Math problem -- triangle area?

Question

If x denotes the length of a side of an equilateral triangle, express the area of the triangle as a function of x.

please explain

Answer 1

All angles are 60°
All sides are x
height = h
h = x sin 60°
Area = (1/2) (x) (h)
Area = (1/2) (x) ( x sin 60°)
Area = (1/2) x ² (√3 / 2)
Area = (√3 / 4) x ²

Answer 2

You need to do this in two steps:
1) find the value of the "height" of the triangle - distance from top point down to the base.
Use Pythagoras as you are creating a right angle triangle with hypotenuse = x² and one side with length (x/2)² (equilateral triangle so splitting base exactly in half)
if h= height of traiangle
x² = (x/2)² + h²
=> h² = 3x²/4 and h= x.√(3/4) = (x√3)/2

Area of triangle = 1/2(base x height) = 1/2(x * (x.√3)/2)
area = (√3/4).x²


Alternatively if you are happy using trigometry you can solve this by saying
Area = 1/2(ab)sin(γ) where a and b are the lengths of two sides of the triangle and γ is the angle between them
a=x, b=x, γ=60 (equilateral trianle : all sides are equal, all angles = 60 degrees)

So Area = 1/2.x.x.sin(60) // sin(60) = √3 / 2

So area = (√3/4).x²

Answer 3

The area of a rt triangle is = 1/2 base x height you are given the base = 16 ft. you must solve for the height by using the pythagorean theorem h^2(hypotenuse) = b^2(base) + x^2 where x is the unknown side therefore: (34)^2 = (16)^2 + (x^2) X= sqrt((340^2 - (16)^2) = sqrt(1156 -256) sqrt(900) X = 30 and the area then is: A= 1/2 x 16 x 30 = 240 or answer c

Answer 4

SInce your triangle is an equilateral triangle with side x and area of a triangle is equal to one half times base times height.

To solve for the base you should try to illustrate first your triangle so:

base = (1/2)(x)
but this is only half of your equilateral triangle so:
base = x

height will be equal to using Pythagorean theorem:

height = (x² - (x/2)²)^ (½)

So the equation for the area of the equilateral triangle:

A(x) = (1/2) x base x height
A(x) = (1/2) (x) (x² - (x/2)²)^ (½)
A (x) = (1/2) (x) (x² - (x²/4))^ (½)
A (x) = (1/2) (x) (3x²/4))^ (½)
A (x) = (1/2) (x) (x/2)(3^ (½))
A (x) = ((√3)/4) x²

Answer 5

from the basic equation

1/2 * base * height

base = x

height = 1/2 * x * tan 60 ===> formed from a right triangle dividing the equilateral triangle to two and therefore acquiring height of the triangle

substituting to the basic equation of the triangle, that is

Area = 1/4 * x^2 * tan60 ===1/4 tan60 = 0.433

Area = 0.433 * x^2

Answer 6

X=60

Answer 7

A=½bh Since the altitude to the base makes a 30-60-90 triangle h=½√3x
A=½x(½√3x)
f(x)=¼√3x²

Answer 8

Area of triangle = 1/2 (base)(height); an equilateral triangle has 3 equal sides, so if one side is x, then

1/2(x)(x) or x^2/2

Answer 9

h=sqrt(3).x/2
A=sqrt(3).x^2/4