How do I find the angles of a Triangular Prism?

Question

Say I have a triangular prism whose triangluar faces are right iscosoles triangles with the side opposite the right angle being 4in. Say the angle of the 4in side of the triangle connecting to the rectangular face is 45 degrees. How do I find the angles between the rectangular faces of the prism, and the angle in relation to both the rectagular faces and the triangular faces?

I am not sure if I am asking that in a way that is understandable, i know what I'm asking so feel free to ask for clarification.

Answer 1

Hi Hector,

This is a great question, especially one that is fun to think about. Imagine that your triangular prism has a triangular face with a hypotenuse called side b and legs of lengths a and c. In your example, the angle between sides a and c are 90 degrees, and the angle between each leg and side b is 45 degrees, with side b = 4 inches. Imagine that your triangular prism also has a height h (which would be the height of each rectangular face).

From this prism, we can see that there are 3 rectangular faces, each parallel to a side of a triangular face (a, b, and c being the side lengths). Upon inspection, the angles between each rectangular face are the same as the angles between each side. For example, let the face with height h and width b be called face BH, and the face with height h and width a be called face AH, and so on. We observe that the angle between faces BH and AH is the same as the angle between side a and b, or just 45 degrees. The angle between faces AH and CH is the same as the angle between sides a and c, which is 90 degrees. This is the same for any triangular prism with rectangular faces.

The angles between the rectangular and triangular faces are always 90 degrees. This is because the side h of the prism is perpendicular to the triangular faces. Perpendicular is nothing but 90 degrees. You can also think about this in 3D geometry. Say legs a and c of the triangle lay on the x and y axes on a single plane. Side h extends above the triangle, and lies on the z-axis. All axes are perpendicular to each other, meaning that the angle between a rectangular and triangular face is 90 degrees.

I hope this answers your question. Feel free to ask for clarification if something does not make sense.

Akshat Y.