Prisms: Area & Volume
"Trapped in a Prism"
This song uses the story of a man unjustly imprisoned to teach students how to calculate the area and volume of a prism. The volume is the area of the base times the height, or length times width times height. The surface area is the total area of all of the faces, or 2(lw + lh + wh). The song explains the math behind these formulas and gives students a catchy way to remember them.
In a distant, dark future, the corporation rules all. One man fights against this faceless organization.
They snatched me off the streets, put me in a van,
Blindfolded, rope tied around both my hands.
I’ve been imprisoned in a rectangular prism,
It’s a 3-D rectangle, and listen:
They told me each day, they’d add 90 ft3 of water
To the cell, until it fills and I’m a goner.
I need to know just how long that will take 'em,
So I need to figure out the volume of the prism.
I’ll start by getting the area of the floor space,
That’s length times width, the area of the base.
I pace along the wall, it’s about 12 ft,
I walk along the other wall, that’s 6 ft.
So Base Area = lw,
I won’t call it quits,
That’s just 12 ft · 6 ft = 72 ft2,
I measured it right,
But it’s gonna be tough to measure the height.
Until I see stones that are placed in the wall,
Each stone is about 2 ft tall.
I count the stones to the top, I count
5 · 2 ft = 10 ft high.
The volume of a prism is area of the base times the height,
I know it so I get it right.
That’s 72 ft2 · 10 ft, are you ready?
In cubic feet, that’s 720 ft3.
If they add 90 ft3 of water every day,
720 ft3 ÷ 90 ft3/day = 8 days.
That’s it, how long I have to escape,
From the prism, so I need to find a way...
You’re in a prison and you’re scared like it’s all doom,
Well I guess you need to figure out
The volume of a rectangular prism or cube,
Volume = lhw.
You’re in a prison, yeah there’s nothing scarier,
If you need to figure out the surface area.
Add the area of all the faces,
Face it, they build a lock, you’re gonna break it.
So you all know I was trapped in a box,
But I escaped, now I’m out wrapping a box.
A present for my enemies, nothing’s scarier,
I need to measure the outside, the surface area.
It’s just all the faces added,
And there are 6 faces, but here’s the magic:
3 different faces and each has a twin,
We only need to measure 3 faces to begin.
Length times width plus length times height,
Plus width times height, but to make it right,
Multiply that by 2 for the opposite faces,
A gift for the guys who left me in this place, yes.
Make no mistake:
Surface Area = 2(lw + lh + wh).
I’m not wrapping a bomb, I’m wrapping a note,
Telling 'em that I’ll die for my freedom, no joke!
You’re in a prison and you’re scared like it’s all doom,
Well I guess you need to figure out
The volume of a rectangular prism or cube,
Volume = lhw.
You’re in a prison, yeah there’s nothing scarier,
If you need to figure out the surface area.
Add the area of all the faces,
Face it, they build a lock, you’re gonna break it.
What’s a rectangular prism? A box, basically. A prism is a stack of polygons. (Remember: a polygon is a two-dimensional shape with three or more sides. So a prism takes it into the third dimension.) A rectangular prism is a stack of rectangles.
How do we measure three dimensional space? Using volume. Volume is how much three dimensional space something takes up. We usually measure it in cubic meters or cubic feet. 90 ft3 is a measure of volume.
The volume of a prism is calculated by multiplying the area of the base times the height. For a rectangular prism, the area of the base is length times width, so the volume is lwh, or length × width × height.
Surface area is the total area of all of the faces of the prism. It’s like taking apart a cardboard box and laying it flat. Surface area measures the two dimensional space that all the sides take up, so we use two dimensional units to measure it like square feet or square meters.
A rectangular prism has six faces, and each face is identical to the one opposite. The top and bottom are the same, the front and back are the same, and the left and right sides are the same. (One version of a rectangular prism is a cube, which has sides of all the same length. What would be a simplified way of writing the formula for measuring the surface area of a cube?)
In this formula, everything within the parentheses is multiplied by two. 2lw measures the top and bottom, 2lh measures the left and right sides, and 2wh measures the front and back.
Why do we multiply each surface by two? Because each face has a twin on the opposite side. So you only have to measure one of them.
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