# A square and an equilateral triangle have the same perimeter. If the diagonal of the square is 12sqrt2 cm, what is the area of the triangle?

Dec 9, 2015

$64 \sqrt{3} {\text{ cm}}^{2}$

#### Explanation:

Let ${A}_{s}$ be the area of the square, ${P}_{s}$ be the perimeter of the square and ${a}_{s}$ be the length of a side of the square. (All sides have equal lengths.)

Let ${A}_{t}$ be the area of the triangle, ${P}_{t}$ be the perimeter of the triangle and ${a}_{t}$ be the length of a side of the triangle. (All sides have equal lengths.)

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1) As we know the length of the diagonal of the square, we can compute the length of a side of the square using the Pythagoras formula:

${a}_{s}^{2} + {a}_{s}^{2} = {d}^{2}$

$\implies 2 {a}_{s}^{2} = {\left(12 \sqrt{2}\right)}^{2}$

$\implies {a}_{s}^{2} = {12}^{2}$

$\implies {a}_{s} = 12 \text{cm}$

2) Knowing the length of one side of the square (and thus knowing all lengths of a square), we can easily compute the square's perimeter:

${P}_{s} = 12 \cdot 4 = 48 \text{cm}$

3) We know that the square and the equilateral triangle have the same perimeter, thus

${P}_{t} = 48 \text{cm}$

4) As all sides have the same length in an equilateral triangle, the length of one side is

${a}_{t} = {P}_{t} / 3 = \text{16 cm}$

5) Now, to compute the area of the equilateral triangle, we need the height $h$ which can be computed with the Pythagoras formula again:

${h}^{2} + {\left({a}_{t} / 2\right)}^{2} = {a}_{t}^{2}$

$\implies {h}^{2} + {8}^{2} = {16}^{2}$

$\implies {h}^{2} = 192 = 64 \cdot 3$

$\implies h = 8 \sqrt{3} \text{cm}$

6) At last, we can compute the area of the triangle:

${A}_{t} = \frac{1}{2} \cdot h \cdot {a}_{t} = \frac{1}{2} \cdot 8 \sqrt{3} \cdot 16 = 64 \sqrt{3} {\text{ cm}}^{2}$