Area and perimeter

We need the equation for the total surface area ($A$) of a sphere. From Geometry of Shapes, Surface Area and Volume, the surface area is:

$A=4\pi r^2$

where $r$ represents the sphere’s radius.

The radius is multiplied by a factor of $3$.

If the original radius is $r$, then the new radius ($r_{\text{new}}$) is:

$r_{\text{new}}=3r$

Replace $r$ with $3r$ in the surface area formula. Make sure the full expression is squared:

$\begin{array}{rl}{A}_{\text{new}}& =4\pi (3r{)}^2\\ A_{\text{new}}& =4\pi (3^2\cdot r^2)\\ A_{\text{new}}& =4\pi (9r^2)\end{array}$

So we can rewrite this as:

$A_{\text{new}}=9\times (4\pi r^2)$

Our algebra shows the surface area becomes $9$ times the original. Compare that with the options:

• Option C is wrong because it treats the change as linear ($3^1$), but area depends on $r^2$.
• Option A is wrong because $27$ is $3^3$, which matches how volume scales, not surface area.
• Option D is wrong because $6$ often comes from mistakenly doing $3\times 2$ instead of squaring the scale factor.

Because $A_{\text{new}}=9\times A_{\text{original}}$, the total surface area increases by a factor of $9$.

In scaling problems, the number that multiplies a length (like a radius) is the linear scale factor $k$.

Since the radius is multiplied by $3$: $k=3$

When a 3D shape is enlarged by a factor $k$:

• Lengths scale by $k$.
• Surface Areas scale by $k^2$.
• Volumes scale by $k^3$.

Because the question asks about total surface area, we use $k^2$.

With $k=3$, the surface area factor is:

$\text{Area Factor}=k^2=3^2$

And: $3\times 3=9$

Using the value $9$:

• Option A gives $27$, which equals $3^3$ (that’s the volume scaling).
• Option C gives $3$, which is only the linear factor $k$.
• Option D gives $6$, which does not follow the power scaling rules.

Option B is the only choice that matches $9$.

To make the comparison simple, start with a sphere of radius $1$ unit.

Let $r_{\text{initial}}=1$.

The surface area of a sphere is:

$A=4\pi r^2$

Substitute $r=1$:

$A_{\text{initial}}=4\pi (1{)}^2=4\pi$

The radius is multiplied by $3$:

$r_{\text{new}}=1\times 3=3$

Now substitute $r=3$ into the formula:

$A_{\text{new}}=4\pi (3{)}^2$ $A_{\text{new}}=4\pi (9)=36\pi$

The factor increase is the ratio:

$\text{Factor}=\frac{A_{\text{new}}}{A_{\text{initial}}}=\frac{36\pi }{4\pi }=9$

Check the options against $9$:

• Option A ($27$) equals $3^3$, which relates to volume, not surface area.
• Option C ($3$) is only the radius scale factor.
• Option D ($6$) comes from an incorrect add/multiply-by-2 idea instead of squaring.

The calculations show the surface area is multiplied by $9$.

We need the correct geometry vocabulary for parts of 3D solids. A pyramid has one base (a polygon) and its side surfaces are triangular faces. The question is asking for the one point where those triangular faces all come together at the top.

This idea appears in Geometry of Shapes, Surface Area and Volume.

In coordinate geometry, the Origin is the point $(0,0)$ or $(0,0,0)$ where the axes cross. That word names a location on a graph, not the top point of a pyramid.

A Centroid means the balance point or geometric center of a figure. For a pyramid, this point lies inside the solid, so it cannot be the point where the triangular faces meet.

A Vertex is any corner point where edges meet, and pyramids have several vertices (around the base and one at the top). Since the question asks for the special top point where all the triangular faces meet, we want the more precise name for that top vertex: the Apex.

By definition, the Apex is the vertex opposite the base of a pyramid (or cone). It is the single point where all the lateral triangular faces intersect.

We are looking for the special name of the top point of a pyramid—the point where all the lateral triangular faces meet. We can find it by eliminating choices that don’t match that description.

The origin is the reference point of a coordinate grid: $(0,0)$ in 2D or $(0,0,0)$ in 3D. Even if you place a pyramid on a graph, “origin” is not the name of a pyramid’s top point.

The centroid is the center/balance point of a shape. In a 3D pyramid, that point sits inside the solid, so it does not describe where the triangular faces come together.

A vertex is any corner where edges meet. A pyramid has multiple vertices, so the word is too general for a question asking for the one particular meeting point of the triangular faces at the top.

The Apex is the specific term for the topmost vertex of a pyramid or cone. It is opposite the base and is exactly where all the triangular side faces meet.

This matches the question, so it is the correct choice.