**Quote:**

Why is it that the large cone = (π(2*x*)²) × (2AB)/3= 8π*x*²AB/3. Specifically, how do you derive an 8 out of the preceding equation?

We get 8 by multiplying 2² from the term (π(

**2***x*)

**²**) and 2 from the term (

**2**AB)/3.

2² × 2 = 4 × 2 = 8

or, watch twos in the whole expression:

(π(

**2***x*)

**²**) × (

**2**AB)/3 = π ×

**2²** ×

*x***²** ×

**2**AB/3 = (

**4** ×

**2**) × π

*x*² × AB/3 =

**8**π

*x*² × AB/3

Now, let's deal with another part of your question –

*Why is it that the large cone = (π(2x)²) × (2AB)/3?*We use the provided formula, S × H/3, where S is the area of the base and H is the height.

Knowing that AB = BC, we easily calculate the height H = AC = AB + BC = AB + AB = 2AB .

For the base we use the formula for the area of a circle. S = π × R²

And now comes the last part –

*How do we know that * R = 2

*x*?

We know this from similarity of triangles ABD and ACE:

They are similar, because they have equal angles: they have angle A in common, and angle B = angle C as they are both right angles (AB and AC are heights of the cones and therefore are perpendicular to the bases).

We already know that AC = 2AB, so R = 2

*x*.