棱柱的表面积

To find the surface area of a prism, we just add up all the faces' areas. However, this may be tricky to visualize at first, so let's take an equilateral triangular prism first given the side length and height.

First, let's find the lateral surface area. This is simply obtained by adding the three lateral faces' area together. Since all of the lateral faces are rectangles with dimensions $s$ and $h$, the lateral surface area is just: $$3sh$$ To find the total surface area, we need to add the areas of the two bases. These are equilateral triangles, so we have the total surface area: $$3sh + \dfrac{s^2\sqrt 3}2$$

Now, we generalize. Let's take a general prism with $n$-gons as bases and height $h$. Let the side lengths of the $n$-gon be $a_{1,2,\dots,n}$. Then, we can calculate the lateral surface area: $$ha_1 + ha_2 + \cdots + ha_n = h(a_1 + \cdots + a_n)$$ To make the formula nicer, we note that the sum of the side lengths is simply the perimeter, so letting $p$ be the perimeter, we have: $$S_{\text{lateral}} = hp$$ Now, for the total surface area. We simply add the areas of the two bases. Thus, letting the area of the base be $B$, we have: $$S_{\text{total}} = 2B + hp$$ So to recap, given the areas of the bases, the perimeter, and the height of a prism, we can easily find the lateral and total surface areas.

The nice thing about calculating the surface area of a prism, as compared to that of shapes like cylinders/spheres, is that a prism is composed entirely of flat, polygonal faces.

Often, this means we can just find the surface area by just finding the area of each face and adding it up.

In general, if a rectangular prism has side lengths $x,y,z$, what is its surface area? Write down a formula.

$A = 2(xy+yz+zx)$ (there are two faces of area $xy$, etc.)

What about non-rectangular prisms?

OK, that wasn't too hard. But what if the lateral faces aren't all the same?

So the lateral surface area is $h(s_1+...+s_6)$. Do you see a simpler way of writing this?

$s_1+...+s_6$ is the perimeter of the hexagon! Hopefully this makes sense -- if you imagine unfolding the green rectangles and then laying them flat on a table, it forms a larger rectangle with height $h$ and width the perimeter $p$ of the hexagon.

So this allows us to give a general formula for the total surface area for a polygonal prism.

So the formula is $\boxed{A=2b+p\cdot h}$, where $h$ is the height, and $p,b$ is the perimeter of the base.

But wait, there's more! There's another common shape very similar to a polygonal prism.

So the surface area of a cylinder is $A=2\pi rh + 2 \pi r^2 = \boxed{2\pi r (r + h)}$.

Quick prism surface area formula reference sheet

(It's hidden so you don't ruin the other explanations for yourself -- you should read those first if you haven't already!)

Rectangular prism of side lengths $x,y,z$: $A=2(xy+yz+zx)$.

Prism with base area $b$, base perimeter $p$, height $h$: $A = 2b+ph$

Cylinder of radius $r$, height $h$: $A = 2\pi r(r+h)$