Surface area and volume

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

$A=4\pi r^2$

where $r$ represents the radius.

The radius is multiplied by a factor of 3.

If the original radius is $r$, then the new radius is:

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

Substitute $3r$ into the surface area formula. Make sure the entire $3r$ 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 it as:

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

This shows the new surface area is $9$ times the original. Compare that with the options:

• Option C is wrong because it treats the change as linear ($3^1$), but surface area depends on $r^2$.
• Option A is wrong because $27$ is $3^3$, which corresponds to how volume scales, not surface area.
• Option D is wrong because it comes from incorrectly 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, the number that multiplies a length (like a radius) is the linear scale factor $k$.

Here, the radius is multiplied by a factor of 3, so: $k=3$

When a 3D shape is enlarged or reduced:

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

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

With $k=3$:

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

And: $3\times 3=9$

Our factor is $9$, so:

• Option A gives $27$ ($3^3$), which is the volume factor.
• Option C gives $3$, which is only the linear factor.
• Option D gives $6$, which does not follow the scaling powers.

Option B is the one that matches $9$.

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

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

Use the sphere surface area formula from the data booklet:

$A=4\pi r^2$

Substitute $r=1$:

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

The radius is multiplied by a factor of 3:

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

Now use $r=3$:

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

Compare new to original:

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

Using the result $9$:

• Option A ($27$) equals $3^3$, which is for volume changes.
• Option C ($3$) is just the radius scale factor.
• Option D ($6$) comes from incorrect operations like adding or doubling instead of squaring.

So the total surface area becomes $9$ times as large.

We need to remember the correct vocabulary for 3D solids. A pyramid has a polygon base, and every side (lateral) face is a triangle. The question is asking for the single point where those triangular faces all come together at the top.

This concept is part of your studies 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 describes a location on a graph, not the named top point of a pyramid.

A Centroid means the geometric center (often like a balance point). For a pyramid, that point lies inside the solid, not at the tip where the triangular faces meet.

A Vertex is any corner where edges meet, and a pyramid has multiple vertices around its base. The question is asking for the one specific top point. That top point is a vertex, but the more exact name for that particular point on a pyramid (or cone) is the Apex.

By definition, the Apex is the vertex opposite the base. It is the unique point where all the lateral triangular faces of the pyramid meet.

We are looking for the special name of the top point of a pyramid—the place where every triangular side face meets. We can find it by using the process of elimination with the answer choices.

The origin is a coordinate-geometry term meaning $(0,0)$ in 2D or $(0,0,0)$ in 3D. Even if you place a pyramid on axes so its tip is at the origin, that still wouldn’t make “origin” the structural name of that point on the solid.

The centroid is the geometric center or average location of points in a shape. In a 3D pyramid, it is found inside the pyramid and relates to balance, so it does not describe the point where the triangular faces join.

A vertex is any corner point where edges meet. Since a pyramid has several vertices (one at the top and others around the base), “vertex” is not specific enough for the exact meeting point of all the triangular faces.

The Apex is the specific term for the topmost vertex of a pyramid or cone. It is the single point opposite the base where all lateral triangular faces intersect.

This fits the requirements of the question perfectly.

We need to decide what must be included in the total when calculating the surface area of a complex (composite) 3D solid. In MYP Geometry, a composite solid is built by combining two or more familiar 3D shapes.

By definition, the surface area of a solid is the combined area of every outer face/surface you can see on the outside.

Mathematically, this relationship is expressed as: $\text{Total Surface Area}=\sum (\text{Area of each individual surface})$

So, to get the correct total, you must include every exterior surface in the sum.

Option A does not fit. Vertices (corner points) are where edges meet. They are 0-dimensional locations, so they do not contribute any area that could be added to find surface area.

Option B is not correct because volumes describe how much 3D space a solid contains. Surface area is about the 2D covering on the outside. Since they measure different things (and use different units like $c{m}^3$ versus $c{m}^2$), volumes cannot be summed to get surface area.

Option D is incorrect because slant heights are just lengths (1D measurements) used in some formulas for cones or pyramids. They can help you calculate a face’s area, but they are not the faces/surfaces being added.

Surface area literally means adding up the areas of all the outside surfaces (faces). Listing each surface first makes sure none are missed when forming the final total.

The blank refers to what we add together to get the surface area of a composite solid. When we build a total by adding parts, each part must have the same dimension as the final result.

Surface area is a two-dimensional (2D) measure, usually in units like $c{m}^2$ or $m^2$.

To end with a 2D total, the things we add must also be 2D. Consider the dimensional type of each choice:

• 0D (Points): Vertices
• 1D (Length): Edges, heights, slant heights
• 2D (Area): Surfaces, faces
• 3D (Space): Volumes

We can immediately discard anything that isn’t 2D:

• Vertices are 0D points, and adding points cannot produce an area.
• Volumes are 3D measures. Adding a volume ($L^3$) cannot give an area ($L^2$) because they represent different physical quantities.

Slant heights are 1D lengths ($1D$). Even though they may appear in formulas used to find the area of a face, they are not the pieces you add at the end to get surface area. Adding lengths would relate more to distance or perimeter, not area.

Surfaces (faces) are 2D parts of a solid. The surface area of a composite solid is found by adding the areas of all its individual surfaces.

$\text{Total Surface Area}={\text{Area}}_1+{\text{Area}}_2+...+{\text{Area}}_n$