Inscribed circle

A circle is inscribed at the bottom wall of the cube with an edge (a = 1). What is the radius of the spherical surface that contains this circle and one of the vertex of the top cube base?

Correct result:

r =  0.8004

Solution:

$$\begin{gathered} a=1 \\ {r}_1=a\mathrm{/}2=1\mathrm{/}2={\displaystyle \frac{1}{2}}=0.5 \\ u=\sqrt{2}\cdot a=\sqrt{2}\cdot 1=\sqrt{2}\doteq 1.4142 \\ {r}_2=u\mathrm{/}2=1.4142\mathrm{/}2\doteq 0.7071 \\ {r}^2=x^2+r_1^2 \\ {r}^2=(a-x{)}^2+r_2^2 \\ {x}^2+r_1^2=(a-x{)}^2+r_2^2 \\ {r}_1^2=-2xa+a^2+r_2^2 \\ x={\displaystyle \frac{a^2+r_2^2-r_1^2}{2\cdot a}}={\displaystyle \frac{1^2+0.7071^2-0.5^2}{2\cdot 1}}={\displaystyle \frac{5}{8}}=0.625 \\ {r}_3=\sqrt{x^2+r_1^2}=\sqrt{0.625^2+0.5^2}\doteq 0.8004 \\ {r}_3=r \\ r=\sqrt{(x-a{)}^2+r_2^2}=\sqrt{(0.625-1{)}^2+0.7071^2}=0.8004 \end{gathered}$$

Try another example

Showing 0 comments:

Next similar math problems:

• Top of the tower
  The top of the tower has the shape of a regular hexagonal pyramid. The base edge has a length of 1.2 m, the pyramid height is 1.6 m. How many square meters of sheet metal is needed to cover the top of the tower if 15% extra sheet metal is needed for joint
• Billiard balls
  A layer of ivory billiard balls of radius 6.35 cm is in the form of a square. The balls are arranged so that each ball is tangent to every one adjacent to it. In the spaces between sets of 4 adjacent balls other balls rest, equal in size to the original.
• 9-gon pyramid
  Calculate the volume and the surface of a nine-sided pyramid, the base of which can be inscribed with a circle with radius ρ = 7.2 cm and whose side edge s = 10.9 cm.
• Pyramid in cube
  In a cube with edge 12 dm long we have inscribed pyramid with the apex at the center of the upper wall of the cube. Calculate the volume and surface area of the pyramid.
• Truncated cone 6
  Calculate the volume of the truncated cone whose bases consist of an inscribed circle and a circle circumscribed to the opposite sides of the cube with the edge length a=1.
• Right triangular prism
  We have cuboid with a base and dimensions of 12 cm and 5 cm and height of 4 cm. The tablecloth cut it into two identical triangular prisms with right triangular bases. The surface of the created prisms was painted with color. Calculate the surface area of
• MO SK/CZ Z9–I–3
  John had the ball that rolled into the pool and it swam in the water. Its highest point was 2 cm above the surface. Diameter of circle that marked the water level on the surface of the ball was 8 cm. Determine the diameter of John ball.
• Spherical cap
  Place a part of the sphere on a 4.6 cm cylinder so that the surface of this section is 20 cm². Determine the radius r of the sphere from which the spherical cap was cut.
• Cube in a sphere
  The cube is inscribed in a sphere with volume 7253 cm³. Determine the length of the edges of a cube.
• Quadrangular pyramid
  Calculate the surface area and volume of a regular quadrangular pyramid: sides of bases (bottom, top): a1 = 18 cm, a2 = 6cm angle α = 60 ° (Angle α is the angle between the side wall and the plane of the base.) S =? , V =?
• A cylinder
  A cylinder 108 cm high has a circumference of 24 cm. A string makes exactly 6 complete turns around the cylinder while its two ends touch the cylinder's top and bottom. (forming a spiral around the cylinder). How long is the string in cm?
• Hexagonal pyramid
  Calculate the surface area of a regular hexagonal pyramid with a base inscribed in a circle with a radius of 8 cm and a height of 20 cm.
• Sphere in cone
  A sphere is inscribed in the cone (the intersection of their boundaries consists of a circle and one point). The ratio of the surface of the ball and the contents of the base is 4: 3. A plane passing through the axis of a cone cuts the cone in an isoscele
• Sphere parts, segment
  A sphere with a diameter of 20.6 cm, the cut is a circle with a diameter of 16.2 cm. .What are the volume of the segment and the surface of the segment?
• Cuboid face diagonals
  The lengths of the cuboid edges are in the ratio 1: 2: 3. Will the lengths of its diagonals be the same ratio? The cuboid has dimensions of 5 cm, 10 cm, and 15 cm. Calculate the size of the wall diagonals of this cuboid.
• Spherical cap
  The spherical cap has a base radius of 8 cm and a height of 5 cm. Calculate the radius of a sphere of which this spherical cap is cut.
• Cubes
  One cube is inscribed sphere and the other one described. Calculate difference of volumes of cubes, if the difference of surfaces in 257 mm².