# geometry

• Oct 6th 2006, 11:43 PM
zasi
geometry
i have some questions help
1.The formula for the area of a circle and the formula for the circumference of a circle.Write a formula for the area of a circle in terms of its circumference.

2. The formula for the height h of an equilateral triangle.where b is the lenght of a side . write a formula for the area of an equilateral traingle in terms of the following.
a.the lenght of a side only
b. the height only
• Oct 7th 2006, 12:12 AM
CaptainBlack
Quote:

Originally Posted by zasi
i have some questions help
1.The formula for the area of a circle and the formula for the circumference of a circle.Write a formula for the area of a circle in terms of its circumference.

The area of a circle in terms of its radius is:

A=pi*r^2

The circumference of a circle in terms of its radius is:

C=2*pi*r,

rearranging this last equation gives:

r=C/(2*pi),

now sumstituting this into the equation for the area gives:

A=pi*[C/(2*pi)]^2=C^2/(4*pi).

RonL
• Oct 7th 2006, 12:28 AM
CaptainBlack
Quote:

Originally Posted by zasi
2. The formula for the height h of an equilateral triangle.where b is the lenght of a side . write a formula for the area of an equilateral traingle in terms of the following.
a.the lenght of a side only
b. the height only

Drop a perpendicular from a vertex onto the opposite side, then as this
is an equilateral triangle this perpendicular is an altitude, and the point
at which it meets the oposite side bisects that side.

Applying Pythagoras's theorem to one half of the equilateral triangle,
we have the sides have length h, b, and b/2 (sketch a diagram at this point
to meke things clear), we see that:

b^2=(b/2)^2+h^2,

so:

h=sqrt(b^2-b^2/4)=b*sqrt(3/4)=b*sqrt(3)/2

and so the area of the equilateral triangle is:

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

If we want it in terms of h rather than b, we note that:

b=2*h/sqrt(3), so:

A = h*b/2 = h^2 * sqrt(3).

RonL