Lateral/total area for a prism with a polygon base
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mmm4444bot
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Pyro said:what changes when figuring [prism areas with] a polygon shaped base
Just more faces (shapes) for which the area must be calculated, I think.
Oh, and working with some triangles, too, sometimes.
A square is a polygon; it's a polygon with four sides.
A triangle is a polygon with three sides.
I'm guessing that you're talking about polygons with more than four sides.
In your exercise, it seems to me that you can "unwrap" the lateral surface into a "flat" rectangle that measures 125 by 315 centimeters (like taking the label off a soup can). That area is 39,375 cm^2.
The prism has a top, yes? So, the area of the top and bottom combined is twice the bottom area (827). That's 1,654 cm^2 for the top&bottom.
The total area is the sum of the lateral and top&bottom areas calculated above.
Otherwise, I'm going to assume that the top and base in your exercise are identical regular polygons, and describe another method.
How many sides does the polygon have ? That's the number of triangles into which you will partition the polygons.
I'm also going to assume that the lateral faces of the prism are perpendicular to both the top and the base. (In other words, a right-prism.)
Here's a simple example.
Let's say we have a right-prism with a hexagonal base (that's a six-sided regular polygon).
Each side of the hexagons is 5 centimeters.
The height of the prism is 20 centimeters.
Therefore, the lateral surface is composed of 5-cm by 20-cm rectangles, and there are six of them.
The base of these rectangles is 5 and their height is 20.
6(5)(20) = 600
The six rectangular areas combine for a total of 600 square centimeters of lateral surface area.
Next, we calculate the total surface area.
The top and bottom are identical hexagons. Each of them can be partitioned into six identical equilateral triangles (that is, triangles with three equal sides and three equal interior angles).
If readers need help understanding this, then draw a hexagon and put a dot in the center. Connect the dot to each of the six "corners", and you will have partitioned the hexagon into six identical equilateral triangles.
There's a "shortcut" formula for the area of an equilateral triangle, using the symbol s to represent the length of the sides:
Area = s^2 * sqrt(3)/4
In my example, s = 5.
Area = 5^2 * sqrt(3)/4 =
25/4 * sqrt(3)
That's about 10.825 square centimeters of area for each triangle, and there are 12 of them.
12(10.825) = 129.9 square centimeters of surface area on the top and bottom combined.
Again, the lateral area is 600 cm^2.
That makes the total area 729.9 cm^2.
Please verify my math. (I make a lot of mistakes.)
If you're working with triangles that are not equilateral (i.e., the polygon is not six-sided), then use this formula for the area of any triangle:
Area = 1/2 * Base * Height
Draw a picture. Make right-triangles. If this exercise comes from a trigonometry course, then expect to use sine, cosine, or tangent functions to solve for a triangle's height (and maybe Pythagorean Theorem, sometimes, too).
If you do not understand my example, let me know, and I will draw you a picture.
Otherwise, please provide more detail about your specific prism, ask specific questions, and show whatever work that you can, if you still need help.
Cheers ~ Mark
MY EDIT: clarified that my post deals with polygons with five sides or more
mmm4444bot
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Denis makes a good point, a square is a polygon.
I edited my first post.