Answer to Question #148945 in Geometry for solid mensuration

Question #148945
The surface area of a sphere inscribed in a regular tetrahedron is 144 cm2

21. What is the radius of the sphere?
22. What is the altitude of the tetrahedron?.
23. If a sphere is inscribed in a cube of side What is the volume of the sphere?
1
Expert's answer
2020-12-17T06:45:14-0500


Let "a=" base of a regular tetrahedron, "l=" slant height of a a regular tetrahedron, "r=" radius of a sphere inscribed in a regular tetrahedron.

21. The surface area of a sphere


"A=4\\pi r^2"

Given "A=144cm^2"

"4\\pi r^2=144""r^2=\\dfrac{36}{\\pi}"

"r=\\dfrac{6\\sqrt{\\pi}}{\\pi}\\approx3.4(cm)"

22.


"SD=l=a\\dfrac{\\sqrt{3}}{2}, ED=\\dfrac{a}{2\\sqrt{3}}"

From the right triangle "SED"


"SD^2=SE^2+ED^2"

"l^2=h^2+(\\dfrac{a}{2\\sqrt{3}})^2"

"\\dfrac{3}{4}a^2=h^2+\\dfrac{1}{12}a^2"

"h=\\dfrac{\\sqrt{6}}{3}a"

"\\angle ESD=30\\degree, \\angle EDS=60\\degree"

Right triangle "SOF"


"SO=2OF=2r"

"h=SE=SO+OE=2r+r=3r"

"h=3r=\\dfrac{18\\sqrt{\\pi}}{\\pi}(cm)\\approx10.155(cm)"

The altitude of the tetrahedron is "h=\\dfrac{18\\sqrt{\\pi}}{\\pi}cm\\approx10.155cm"


23. If a sphere is inscribed in a cube of side "a", then


"r=\\dfrac{a}{2}, V_{sphere}=\\dfrac{4}{3}\\pi r^3 =\\dfrac{\\pi a^3}{6}"


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