Cylinders, cones and spheres by vector tools · Grade 12 · 5 units
In Cylinders, cones and spheres by vector tools (Grade 12 · 5 units) you practise 4 skills: The cone and its inscribed sphere · A sphere cut by a plane · A cylinder inscribed in a sphere or a cone · The box in coordinates. Cylinders, cones and spheres are round solids. The volume of a cylinder is the base area times the height, and the volume of a cone is a third of that. The curved surface area of a cone is pi times the radius times the slant height, and the slant height is the hypotenuse of the right triangle whose legs are the radius and the height. When the angle between the slant height and the height is given, trigonometry in that triangle gives the slant height.
To solve a problem about a round solid, look at its section through the axis. The section of a cone is an isosceles triangle, the section of a cylinder is a rectangle, and the section of a sphere through its centre is a great circle. A sphere inscribed in a cone shows as a circle inscribed in a triangle, and a cylinder inscribed in a sphere shows as a rectangle with its corners on the circle. A cylinder standing inside a cone shows as a rectangle with its top corners on the sides, and above it is a small triangle similar to the whole one. A plane that cuts a sphere cuts it in a circle. The perpendicular from the centre of the sphere to the plane meets it at the centre of that circle, so the radius of the sphere, the distance of the plane from the centre and the radius of the section are the sides of a right triangle. That gives the radius of the section, and then its circumference and its area. In a box whose edges are parallel to the axes, the coordinates of the corners give the vectors of the diagonals. The length of a diagonal is the size of its vector, and the angle between two diagonals leaving the same corner comes from the scalar product. In a cube, two face diagonals leaving the same corner make an angle of sixty degrees, since the segment joining their ends is a face diagonal too, so the triangle is equilateral.
Tip: Before you calculate, draw the section through the axis and mark the right triangle in it.
What you learn
The cone and its inscribed sphere
For example: The base of a cone has radius 1 , and its height is 9 . What is the volume of the cone? Give your answer in terms of π .
Answer and solution
The answer: 3π
The volume is V = π × 1² × 9 ÷ 3 , that is, V = 3π .
A sphere cut by a plane
For example: A plane cuts a sphere of radius 5 at a distance 4 from the centre of the sphere. What is the radius of the section?
Answer and solution
The answer: 3
The centre of the section is the foot of the perpendicular from the centre of the sphere, so by Pythagoras’ theorem r = √(5² − 4²) = 3 .
A cylinder inscribed in a sphere or a cone
For example: A cylinder of height 8 is inscribed in a sphere of radius 5 : the circles of both bases lie on the sphere. What is the radius of the base of the cylinder?
Answer and solution
The answer: 3
Half the height is 4 , so by Pythagoras’ theorem r = √(5² − 4²) = 3 .
The box in coordinates
For example:
In the box ABCDA′B′C′D′ the edges are parallel to the axes. The corners A(0, 3, 2) and C′(2, 4, 4) are given. What is the length of the diagonal AC′ ?Answer and solution
The answer: 3
From the coordinates AC′⃗ = (2, 1, 2) , so AC′ = √9 = 3 .
Not on the site yet
Parts of this topic that are not built yet: A pyramid inscribed in a cube.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).