Rhombus confused

Loki123

Junior Member
Joined
Sep 22, 2021
Messages
174
The base of an oblique prism is an equilateral triangle of side a, one side is normal to the base and has the shape of a rhombus with a smaller diagonal c. Calculate the volume.

After some thinking I have visualized this problem. Basically one side goes from a square to a rhombus and that's how it has 90 degrees to the base. At least that's how I understood. The answer has to include only a and c and so far the only thing I am missing is H, height of the prism, which is also h, height of the rhombus. So If I am right the question is basically, how to calculate rhombus height with a side and a smaller diagonal?

Help

IMG_20220114_200541.jpg
 

lev888

Elite Member
Joined
Jan 16, 2018
Messages
2,665
The base of an oblique prism is an equilateral triangle of side a, one side is normal to the base and has the shape of a rhombus with a smaller diagonal c. Calculate the volume.

After some thinking I have visualized this problem. Basically one side goes from a square to a rhombus and that's how it has 90 degrees to the base. At least that's how I understood. The answer has to include only a and c and so far the only thing I am missing is H, height of the prism, which is also h, height of the rhombus. So If I am right the question is basically, how to calculate rhombus height with a side and a smaller diagonal?

Help

View attachment 30684
Express the area of the rhombus 2 ways, solve the resulting equation for h.
 

Loki123

Junior Member
Joined
Sep 22, 2021
Messages
174
Side times height. That's what it says in the book. Is it wrong cause I got a semi-correct answer.

C^2+d^2=4a^2
D=sqrt(4a^2-c^2)

ah/2=cd/2
ah=c*sqrt(4a^2-c^2)
h=c*sqrt(4a^2-c^2)/a

V=a^2*sqrt(3)/4 * c*sqrt(4a^2-c^2)/a
V=ac*sqrt(12a^2-3c^2)/4

It's supposed to be
h=c*sqrt(4a^2-c^2)/2a
V=ac*sqrt(12a^2-3c^2)/8
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
12,914
Side times height. That's what it says in the book. Is it wrong cause I got a semi-correct answer.

C^2+d^2=4a^2
D=sqrt(4a^2-c^2)

ah/2=cd/2
ah=c*sqrt(4a^2-c^2)
h=c*sqrt(4a^2-c^2)/a

V=a^2*sqrt(3)/4 * c*sqrt(4a^2-c^2)/a
V=ac*sqrt(12a^2-3c^2)/4

It's supposed to be
h=c*sqrt(4a^2-c^2)/2a
V=ac*sqrt(12a^2-3c^2)/8
You're just off by a factor of 2; your formula cd/2 for area of a rhombus is correct.

But look again at ah/2 ! Isn't that the formula for a triangle??
 
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