the figure below shows 3 similar circles that just touch one another.A,Band C are the centres of the circles.If the perimeter of the triangle ABC is 42cm, what is the total area of the unshaded part of the 3 circles. (take pie = 22/7)
what are "similar" circles? If you mean they have the same radius, then
ABC is equilateral, with side 14, circles have radius 7.
Area of ABC is 49/2 √3
Area of circular sectors included in ABC is
3 * 1/2 (40)(π/3) = 20π
Dunno what's shaded or not, but that should give you the numbers to work with.
To find the total area of the unshaded part of the three circles, let's break it down into smaller steps:
Step 1: Find the radius of the circles
Since the circles are similar and touch each other, the radius of the circles will be the same. Let's call it "r".
Step 2: Find the side length of the equilateral triangle ABC
Since the perimeter of the triangle ABC is given as 42 cm, we know that the sum of the sides of an equilateral triangle is equal to three times the side length. Therefore, the side length of the equilateral triangle is 42 cm / 3 = 14 cm.
Step 3: Find the height of the equilateral triangle
The height of an equilateral triangle can be found using the equation h = (√3/2) * s, where "h" is the height and "s" is the side length. Plugging in the values, we get h = (√3/2) * 14 cm = 7√3 cm.
Step 4: Find the area of the equilateral triangle
The area of an equilateral triangle can be found using the equation A = (√3/4) * s^2, where "A" is the area and "s" is the side length. Plugging in the values, we get A = (√3/4) * (14 cm)^2 = 42√3 cm^2.
Step 5: Find the area of one circle
The area of a circle can be found using the equation A = π * r^2, where "A" is the area and "r" is the radius. Plugging in the values, we get A = (22/7) * (r cm)^2.
Step 6: Find the total unshaded area of the three circles
Since the three circles touch each other at their centers, they form an equilateral triangle with sides equal to the sum of the diameters of the circles. The sum of the diameters is equal to three times the radius, so the side length of the equilateral triangle formed by the circles is 3 * 2r = 6r cm.
The unshaded area is the difference between the area of the equilateral triangle and the combined areas of the three circles. Therefore, the unshaded area = (42√3 - 3 * (22/7) * r^2) cm^2.
Please note that we cannot solve for the exact value of the unshaded area without knowing the value of the radius "r" or any other specific measurements.
To find the area of the unshaded part of the three circles, we first need to determine the radius of each circle.
Since the figures are described as "similar circles," it means that they have the same shape, but different sizes. This further implies that the ratio of the radii of the circles is the same as the ratio of their perimeters.
Let's denote the radius of circle A as r. Given that the perimeter of triangle ABC is 42 cm, we can use this information to find the perimeter of triangle ABC in terms of the radius r.
The perimeter of a triangle is the sum of the lengths of its three sides. In this case, the three sides of triangle ABC are the circumferences of the three circles. Since the circles are touching each other, their circumferences will be equal to the sum of the radii of two circles multiplied by pi (π).
The perimeter of triangle ABC is given as 42 cm. This is equal to the sum of the circumferences of the three circles.
Perimeter of triangle ABC = Circumference of circle A + Circumference of circle B + Circumference of circle C
Perimeter of triangle ABC = 2πr + 2πr + 2πr
Perimeter of triangle ABC = 6πr
We know that the perimeter of triangle ABC is 42 cm, so:
6πr = 42
Now we can solve this equation to find the value of r:
r = 42 / (6π)
r = 7 / π
r = 7 / (22/7) [Using the value of π as 22/7]
r = 1 cm
Now that we know the radius of each circle is 1 cm, we can calculate the area of the unshaded part of the circle.
The total area of the unshaded part of the three circles is the area of the entire circle minus the area of the shaded part. Since the unshaded area consists of three identical parts, we can calculate the area of one part and then multiply by 3.
The area of a circle is given by the formula A = πr^2.
For each circle, the area of the shaded part is given by:
Area of shaded part = Area of circle - Area of the equilateral triangle
Area of shaded part = πr^2 - √3/4 * (2r)^2
Area of shaded part = πr^2 - √3/4 * 4r^2
Area of shaded part = πr^2 - √3r^2
Area of shaded part = r^2 (π - √3)
Now we can calculate the total area of the unshaded part of the three circles:
Total area of unshaded part = Area of shaded part * 3
Total area of unshaded part = 3 * r^2 (π - √3)
Total area of unshaded part = 3 * (1^2) (22/7 - √3)
Total area of unshaded part = 3 * (22/7 - √3) square cm
Thus, the total area of the unshaded part of the three circles is 3 * (22/7 - √3) square cm.