Video Transcript
The area of an equilateral triangle
is 947 times the square root of three centimeters squared. Find the side length using Heron’s
formula, giving the answer to the nearest centimeter.
Since our triangle is equilateral,
all three sides have the same length, so let’s call this length 𝑥. We’re given the area of the
triangle, that’s 947 times root three centimeters squared, and asked to use Heron’s
formula to calculate the side length which we’ve labeled 𝑥. This tells us that for a triangle
with side lengths 𝑎, 𝑏, and 𝑐, the area 𝐴 is equal to the square root of 𝑠
times 𝑠 minus 𝑎 times 𝑠 minus 𝑏 times 𝑠 minus 𝑐, where 𝑠 is the semiperimeter
of the triangle; that’s the sum of the side lengths over two. Now in our case, 𝑎, 𝑏, and 𝑐 are
all equal to 𝑥, so the semi perimeter 𝑠 is 𝑥 plus 𝑥 plus 𝑥 over two; that’s
three 𝑥 over two.
In our formula for the area 𝐴, we
have 𝐴 equal to the square root of 𝑠 times 𝑠 minus 𝑥 times 𝑠 minus 𝑥 times 𝑠
minus 𝑥. That’s the square root of 𝑠 times
𝑠 minus 𝑥 cubed. We already have 𝑠 in terms of 𝑥,
and remember it’s 𝑥 we want to find. So substituting 𝑠 equals three
over two 𝑥 into our expression for 𝐴, we have 𝐴 equal to the square root of three
over two times 𝑥 times three over two 𝑥 minus 𝑥 cubed. Now, three over two 𝑥 minus 𝑥 is
equal to 𝑥 over two. And so, our area is the square root
of three over two times 𝑥 times 𝑥 over two cubed. Using the power law for indices on
our cube inside the square root, we have the square root of three over two 𝑥
multiplied by 𝑥 cubed over two cubed, which simplifies to the square root of three
over two to the power four times 𝑥 to the power four.
And now making a little more room
and using the fact that by the laws of indices, the positive square root of a number
or expression to the fourth power is that number or expression squared, and now
since area is always positive, we have the area equal to root three over two squared
multiplied by 𝑥 squared. That’s root three over four times
𝑥 squared. Now remember, we’re actually given
that the area is equal to 947 root three centimeters squared. And if we divide both sides by root
three and multiply both sides by four, we have 𝑥 squared is equal to 947 times
four. That’s 𝑥 squared is equal to
3788.
Taking the positive square root on
both sides, since length is always positive, we have 𝑥 equal to the square root of
3788, which to four decimal places is 61.5467. We’re asked for the side length to
the nearest centimeter, and that’s 62 centimeters. Hence, the side length to the
nearest centimeter of the equilateral triangle with area equal to 947 root three
centimeters squared is 62.
