Question

A `5 cm` high prism has a right triangle base with cateti `6 cm and 8 cm` long. Calculate the surface area using a perimeter construct and a single formula. Calculate also the volume?

Answer 1

Surface Area: `168` sq.cm.
Volume: `120` cu.cm.

Explanation:

Here is the prism as I believe it has been described:

The hypotenuse length of `10` is calculated based on the Pythagorean Theorem: `sqrt(6^2+8^2)`

Since we were asked to use a perimeter construct in calculating the surface area, we might note that the vertical surface area is:
`"perimeter of right-triangular base" xx "prism height"`
`=24xx5`
`=120`

The total surface area is the vertical surface area plus the area of the top and the bottom right triangles:
`120+(6xx8)/2+(6xx8)/2`
`=168` (sq.cm.)

The volume of a prism is the area of its base times its height:
`(6xx8)/2xx5=120` (cu.cm.)

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As a side note:
The term cateti (singular catetus, if I remember correctly) it a term that refers to the non-hypotenuse sides of a right triangle. It is not a term we often see (at least in English, I'm not sure that I have encountered it in over fifty years) but I think it is a useful one and would encourage it as more convenient that saying "the non-hypotenuse sides". (I think the term may be Italian, if not Latin; could someone verify or correct this?)

Answer 2

The term is actualli latin "catetus". the plural "cateti" us also used in italian as is. As an aside most of the Italian was derived from latin hence the close similarity... In general an ending like "us" is almost never italian. Italian nouns for the most part end with vowels - singular a or o and plural e or i depending on gender... Cheers