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Tom
Registered Member

Date Joined Jul 2001
Total Posts : 10353

Posted 1/28/2002 4:07 AM (GMT -6)

I apologize for asking this here, but I can't track the answer down on the Net and our resources are limited overseas. My daughter is doing 10th grade geometry and she's into finding the area of polygons right now. We're stumped with the following problem:

You have an equilateral triangle with an apothem (a line drawn from the center of the triangle to the middle of one of the sides) of 6 inches. What's the area of the triangle?

Tom

Edited by - TGB on 01/28/2002 09:21:47 AM

June
Registered Member

Date Joined Dec 2001
Total Posts : 105

Posted 1/28/2002 5:59 AM (GMT -6)

TGB,

somebody please verify this for me, but the area in 36 sq. in.

The area of a triangle is .5(base)(height).

the height of the triangle is 12 in. (the apothem times 2).

the base of the triangle is half of the height because the triangle is equilateral. (12 divided by 2).

.5 x 12in. x 6in. = 36in.

Hope that's right!

June

whasssuuuppp
Registered Member

Date Joined Aug 2001
Total Posts : 210

Posted 1/28/2002 9:19 AM (GMT -6)

Hmm, I get a different answer, but it's been a loooonnnnggg time since I've had math, so bear with me.

I think you need to use some trig. to get the answer.

The apothem divides the equilateral triangle (a triangle with 3 60-degree angles) into 2 smaller trianges of equal size. Therefore, it also divides one of the 60 degree angles into 2 30-degree angles. So now you know TWO things about each of these smaller triangles--which is enough to figure out the area. Here's how:

The TANGENT of an angle equals the length of the size OPPOSITE the angle divided by the side ADJACENT to the angle. So, in this case, the tangent of 30 degrees equals the opposite side (a length we DON'T know) divided by the adjacent side (a length we DO know--it's 6 inches). It looks like this (use a scientific calculator):

tan 30= x/6
0.577= x/6
x= 0.577 * 6
x=3.464

Now we know the lengths of two sides of the triangle: 6 inches and 3.464 inches.
The formula for the area of a triangle is:
1/2 * base * height

So the area of one of the triangles is:
1/2 * 6 * 3.464
=10.392

And, because we divided the original triangle into 2 triangles, you need to multiply the above answer by 2:
=10.392 * 2
=20.785 square inches

Yes? No? Hope that helps!

**************************

Tom
Registered Member

Date Joined Jul 2001
Total Posts : 10353

Posted 1/28/2002 12:03 PM (GMT -6)

Oh boy, I thought this would be easy!

An “apothem” is a line that runs from the center of a regular polygon directly perpendicular to one of the sides.

For example: Take a square with 1” sides. This square has four apothems of equal length, each one running from the center of the square directly to each side of the square forming right angles. See it? If the sides are each 1” in length, what is the length of one apothem? Easy, ½ an inch.

June said the height is twice the apothem. And in the case of regular polygons of four sides or more (squares, hexagons, octagons, etc.), she’s right. But in the case of an equilateral triangle, the height is three times the apothem.

Wassuuup, you said the apothem divides our triangle into two equal halves. Not according to my daughter’s Geometry book. The apothem is a line which begins at the center of the triangle and runs perpendicular to one side, forming a right angle. So it doesn’t divide our angle into two smaller but congruent triangles.

So an equilateral triangle, obviously, has three apothems of equal length, each running from the same center point out to one of the three sides. I wish I could draw it and post it. Stink! If you draw an equilateral triangle and find the center, from which point three lines of equal length may originate, each line running straight to a side (dividing the length of that side in two) and thus forming a right angle, you got it. See it?

OK now, you should also see that each apothem, if extended on through the rest of the triangle, runs directly into the vertex of the opposite angle. See it? This line (running from the midpoint of any one side straight to the vertex of the opposite angel) is the height of the angle. It is three times the length of the apothem (and includes the apothem of course). This, Wassuuup, is what you thought was the apothem.

OK, back to my problem that I can’t figure out. All I know is that the apothem of an equilateral triangle is 6 inches. I also know, then, that the height is 18 inches (what Wassuuuuup thought was the 6 inch dividing line) and the radius (a line running from the center point [where the apothem originates] to any vertex [as opposed to a side]) is 12 inches. That may or may not be important. Anyhow, if area is ½ base x height, what’s the base (or the length of any side)?

If nobody wants to try again, that’s OK. It’s a theological discussion group, not a math help group.

Sorry Pia!

Tom

Edited by - TGB on 01/28/2002 5:05:53 PM

Edited by - TGB on 01/28/2002 5:07:35 PM

whasssuuuppp
Registered Member

Date Joined Aug 2001
Total Posts : 210

Posted 1/28/2002 12:21 PM (GMT -6)

Ok, let me try again.

Actually, I think you still need to use trig. to get the answer.

So the apothem is 6 inches, and the height of the triangle is 18 inches. It it THIS 18-inch line that divides the original triangle into 2 smaller, equal-sized triangles. Now you can figure out the area of one of these triangles and multiply it by 2, like before.

Use tangent again (the tangent of an angle equals the length of the opposite side divided by the length of the adjacent side). You could use sine or cosine, too. Each angle of the equilateral triangle is 60 degrees. So here's the equation this time (with x being the length of the "base"):

tan 60= 18/x
1.732= 18/x
1.732x= 18
x= 18/1.732
x=10.393

So the area of the triangle is 1/2 * base * height

=1/2 * 10.393 * 18
=93.533

And then multiply by 2, because this is the area of only half the original triangle.

=187.067

Whattdya think?
**********************

Iconoclast
Registered Member

Date Joined Oct 2001
Total Posts : 1227

Posted 1/28/2002 2:21 PM (GMT -6)

The first thing that comes to my mind is to use Pythagorus' (sp?) theorem:

The apothem line is 18" and makes a 90 degree triangle that we need to multiply the area times 2 to get the whole area. Lets lable the length of a side as X. Then we have a triangle of 3 sides length X, X/2 and 18 inches. It is a right triangle and therefore Hypotenuse(Squared) = Side(Squared) + Side(Squared)

So: X(squared) = 18(squared) + (X/2)(squared)

From here - just need to solve for X and then the area is:

Area = (X * 18)/2

thanks,
Iconoclast

"being weird is rarely a total waste of time" - Greg Boyd

Iconoclast
Registered Member

Date Joined Oct 2001
Total Posts : 1227

Posted 1/28/2002 2:25 PM (GMT -6)

I came up with: 187.06148

Looks like we have convergent evidence that closely agrees. Must be right.

thanks,
Iconoclast

"being weird is rarely a total waste of time" - Greg Boyd

(o)
Registered Member

Date Joined Jan 2002
Total Posts : 9

Posted 1/31/2002 7:46 AM (GMT -6)

You guys are so smart.

Tom
Registered Member

Date Joined Jul 2001
Total Posts : 10353

Posted 1/31/2002 8:13 AM (GMT -6)

Actually the correct answer is: 108 times the square root of 3. Gotta run now. I'll post the "how to" later!

Tom

Iconoclast
Registered Member

Date Joined Oct 2001
Total Posts : 1227

Posted 1/31/2002 3:35 PM (GMT -6)

Square Root of 3 = 1.7320508075688772935274463415059

(Square Root of 3) x 108 = 187.06148721743874770096420488263

Hmmmm...guess I should have rounded the 8 to a 9.

thanks,
Iconoclast

"being weird is rarely a total waste of time" - Greg Boyd

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