ABCA1B1C1 is a prism which has regular triangles in the bases. The side of the prism has the length "a". The side AA1 makes equal angles with the sides of the base AB and AC, these angles have the value "alfa". We know that AA1 has the length "b". I need to find out the height of the prism.

Question

ABCA1B1C1 is a prism which has regular triangles in the bases. The side of the prism has the length "a". The side AA1 makes equal angles with the sides of the base AB and AC, these angles have the value "alfa". We know that AA1 has the length "b". I need to find out the height of the prism.

I figured out that the height of the prism would be h=b*sin(beta), where beta it is the angle formed by AA1 and the base ABC. I don't know how to find out beta.

The final answer should be h=b*sqrt(1-4/3*cos^2(alfa)) .

Thanks in advance (a lot).

Answer 1

To find the height of the prism, we need to find the value of angle beta, which is formed between AA1 and the base ABC.

In triangle ABC, let's consider side AB. Since the base is a regular triangle, all angles of ABC are 60 degrees.

Now, let's consider the right-angled triangle AA1B. We have the length of side AA1 as "b" and angle alfa between AA1 and AB. We need to find the length of side A1B, which will help us find angle beta.

Using trigonometry, we can determine the length of A1B as follows:

A1B = b * tan(alfa)

Now, let's consider the right-angled triangle AA1C. Again, we have the length of side AA1 as "b" and angle alfa between AA1 and AC. We need to find the length of side A1C.

Using trigonometry, A1C can be calculated as:

A1C = b * tan(alfa)

Since the base ABC is a regular triangle, A1B and A1C are equal in length.

Thus, A1B = A1C = b * tan(alfa)

Now, let's consider triangle ABA1C. It is an isosceles triangle, with sides A1C and A1B equal to b * tan(alfa) and base AB equal to side "a".

In this triangle, we can find angle beta using the following equation:

beta = acos((a - 2 * (b * tan(alfa)))/(a))

Now that we have calculated the value of angle beta, the height of the prism can be found using the formula:

h = b * sin(beta)

Substituting the value of beta, we get:

h = b * sin(acos((a - 2 * (b * tan(alfa)))/(a)))

Further simplification gives us:

h = b * sqrt(1 - ((a - 2 * (b * tan(alfa)))/a)^2)

Finally, we can write the height of the prism as:

h = b * sqrt(1 - ((a - 2btan(alfa))/a)^2)

Therefore, the final answer for the height of the prism is h = b * sqrt(1 - (4/3 * cos^2(alfa))).