Learning Resources
Loci & 3D Solids
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Construct and describe loci in 2D and 3D, calculate 3D distance between coordinates, and evaluate surface areas and volumes of 3D solids.
A locus (plural: loci) is the path or region formed by all points satisfying a given geometric condition.
(a) Common Geometric Loci
- Fixed Distance from a Point: Circle in 2D; Sphere in 3D (\( r = \text{radius} \)).
- Fixed Distance from a Line: Pair of parallel lines in 2D; Cylindrical surface in 3D.
- Equidistant from Two Points: Perpendicular bisector in 2D; Bisector plane in 3D.
(b) 3D Distance Formula
Extending Pythagoras to three dimensions gives the straight-line distance between \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \):
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Key Formulas
\[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\] — 3D Distance Formula: Distance between two coordinate points in 3D space.
\[(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2\] — Sphere Equation: Locus of points at distance \( r \) from center \( (h,k,l) \).
\[V_{\text{prism}} = l \times w \times h, \quad A_{\text{prism}} = 2(lw + wh + hl)\] — Rectangular Prism: Volume and total surface area equations.
\[V_{\text{cyl}} = \pi r^2 h, \quad A_{\text{cyl}} = 2\pi r h + 2\pi r^2\] — Cylinder: Volume and total surface area equations.
Worked Examples
Example 1 (Easy - 3D Distance Between Points): Find the distance between \( A(1, 2, 3) \) and \( B(4, 6, 3) \) in 3D space.
- Apply formula: \( d = \sqrt{(4-1)^2 + (6-2)^2 + (3-3)^2} \).
- Simplify: \( d = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16 + 0} = \sqrt{25} \).
- Calculate root: \( d = 5 \).
Answer: 5
Example 2 (Medium - Rectangular Prism Surface Area): A rectangular prism has length \( 8\text{ cm} \), width \( 5\text{ cm} \), and height \( 3\text{ cm} \). Calculate its total surface area in \( \text{cm}^2 \).
- Apply surface area formula: \( A = 2(lw + wh + hl) \).
- Substitute values: \( A = 2(8 \times 5 + 5 \times 3 + 3 \times 8) = 2(40 + 15 + 24) \).
- Sum terms: \( A = 2(79) = 158\text{ cm}^2 \).
Answer: 158
Example 3 (Hard - Radius from Cylinder Volume): A right circular cylinder has volume \( 2000\text{ cm}^3 \) and height \( 20\text{ cm} \). Calculate the radius \( r \) of its base to two decimal places.
- Use formula \( V = \pi r^2 h \implies 2000 = \pi \cdot r^2 \cdot 20 \).
- Divide by \( 20\pi \): \( r^2 = 2000 / (20\pi) = 100 / \pi \approx 31.831 \).
- Take square root: \( r = \sqrt{31.831} \approx 5.64\text{ cm} \).
Answer: 5.64
Common Mistakes
Mistake Confusing a solid's net with its cross-section.
Correction A net is a flattened 2D pattern that folds to form a 3D solid; a cross-section is the 2D face formed by slicing through a solid.
Why it feels right Both represent 2D plane figures derived from a 3D object.
Mistake Assuming the 3D locus of points at a fixed distance from a straight line is a circle.
Correction In 3D space, points equidistant from a line form a continuous cylinder surface.
Why it feels right Slicing perpendicular to the line creates a circular cross-section, causing students to omit the continuous length.
Real World
Telecommunications Network Coverage: Safaricom network towers radiate signals uniformly in 3D space, creating spherical loci boundaries for 4G and 5G coverage.
Packaging & Container Manufacturing: Engineers at Bidco Africa calculate exact surface areas from 2D nets to minimize raw sheet metal costs for oil cans and packaging boxes.
Practice
Find the distance between points A(1, 2, 3) and B(4, 6, 3) in three-dimensional space.
(Type only the number, e.g., 5)
Review the concepts above.
A rectangular prism has a length of 8 cm, a width of 5 cm and a height of 3 cm. What is the total surface area of the prism in square centimetres?
(Type only the number, e.g., 158)
Review the concepts above.
A rectangular prism has a length of 12 cm, a width of 5 cm and a height of 8 cm. What is its volume in cubic centimetres?
(Type only the number, e.g., 480)
Review the concepts above.
A right circular cylinder has a volume of 2000 cm³ and a height of 20 cm. What is the radius of its base in cm? (Give your answer to two decimal places.)
(Type only the number, e.g., 5.64)
Review the concepts above.
A sphere has a radius of 6 cm. What is its volume in terms of pi? (Calculate V / pi)
(Type only the number, e.g., 288)
Review the concepts above.
A cube has a total surface area of 150 cm². What is the volume of the cube in cubic centimetres?
(Type only the number, e.g., 125)
Review the concepts above.