Question

three bottle of perfume come in a special gift set. Each bottle in its own individual canister.Then the three canisters are placed in one gift set. Each canister has a radius of 5cm. and all three canisters touch each other in the gift box. what is the are of the space between the canisters?.

Answer 1

three bottle of perfume come in a special gift set. Each bottle in its own individual canister. Then the three canisters are placed in one gift set. Each canister has a radius of 5cm. and all three canisters touch each other in the gift box. what is the are of the space between the canisters?.

Solution:

We have next diagram:

Step 1: Making a connection between centers of circles.

As for as we can see each side of triangle equals to 2 radius of circle. Therefore we make a conclusion that triangle is equilateral. Each angle of it equals to 60 degree and side $= 2^{*}5 = 10(\mathrm{cm})$

Step 2: Calculate area of triangle.

a - Side of triangle

$S = 43.3cm^2$

Step 3: Calculate area of circle's sector.

$$S = \frac {\pi r ^ {2} \alpha}{360 {}^ {\circ}}$$

$\alpha$ -angle of triangle $= 60$

$r$ - radius $= 5$

$S = 13.1(cm^2)$

As for as we can see triangle has 3 sectors inside, so area of sectors $= 13.1 * 3 = 39.3 (cm^2)$

Step 4: Calculate space between circles.

Area of triangle – area of sectors

$S = 43.3 - 39.3 = 4(cm^2)$

Answer: $4cm^2$