An equilateral arch ABC is drawn about an equilateral triangle having side of length 7.?

Question

Ex: AC is a circular arc having center B. Calculate the area of the equilateral arch.
Picture: http://img512.imageshack.us/img512/258/archik1.jpg

Answer 1

Area of figure = Area of equilateral triangle + Area of arc AB + Area of arc BC.
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Now, to find the area of an equilateral triangle, use the formula

A = ((s^2)/4)*(sqrt3) where s is the length of the side. Here, S= 7, so now solve the equation.

((s^2)/4)*(sqrt3)
=((7^2)/4)*(sqrt3)
=(49/4)*(sqrt3)
=12.25*sqrt3
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To find the area of the arc, you need to see angle ABC as a central angle for a circle. Basically, see ABC as a sector of a circle where AB is the radius. Each angle in an equilateral triangle is 60 degrees.

In order to find the area of a sector, use the formula

a = (area of circle)*(d/360) where d is the number of degrees. Here, d equals 60. The area of a circle is (pi)(r^2) where r is the radius of a circle. Here, r equals 7. Now solve.

a = ((pi)(r^2))*(d/360)
a = ((pi)(7^2))*(60/260)
a = ((pi)(49))*(1/6)
a = 49pi * (1/6)
a = (49(pi))/6

The area of the sector is (49pi)/6. This area, however, includes the equilateral triangle. So, in order to find the area of the arc, we need to subtract the area of the triangle from the area of the sector.

a = ((49pi)/6) - 12.25(sqrt3)
Simplify this on paper to get
a = (98pi - 147sqrt3)/12

The area of arc AB equals the area of arc BC. So, to get the total area of the two arcs, multiply by two to get

area of two arcs = (98pi - 147sqrt3)/6
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Now add the area of the triangle and the two sectors.

A = (12.25pi * sqrt3) + (98pi - 147sqrt3)/6
Simplify this to get a total area of the figure...

A = (196pi - 147sqrt3) / 12
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Answer 2

Hi. The radius of the circle is 7 (Radius BA, center B). Does this get you started? The chord is obviously 7 as well. So you calculate the area of the circular section (x2) then the triangle's area. Add.

Answer 3

Approx. 4.426 if I did it right...

pie r squared: r=7 gives.. 153.86

Divided by 6 (360/6 = 6) gives 25.64333
That is the area of 60 degrees of the circle.
(Equalateral triangles have 60 degree angles..).
1/2 base times height = area of the triangle.
height is 7 * sin(60) = 7 * .866 = 6.0621778
1/2 * base = 3.5
Area of triangle = 3.5 * 6.0621778 = 21.2176

Arc area = 25.64333 - 21.2176 = 4.426 (rounded)

Answer 4

well, first, find the area of that half oval by finding the area of an oval and dividing it by two, and then, find the area of the triangle and subtract it from the area of the half oval.

Answer 5

what is an equilateral arc?