A circle has a radius of 6 in. The inscribed equilateral triangle will have an area of: ?
The height of the equilateral triangle will be a median in that triangle, and the centroid will be the centre of the circle.
That centroid splits the median in the ratio of 2 : 1
so the whole median, or the height of the triangle will be 9 in.
Half of the triangle will be a 30-60-90 triangle whose sides are in the ration of 1 : √3 : 2
9/√3 = b/1
so b = 9/√3 or the base is 18/√3
area of triangle = (1/2)(18/√3)(9) = 81/√3 or 27√3 or 46.765
Just thought of an easier way.
From the centre draw the 3 radii, giving us 3 congruent triangle with sides 6 and 6 and contained angle of 120°
Area of one of these = (1/2)(6)(6)sin120°
= 18(√3/2) = 9√3
but we have 3 of them, so the total equilateral triangle is 27√3
i believe it would be 56.52 (not 100% positive though) because if the radius is 6, then the circumference would be 3.14(6^2)= 3.14(36) which is 113.04 then you would put that over 360 and set up a proportion:
113.04/360 = x/180 (simplified to 20347.2/360) which gives you a final answer of 56.52
hope this helps!
To find the area of an equilateral triangle inscribed in a circle, we can use the following formula:
Area of inscribed equilateral triangle = (√3 / 4) * (side length)^2
In this case, the radius of the circle is given as 6 inches. The side length of the inscribed equilateral triangle will be equal to the diameter of the circle, which is twice the radius.
Side length of the triangle = 2 * radius = 2 * 6 inches = 12 inches
Now we can substitute the side length into the formula:
Area of inscribed equilateral triangle = (√3 / 4) * (12 inches)^2
Calculating this, we get:
Area of inscribed equilateral triangle ≈ 62.35 square inches
Therefore, the area of the inscribed equilateral triangle is approximately 62.35 square inches.
To find the area of an equilateral triangle inscribed in a circle, we need to use a few geometric properties.
First, let's draw the circle with radius 6 in:
```
O (Center of the circle)
/ \
/ \
/ R \ (R = 6 in, Radius)
```
Next, we draw the equilateral triangle inscribed in the circle, where each side of the triangle touches the circle at a point:
```
O
/ | \
/ | \
/____|____\ (Sides of the triangle are touching the circle)
```
We know that in an equilateral triangle, all sides are equal, and each angle is 60 degrees.
Now, let's find the side length of the equilateral triangle. The diameter of the circle is equal to the side length of the equilateral triangle. Since the radius is 6 in, the diameter is 2 * 6 = 12 in.
The formula to find the area of an equilateral triangle is:
Area = (sqrt(3)/4) * (side length)^2
Substituting the values we know:
Area = (sqrt(3)/4) * (12 in)^2
Calculating it further:
Area = (sqrt(3)/4) * 144 in^2
Area = 36sqrt(3) in^2
Therefore, the area of the equilateral triangle inscribed in the circle is 36sqrt(3) square inches.