Area of an Equilateral Triangle

[SumDood]

So my fifth grade son had this problem on his Math Team placement test:

31. An equilateral triangle has a perimeter of 15 inches. What is its area?
a. 25 sq. in.
b. 7.5 sq. in.
c. 15 sq. in.
d. 12.5 sq. in.

He missed the problem and his homework was to have a parent explain the problems to him. Geometry was one of my favorite subjects so I had fun explaining to him the process I would use to find the answer. First I had to use the pythagorean theorem to find the height of the triangle and then used that that value in the area of a triangle formula (A = 1/2 x B x H). Since the height was the square root of 18.75, the final answer ended up to be 10.8253175473... Which ended up to be the same answer if you use the formula from wiki (Area of a triangle). Obviously, this was not one of the 'given' answers.

Before I send a semi-self-righteous email to the school, can someone check my work and see if I missed something obvious in the problem?

Greatest appreciation in advanced.

 

[Madalch]

Your answer looks correct to me.

 

[ehcks]

Wikipedia has (Sqrt(3)/4)*side² for the area of an equilateral triangle. It also comes up with 10.8...

Wolfram gives A = 1/4 sqrt((a+b-c) (a-b+c) (-a+b+c) (a+b+c)) for the area of a triangle if you know the sides. It also comes up with 10.8...

There has to be a problem with the question.

For some unknown reason I suspect the answer it wanted was b. Do you know what the answer was?

 

[Nowhere Man]

Likewise, 10.825 (roughly). I somehow misread "equilateral" as "right" and was trying to figure out the Diophantane equation a+b+a²+b²=15 .

Fred

 

[GlennB]

Ditto about 10.825 . Given the numbers involved it's never going to be a 'neat' answer like 7.5 as I make h to be ~4.33

 

[Puppycow]

A multiple choice question and all of the answers are wrong?

That's weird. Which one did he answer btw, and did he realize that none of the given answers were right?

 

[Andy_Ross]

Answer is -

e. None of the above.

 

[Brainster]

The correct answer:

Take your kid out of that class.

 

[GlennB]

Just curious .... presumably some of the other kids solved that question. Did anybody kick up a fuss about the choice of wrong answers?

 

[Brian-M]

Where am I going wrong? I get an answer of 11.46

If an equilateral triangle has a perimeter of 15 inches, then each side must be 5 inches.

Now I don't know how to calculate the area of an equilateral triangle directly, but I do know how to calculate the area of right-angle triangles, so I divide the triangle in half to get two right-angle triangles with a hypotenuse of 5 inches, base of 2.5 inches and a height of sqrt(hypotenuse² - base²) = 4.5826 inches

The area of each right-angle triangle is (2.5*4.5826)/2 = 5.728

Multiply by two (because each right-angle) triangle is half the original equlilateral triangle, and the total area is 11.46

How come I get this when everyone else gets 10.8 ?

 

[erwinl]

Where am I going wrong? I get an answer of 11.46

If an equilateral triangle has a perimeter of 15 inches, then each side must be 5 inches.

Now I don't know how to calculate the area of an equilateral triangle directly, but I do know how to calculate the area of right-angle triangles, so I divide the triangle in half to get two right-angle triangles with a hypotenuse of 5 inches, base of 2.5 inches and a height of sqrt(hypotenuse² - base²) = 4.5826 inches

The area of each right-angle triangle is (2.5*4.5826)/2 = 5.728

Multiply by two (because each right-angle) triangle is half the original equlilateral triangle, and the total area is 11.46

How come I get this when everyone else gets 10.8 ?

Not everybody else.
That was my answer as well.

Still not among the answers given though.

 

[Guybrush Threepwood]

Where am I going wrong? I get an answer of 11.46

If an equilateral triangle has a perimeter of 15 inches, then each side must be 5 inches.

Now I don't know how to calculate the area of an equilateral triangle directly, but I do know how to calculate the area of right-angle triangles, so I divide the triangle in half to get two right-angle triangles with a hypotenuse of 5 inches, base of 2.5 inches and a height of sqrt(hypotenuse² - base²) = 4.5826 inches

The area of each right-angle triangle is (2.5*4.5826)/2 = 5.728

Multiply by two (because each right-angle) triangle is half the original equlilateral triangle, and the total area is 11.46

How come I get this when everyone else gets 10.8 ?

How do you get 4.5826 for the height? I get sqrt(25-6.25)=sqrt(18.75) = 4.330127

4.5826^2 is 21 which implies you are taking the base length of your right angled triangle as 2 not 2.5

 

[Brian-M]

How do you get 4.5826 for the height? I get sqrt(25-6.25)=sqrt(18.75) = 4.330127

4.5826^2 is 21 which implies you are taking the base length of your right angled triangle as 2 not 2.5

Damn... I mustn't have pressed the 5 button hard enough when I typed in the 2.5. I only put it in the calculator once, and was going by the 4.58257 I jotted down on a piece of paper.

My mistake, operator error.

 

[elgarak]

I'm suspecting that the kids were given an estimation formula. sqrt(3)/4 = 0.4330, so this is estimated to 0.5.

0.5 * s^2 = 0.5*25 = 12.5

That would explain the answers. Still, the question should indicate that it is only a rough estimate.

EDIT: I speculated pretty much on the estimation, but it fits in that errors could easily result in the other answers:

25 = 5 ^ 2 (the side squared, but forgotten the 0.5)
7.5 = 0.5 * 15 (the kid remembered some of the estimate, but forgot what to half)
15 = ?? (I'm stumped here a bit, but could think that a second estimate was given such that the perimeter can be taken to estimate the area, or some such)

 

[TjW]

Well, if it was a placement test for a math team, it wouldn't be unreasonable for them to be using estimates to eliminate wrong answers.
But it's really stupid to send the kid home to talk to his parents about it without emphasizing to the parent what technique they were trying to use, and why.

 

[CynicalSkeptic]

I'm suspecting that the kids were given an estimation formula. sqrt(3)/4 = 0.4330, so this is estimated to 0.5.

I'm not aware of any math rule that allows .433 to be rounded up to .5.

But at least your answer matches one of the choices, so I suspect you're right.

 

[elgarak]

I'm not aware of any math rule that allows .433 to be rounded up to .5.

But at least your answer matches one of the choices, so I suspect you're right.

Neither am I. I am not saying it's a good estimate (it's awful, as matter of fact, but could work to estimate how many tiles you would need to cover the bathroom floor while you're at the store, but even then, for small sides, it gives grossly wrong estimates).

The question, as given, is an abomination since it's plain wrong. It asks for the area, period, not an estimate, and as such gives only wrong answers.

I'm just trying to find an explanation why such a mess could be given in the first place.

 

[Brown]

Who's the teacher, Jeb Bush, who thought a 3-4-5 triangle was a 30 degree-60 degree-90 degree triangle? Or maybe it was the Scarecrow from the Wizard of Oz: "The sum of the square roots of any 2 sides of an isosceles triangle is equal to the square root of the remaining side."

Edited to add: I may have mis-characterized Jeb Bush's screw-up. He knew that a 3-4-5 triangle had a right angle in it, and he sorta seemed to know that the total of all angles ought to be 180 degrees, but his math was fuzzy. I believe there was an instance on this forum, however, in which someone may have mistakenly believed that a 3-4-5 triangle was a 30 degree-60 degree-90 degree triangle.

 

[malicus]

Looks very much like a question setter error, ie the setter thinks an equilateral triangle with a perimeter of 15 inches has equal sides of 5 inches (true) and splitting the triangle in two to ease calculating the area produces another side, also of 5 inches (not so true).

Therefore I suspect the correct answer is 12.5.

 

[SumDood]

Well the teacher who administered the test said they knew about the error. I didn't get a good explanation why they asked a question with no correct answer in the choices given. She was "tickled" that a parent was involved enough to question the test. Sorta sad that I was the only one.