Lessons 41 · 1 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
41
The ball (solid sphere) and its elements
Textbook, Part 2: pp. 146–154
GoalStudy the elements of a ball and a sphere, the equation of a sphere, plane sections, tangent planes and solids inscribed in or circumscribed about a ball.
New words
sphere · sferaball · shargreat circle · katta doiratangent plane · urinma tekislik
Explanation
The solid consisting of the points of space at a distance of at most R from a given point is a ball, and its boundary is a sphere; the given point is the centre and R the radius. The equation of a sphere with centre (a, b, c) is (x − a)² + (y − b)² + (z − c)² = R², and for the ball ≤ R². If we cut the ball with a plane at distance d from the centre, then for d > R there is no common point, for d = R the plane is tangent, and for d < R the section is a disc of radius r = √(R² − d²); a plane through the centre gives the largest section, a great circle. A tangent plane is perpendicular to the radius drawn to the point of tangency; the length of a tangent from an outside point is t = √(m² − R²), where m is the distance from the point to the centre. A polyhedron is inscribed in a ball if all its vertices lie on the sphere (for a cube of edge a, R = a√3/2); if all its faces touch the ball, the ball is inscribed in the polyhedron (for a cube r = a/2).
Worked examples
The sphere with centre (1, −2, 3) and R = 4: (x − 1)² + (y + 2)² + (z − 3)² = 16. The point (1, 2, 3) lies on it: 0 + 16 + 0 = 16. A plane at distance d = 3 from the centre of a ball with R = 5 cuts it in a disc with r = √(25 − 9) = 4, of area 16π.
A cube of edge 6: the inscribed ball has r = 3; the circumscribed ball has R = diagonal/2 = 6√3/2 = 3√3. A sphere has radius 5 and an outside point M is at distance 13 from the centre: the tangent length is t = √(169 − 25) = 12.
Class activity
“Orange sections”: with an adult’s help slice an orange (or a roughly spherical potato) along parallel planes. Measure the radius R of the largest section, the radii r of other sections and their distances d from the centre, and check that r² + d² = R².
Practice
1
Find the centre and radius of the sphere (x + 2)² + (y − 1)² + z² = 49.
(−2, 1, 0), R = 7
2
A ball of radius 25 is cut by a plane at distance 7 from the centre. Find the area of the section.
r = √(625 − 49) = 24, area 576π
3
A tangent is drawn to a sphere of radius 9 from an outside point 15 from the centre. What is the length of the tangent?
12
4
Why is every plane section of a ball a disc?
By the Pythagorean theorem the distance from the foot of the perpendicular to every point of the boundary of the section is √(R² − d²), the same for all. So the section is a disc centred at that foot with radius r = √(R² − d²).