Semicircle Mania

Let the radius of the smaller semicircles be $r$ . Then the radius of the larger semicircle is $3r$ .

The area in red is then

$A = 3*(\frac{1}{2}\pi r^{2}) = \frac{3}{2}\pi r^{2}$ ,

and the area in blue is

$B = \frac{1}{2}\pi (3r)^{2} - A = \frac{9}{2}\pi r^{2} - \frac{3}{2}\pi r^{2} = 3\pi r^{2}$ .

The ratio of $A:B$ is then $\frac{3}{2} : 3 = \boxed{1 : 2}$ .

The ratio of red:green will be the same if they are semi circles or full circles. I mentally doubled the picture.

The radius of the larger blue circle is three times the radius of the smaller red circles, or a 1:3 ratio. Therefore because area is 2-D, the ratio is squared and becomes 1:9

Call the area of a full red circle 1, the area of the full blue circle would be 9. The area of the blue circle minus the area of 3 red circles is 9-3=6

The ratio of red:blue is therefore 3:6 or 1:2.

Don't need radius length or formulas!

Let r be the radius of bigger semicircle . Then radius of red semicircle = $\frac { r }{ 3 }$

$ar(red semicircle)=\frac { \pi { r }^{ 2 } }{ 9 }$

$ar(red region) = 3[ar(red semicircle)]= \frac { \pi { r }^{ 2 } }{ 3 }$

$ar(big semicircle)= \pi { r }^{ 2 }$

$ar(blue region)=ar(big semicircle)- 3[ar(red semicircle)]=\pi { r }^{ 2 }-\frac { \pi { r }^{ 2 } }{ 3 } =\pi { r }^{ 2 }(1-\frac { 1 }{ 3 } )=\frac { 2\pi { r }^{ 2 } }{ 3 }$

$\frac { ar(red) }{ ar(blue) } =\frac { \frac { \pi { r }^{ 2 } }{ 3 } }{ \frac { 2\pi { r }^{ 2 } }{ 3 } } =1:2$

Let's assign one unit to the radius of the equal small circles. So the area of each small circle is π . The red area is the sum of the areas of the 3 small semicircles, i.e. 3π/2 . Now, the radius of the large circle is 3 units, so the area of the large semicircle is 9π/2 , which is 3 times the red area. So, the requested ratio is 1:2 .

assume the radius of the red semicircle to be 1 unit, then the radius of the big semicircle is 3 units

area of red part = 3 x 1/2 x pi x1^2 = 3/2 pi

area of blue part = 1/2 x pi x 3^2 - 3/2 pi = 4.5pi - 1.5pi = 3pi

red part/blue part = (3/2 pi) / 3 pi = 1/2

Ratio of diameter smaller semi circle to larger semicircle is 1:3 Ratio of area 1:9 Area ratio of three small semi circle to larger circle 3:9 =1:3 Ratio of red region to blue region 1:(3-1)=1:2✓

For semicircle radius $r$ , the Red Area $=$ $\frac{3}{2}\pi{r}^{2}$ . Now, GIVEN the equivalence of the semicircles’ radii , we can derive the diameter, and thus radius, of the larger semicircle to be $3r$ . You know the drill, the total area (Blue + Red) $=$ $\frac{\pi{(3r)}^{2}}{2}$ $=$ $\frac{9}{2}\pi{r}^{2}$ . Now, the Blue Area $=$ $[\frac{9-3(=6)}{2}]\pi{r}^{2}$ $=$ $3\pi{r}^{2}$ . Thus, $\frac{RedArea}{BlueArea}$ $=$ $\frac{\frac{3}{2}\pi{r}^{2}}{3\pi{r}^{2}}$ $=$ $\frac{1}{2}!!!$

Let the radius of a red circle be 1. The total red area is 3pi/2. The radius of the blue circle is 3. The blue area is 9pi/2 - 3pi/2 = 6pi/2. The ratio of the red area to the blue area is 3pi/2 : 6pi/2 or 1: 2.