 3.14(3.14)^2 = 30.96\,cm^2$

Exercise 4: Diagonal of a Field

• Problem: A rectangular field has an area of $1.2\,hectares$ and a length of $150\,m$. Find the width and the length of the diagonal.
• Calculation:
  • Conversion: $1.2\,hectares = 12,000\,m^2$
  • Width $W = \frac{A}{L} = \frac{12,000}{150} = 80\,m$
  • Diagonal (Pythagorean Theorem): $D^2 = L^2 + W^2$
  • $D^2 = 150^2 + 80^2 = 22,500 + 6,400 = 28,900$
  • $D = \sqrt{28,900} = 170\,m$

Exercise 5: Regular Octagon

• Problem: Calculate the area of a regular octagon with side length $5\,cm$ and a width across the flats of $12\,cm$.
• Calculation:
  • The octagon can be divided into 8 triangles with a height (apothem) of half the width: $h = 6\,cm$.
  • Area of one triangle $A_1 = \frac{1}{2}(5)(6) = 15\,cm^2$
  • Total Area $A = 15 \times 8 = 120\,cm^2$

Exercise 6: Material Wasted

• Problem: A circle is cut from a square with an edge length of $4\,in$. Find the wasted material.
• Calculation:
  • Square Area $A_s = 4 \times 4 = 16\,in^2$
  • Circle Radius $r = 2\,in$
  • Circle Area $A_c = \pi(2)^2 = 12.56\,in^2$
  • Wasted Material = $16 - 12.56 = 3.44\,in^2$

Exercise 7: Composite Areas

• Composite Figure Calculation (Rectangle + Semicircle):
  • Rectangle: $20\,m \times 10\,m \rightarrow Area = 200\,m^2$
  • Semicircle (Radius $5\,m$): $A = \frac{1}{2} \pi(5)^2 = 39.27\,m^2$
  • Total Area: $200 + 39.27 = 239.27\,m^2$
  • Perimeter: Rectangle portion ($2L + W) = 2(20) + 10 = 50\,m$. Semicircle arc ($\pi r) = 3.14 \times 5 = 15.71\,m$. Total Perimeter = $65.71\,m$.

Course Information and Syllabus Context

• Subject: Mathematics in the Modern World / Solid Mensuration
• Core Focus: Solid Figures, recall of perimeters and areas, concept of lines, and course policies.
• Personnel:
  • Arthur G. Ibañez, PhD
  • Roger P. Pompon
  • John Michael (Dean of Engineering)