Help with geometry please?

Question

Hi. I need some help with 3 geometry problems. Please explain how to do these?

1) The area of a circle is 16 pi. What is it's circumference?

2) What is the area of a circle with diameter 12?

3) What is the area of an equilateral triangle with perimeter 12?

Any help is greatly appreciated. Thank you very much!

Answer 1

1)
A = πr^2
c = 2πr
r = √(A/π)
c = 2π √(A/π)
c = 2π*√(16π/π)
c = 2π*√(16)
c = 2π*4
c = 8π

2) A = πr^2
d = 2r
r = d/2
r^2 = d^2/4
A = πd^2/4
A = 144π/4 = 36π
3) A = side*perpendicular/2
Now we need to find the perpendicular
As the triangle is equilateral and therefore isosceles, the perpendicular will bisect a side.
Since the perimeter is 12 = 3s, s = 4
Each segment of the bisected side will be 2
In order to find the length of the perpendicular, we can use the Pythagorean theorem. A^2 + B^2 = C^2
2^2 + x^2 = 4^2
4 + x^2 = 16
x^2 = 12
x = √12 = 2√3
The area is 2*√3*4/2 =4*√3

Answer 2

1) Area is pi(r^2). You have the pi part so r^2 must be 16. What number squared is 16? When you find that (which happens to be r), put it in the formula C = 2(pi)(r); multiply those three numbers together for your answer.

2) Area is pi(r^2). If the diameter is 12, the radius is half of that.

3) An equilateral triangle means that all three sides are the same length. If you know the total distance around the triangle is 12, then divide by 3 to get how long each individual side is.

Answer 3

1.) Since the formula to find the area of a circle is A = (pi)(r^2), you can use this to find the circumference as well just by getting the value for r, or the radius. First off, write the equation out using the information you are given:

16pi = (pi)(r^2)

Your first step is to eliminate pi from the equation by dividing both sides of the equation by pi, and you're left with 16 = r^2. Next, take the square root of both sides to get rid of the exponent on the r. Now you're left with 4 = r, or you have found the radius to be 4. Now, to find the circumference of the circle, you need another formula, this time for the circumference, which happens to be C = 2(pi)(r). Now that you know the value for r, plug that into the equation and solve it out. Your answer comes out to be 25.12.

2.) The formula to find the area of a circle is (pi)(r^2). r represents the radius, which is also half the diameter, which comes out to be 6 in this example. Using 3.14 as pi, your formula becomes 3.14 * (6^2) or 3.14 * 36. Your result is 113.04 square units.

3.) Since this triangle is equilateral, all the sides are the same length. The perimeter of the triangle is 12, which must be divided amongst the three sides equally, or each side equals 4. The formula to find the area of an equilateral triangle is (s^2) * (sqrt3) divided by 4, where s is the length of one of the sides. Since the sides all have a length of 4, go ahead and plug that in for s in the formula, and then solve out. The answer becomes 6.9282 square units.

Answer 4

Let s=the circle's area and r its radius
16 pi = s = pi * r^2

r=sqrt(s/pi)

The circumference is 2pi * r = 2*pi* sqrt(s/pi) =
= 2*pi*sqrt(s)/sqrt(pi) = 2 sqrt(s)/sqrt(pi) =
= 2 sqrt(16 pi)/sqrt(pi) = 2 sqrt(16) = 2*4 = 8

2) pi*(12/2)^2 = pi*6^2 = 36pi

3)The side of such a triangle is 12/3=4
The height of such a triangle is
h = 4 cos(30 degrees) = 4*sqrt(2)/2

The area = 4*h/2 = [4*4*sqrt(2)/2]/2 =
= 4*4*sqrt(2)/(2*2) = 4*4sqrt(2)/4 = 4 sqrt(2)