--- title: "Right circular cone, vertical angle 30°" subject: "Mathematics" chapter: 3 key: "semester_2/mathematics:right-circular-cone-vertical-angle" source: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cone-vertical-angle site: BCT Engineering --- # Right circular cone, vertical angle 30° Mathematics · Chapter 3: Three Dimensional Solid Geometry - Asked in: 2074 Bhadra, 5 marks; 2070 Bhadra, 5 marks (asked 2 times) - Video: [The Right Circular Cone problem #1](https://www.youtube.com/watch?v=0efcSCqRHv0) — Pravin Ahire's MATH'S FORUM - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/mathematics/right-circular-cone-vertical-angle - Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3 ## Question Find the equation of right circular cone whose vertex at origin and axis is the line x/1 = y/2 = z/3 with vertical angle 30°. ## Short answer Semi-vertical angle 15°. Direction of axis (1,2,3) and OP make cos15° = (x+2y+3z)/(√14√(x²+y²+z²)). Squaring and using cos²15°=(2+√3)/4 gives 2(x+2y+3z)² = 7(2+√3)(x²+y²+z²). ## Answer **Given.** Vertex O(0, 0, 0), axis x/1 = y/2 = z/3, vertical angle 30°. **Assumption.** The vertical angle is the full angle at the vertex, so the semi-vertical angle is α = 15°. #### Method For a point P(x, y, z) on the cone, the line OP makes angle α with the axis, whose direction is (1, 2, 3): ```text cos α = (x + 2y + 3z) / ( √14 · √(x² + y² + z²) ) (x + 2y + 3z)² = 14 cos²α (x² + y² + z²) ``` #### Value of cos² 15° cos² 15° = (1 + cos 30°)/2 = (1 + √3/2)/2 = (2 + √3)/4. ```text (x + 2y + 3z)² = 14 · (2 + √3)/4 · (x² + y² + z²) 2 (x + 2y + 3z)² = 7 (2 + √3)(x² + y² + z²) ``` #### Expanded form Expanding (x + 2y + 3z)² = x² + 4y² + 9z² + 4xy + 12yz + 6zx and collecting terms: **(12 + 7√3)x² + (6 + 7√3)y² + (7√3 − 4)z² − 8xy − 24yz − 12zx = 0** (If the 30° were the semi-vertical angle, replace cos²15° by cos²30° = 3/4. This gives 2(x + 2y + 3z)² = 21(x² + y² + z²), i.e. 19x² + 13y² + 3z² − 8xy − 24yz − 12zx = 0.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the vertical angle 30 degrees mean, and what is alpha? A. It is the full angle at the vertex, so the semi-vertical angle is 15 degrees B. It is the semi-vertical angle, so alpha is 30 degrees C. It is the angle with the axis, so alpha is 60 degrees D. It is the full angle, so alpha is 30 degrees Answer: A. The vertical angle is the whole angle at the vertex, so alpha = 15 degrees. (Key point: Vertical angle) 2. What condition does a point P on the cone satisfy? A. OP is perpendicular to the axis B. OP makes angle 30 degrees with the x-axis C. OP makes angle alpha with the axis direction (1, 2, 3) D. OP is parallel to the axis Answer: C. Every generator makes the semi-vertical angle with the axis. (Key point: Condition on generator) 3. What is cos^2 15 degrees? A. (2 - sqrt(3))/4 B. (2 + sqrt(3))/4 C. 3/4 D. (1 + sqrt(3))/4 Answer: B. cos^2 15 = (1 + cos 30)/2 = (1 + sqrt(3)/2)/2 = (2 + sqrt(3))/4. (Key point: Value of cos^2 15) 4. Which is the equation before expansion? A. 2(x + 2y + 3z)^2 = 7(2 + sqrt 3)(x^2 + y^2 + z^2) B. (x + 2y + 3z)^2 = 14 (x^2 + y^2 + z^2) C. 2(x + 2y + 3z)^2 = 7(2 - sqrt 3)(x^2 + y^2 + z^2) D. (x + 2y + 3z)^2 = (2 + sqrt 3)(x^2 + y^2 + z^2) Answer: A. Squaring cos alpha gives (x+2y+3z)^2 = 14 cos^2 alpha (x^2+y^2+z^2), with cos^2 alpha = (2+sqrt 3)/4. (Key point: Equation) ## Close variants set in other exams - [Right circular cone with vertex (1,1,1), semi-vertical 30°](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cone-vertex-1) — Mathematics, chapter 3, asked 3× ## Related in Mathematics - [Cone with vertex origin and circle base](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acone-vertex-origin-circle-base) — Mathematics, chapter 3, asked 2× - [Right circular cylinder of radius 2 (axis 2,1,2)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-radius-2) — Mathematics, chapter 3, asked 7× - [Right circular cylinder of radius 2 (axis 1,1,2)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-radius-2-2) — Mathematics, chapter 3, asked 7× - [Right circular cylinder with axis (x−α)/l=(y−β)/m=(z−γ)/n](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aright-circular-cylinder-axis-x) — Mathematics, chapter 3, asked 7× - [Cone with vertex (3,1,2) and given base](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acone-vertex-3-1-2) — Mathematics, chapter 3, asked 2× - [Cone with vertex (α,β,γ) and parabola guiding curve](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acone-vertex-parabola-guiding-curve) — Mathematics, chapter 3, asked 2×