IB MYP 4-5 Maths- Volume of regular polyhedra- Study Notes - New Syllabus
IB MYP 4-5 Maths- Volume of regular polyhedra – Study Notes
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Volume of Regular Polyhedra
Volume of Regular Polyhedra
A regular polyhedron is a solid in three dimensions with all faces being congruent regular polygons and all edges equal in length.
There are 5 regular polyhedra (Platonic Solids):
- Tetrahedron (4 triangular faces)
- Cube (6 square faces)
- Octahedron (8 triangular faces)
- Dodecahedron (12 pentagonal faces)
- Icosahedron (20 triangular faces)
Properties of Regular Polyhedra (Platonic Solids)
| Polyhedron | Faces | Shape of Each Face | Edges | Vertices |
|---|---|---|---|---|
Tetrahedron | 4 | Equilateral Triangle | 6 | 4 |
Cube | 6 | Square | 12 | 8 |
Octahedron | 8 | Equilateral Triangle | 12 | 6 |
Dodecahedron | 12 | Regular Pentagon | 30 | 20 |
Icosahedron | 20 | Equilateral Triangle | 30 | 12 |
Volume depends on edge length \( a \). Each polyhedron has its own formula.
Formulas for Volume:
| Polyhedron | Volume Formula |
|---|---|
| Cube | \( V = a^3 \) |
| Tetrahedron | \( V = \dfrac{a^3}{6\sqrt{2}} \) |
| Octahedron | \( V = \dfrac{\sqrt{2}}{3}a^3 \) |
| Dodecahedron | \( V = \dfrac{15 + 7\sqrt{5}}{4}a^3 \) |
| Icosahedron | \( V = \dfrac{5(3+\sqrt{5})}{12}a^3 \) |
Example:
Find the volume of a cube with side length \( a = 8 \, \text{cm} \).
▶️ Answer/Explanation
Step 1: Formula: \( V = a^3 \).
Step 2: Substitute: \( V = 8^3 = 512 \, \text{cm}^3 \).
Final Answer: \( \boxed{512 \, \text{cm}^3} \).
Example:
If the volume of a cube is \( 27{,}000 \, \text{cm}^3 \), find its side length.
▶️ Answer/Explanation
Step 1: \( V = a^3 \Rightarrow a = \sqrt[3]{V} \).
Step 2: \( a = \sqrt[3]{27{,}000} = 30 \, \text{cm} \).
Final Answer: \( \boxed{30 \, \text{cm}} \).
Example:
Find the volume of a tetrahedron with side length \( a = 12 \, \text{cm} \).
▶️ Answer/Explanation
Step 1: Formula: \( V = \dfrac{a^3}{6\sqrt{2}} \).
Step 2: Substitute: \( V = \dfrac{12^3}{6\sqrt{2}} = \dfrac{1728}{6\sqrt{2}} \).
Step 3: Simplify: \( V \approx 204 \, \text{cm}^3 \).
Final Answer: \( \boxed{204 \, \text{cm}^3} \).
Example:
A regular tetrahedron has volume \( 48 \, \text{cm}^3 \). Find its side length.
▶️ Answer/Explanation
\( V = \dfrac{a^3}{6\sqrt{2}} \Rightarrow a^3 = V \times 6\sqrt{2} \).
So \( a^3 = 48 \times 6\sqrt{2} \approx 407.3 \).
\( a = \sqrt[3]{407.3} \approx 7.4 \, \text{cm} \).
Final Answer: \( \boxed{7.4 \, \text{cm}} \).
Example:
Find the volume of an octahedron with edge length 5 cm.
▶️ Answer/Explanation
Volume:
\( V = \dfrac{\sqrt{2}}{3}a^3 = \dfrac{\sqrt{2}}{3}(5)^3 = \dfrac{\sqrt{2}}{3} \times 125 \approx 58.9\,\text{cm}^3 \)
Example:
Find the surface area of a dodecahedron with edge length 4 cm.
▶️ Answer/Explanation
Surface Area:
\( A = 3\sqrt{25 + 10\sqrt{5}}\,a^2 \)
\( A = 3\sqrt{25 + 10\sqrt{5}}(4)^2 = 3\sqrt{25 + 10\sqrt{5}} \times 16 \)
\( A \approx 166.3\,\text{cm}^2 \)
Example:
Find the volume of an icosahedron with edge length 3 cm.
▶️ Answer/Explanation
Volume:
\( V = \dfrac{5(3 + \sqrt{5})}{12}a^3 \)
\( V = \dfrac{5(3 + \sqrt{5})}{12}(3)^3 = \dfrac{5(3 + \sqrt{5})}{12} \times 27 \)
\( V \approx 58.2\,\text{cm}^3 \)