Question

Find the area of an equilateral triangle with apothem 7 cm. Round to the nearest whole number

Answer 1

In equilateral triangle all sides are equal =x,
from  right triangle that  is formed by apothem , height 7*3= 21
hypotenuse /opposite leg = sin angle,
hypotenuse is a side of triangle =x, opposite leg is apotheme =21,
angle in equilateral triangle =60⁰
21/x= sin 60, x*sin60=21, x=21/sin60, x=21/(√3/2), x=42/√3
Area of triangle =1/2 x*x*sin angle

Area of triangle =1/2 *42/√3*42/√3*sin 60=1/2*(42²/3)*(√3/2) ≈ 255 cm²



this is just second way to do this problem, that I did at first (either way is correct)
 apothem divides one side by half,
so we get small right triangle with sides x hypotenuse, x/2 is one leg , and 21 is another leg
by Pythagorean theorem 
x²=(x/2)² +21²
x²-x²/4=441
3x²/4=441
3x²=441*4=1764
x²=1764/3
x=42/√3
Area of triangle =1/2*base *height =1/2*42/√3 *21≈255 cm²