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[MUSIC] Hello. 
I'm at the Knowlton School of 

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Architecture at the Ohio State 
University. 

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And today, we're going to measure the 
slope of some staircases, namely, this 

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staircase here and this even larger 
staircase. 

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You may think that this larger staircase 
is steeper than this staircase because, 

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well, this staircase is harder to climb. 
But let's actually find out. 

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And we're going to find out by measuring 
how far the staircase goes up by how wide 

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each step is. 
When I measure this step, I see that it 

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goes over twelve inches and it rises 
about six and a quarter inches. 

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On the other hand, we have this larger 
staircase here. 

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Let's see what its slope is. 
So, when I measure, I get 36 inches for 

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the width of each step. 
On the other hand, [SOUND] I find that 

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the height of each step is about 18-3/4 
inches. 

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Now, that we've made the measurements, 
let's go compute the slope. 

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So, we've measured our stairs and we 
found out that the small steps went up by 

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6.25 inches and they were 12 inches wide, 
okay? 

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This gives us a slope for the small steps 
of 6.25 all over twelve. 

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But there were also large steps. 
Let's draw those in. 

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Here we go. 
So, this was one of the large steps and 

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then, we had another one. 
It goes off and it builds, it climbs over 

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there. 
It goes down like this. 

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This is a large step. 
Alright. 

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The height of the large step was this 
place right here. 

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18.75 inches, 
that's the height of the large step. 

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And the width of the large step was 36 
inches. 

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Okay, great. So, the slope of the large 
steps is 18.75 all over 36. 

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These two slopes are the same, because we 
can simplify, we can reduce this fraction 

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to be dividing the numerator by three and 
dividing the denominator by three, we get 

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6.25 / 12. 
And we can see, if we connect the tips of 

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these stairs with a line or this makes a 
line of the desired slope and see how the 

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large stairs and the small stairs all 
meet at the corner of this line. 

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The fact that the staircase with the 
large steps and the staircase with the 

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small steps have exactly the same slope 
is evidenced by the fact that they have 

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the same railing. 
There's the railing for the staircase 

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with the large steps and here's the 
railing for the staircase with the small 

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steps. 
And they're exactly at the same angle. 

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So, we've seen that the staircase with 
the larger steps and the staircase with 

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the smaller steps actually have exactly 
the same slope. 

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Now, let's use the slope to estimate how 
tall the staircase is after you've gone a 

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certain distance in the horizontal 
direction. 

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How high up am I now? 
Here, you can see that we have a tape 

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measure where we're measuring the width 
or the horizontal distance that I 

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traveled here on the staircase. 
And if we get to the point where I was 

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standing, we'll find out that it's about 
23 feet. 

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Now, how high was I standing when I was 
up there? 

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So, now that we've made our calculation, 
let's check our calculation. 

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Okay. Now, we have a question. 
We went to a certain part on the 

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staircase, 
say, right here. 

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And then we measured. 
We, we, we, we marked, 

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we had a point in mind and we measured 
the distance on the ground to that point. 

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We found out that it was 23 feet, 
okay? 

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So, if you remember, the slope of a 
staircase was 6.25 / 12, which is 

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approximately 0.52. 
We want to know this height, x. 

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So, we went x / 23 to be approximately 
0.52. 

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Solve for x. 
Multiply both sides by 23 and we get x is 

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approximately 0.52 times 23. 
0.52 times 23 is 11.96. 

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So, x is approximately 12 feet. 
Now, let's go check our answer. 

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So here we are, checking our measurement. 
[MUSIC] And I can see, it's almost 

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exactly twelve feet. 
[MUSIC]