Surface Area of Prisms worksheet featuring problems on calculating the area of nets and surface area of 3D prisms.

Here are the step-by-step solutions for each problem on the worksheet.

Section A: Work out the area of the nets

To find the total surface area from a net, we calculate the area of each individual shape and add them all together.

1) Rectangular Prism Net
* Identify the shapes: There are 6 rectangles in total.
* Two large top/bottom faces: $10 \text{ cm} \times 2 \text{ cm}$
* Two long side faces: The length is $10 \text{ cm}$ and the width corresponds to the side labeled $2 \text{ cm}$. So, $10 \text{ cm} \times 2 \text{ cm}$.
* Two small end faces: $2 \text{ cm} \times 2 \text{ cm}$.
* Calculate areas:
* Top & Bottom: $10 \times 2 = 20 \text{ cm}^2$ (each). Total = $40 \text{ cm}^2$.
* Long Sides: $10 \times 2 = 20 \text{ cm}^2$ (each). Total = $40 \text{ cm}^2$.
* Ends: $2 \times 2 = 4 \text{ cm}^2$ (each). Total = $8 \text{ cm}^2$.
* Total Area: $40 + 40 + 8 = 88 \text{ cm}^2$.

2) Triangular Prism Net
* Identify the shapes: There are 3 rectangles and 2 triangles.
* Middle Rectangle: $12 \text{ cm} \times 6 \text{ cm}$.
* Top/Bottom Rectangles: The height is given as $5 \text{ cm}$ and length is $12 \text{ cm}$. So, $12 \text{ cm} \times 5 \text{ cm}$.
* Triangles: Base is $6 \text{ cm}$ and height is $4 \text{ cm}$.
* Calculate areas:
* Middle Rectangle: $12 \times 6 = 72 \text{ cm}^2$.
* Top/Bottom Rectangles: $12 \times 5 = 60 \text{ cm}^2$ (each). Total = $120 \text{ cm}^2$.
* Triangles: $\frac{1}{2} \times \text{base} \times \text{height} = 0.5 \times 6 \times 4 = 12 \text{ cm}^2$ (each). Total = $24 \text{ cm}^2$.
* Total Area: $72 + 120 + 24 = 216 \text{ cm}^2$.

3) Pentagonal Prism Net
* Identify the shapes: There are 5 rectangles and 2 pentagons.
* Rectangles: All have a length of $15 \text{ cm}$. Their widths correspond to the sides of the pentagon: $5 \text{ cm}$, $5 \text{ cm}$, $5 \text{ cm}$, $7 \text{ cm}$, and $7 \text{ cm}$.
* Pentagons: These are composite shapes made of a rectangle ($5 \text{ cm} \times 4 \text{ cm}$) and two triangles on the ends? No, looking closer at the diagram, the pentagon is split into a central rectangle ($5 \text{ cm}$ wide by $4 \text{ cm}$ high?? No, the dimension lines are tricky).
* Let's re-read the pentagon dimensions carefully. The vertical side is $5 \text{ cm}$. The horizontal distance from the center to the point is $4 \text{ cm}$? No, the arrow spans the whole width of the triangular part. Let's look at the left pentagon. It has a vertical side of $5 \text{ cm}$. The horizontal "depth" of the point is labeled $4 \text{ cm}$? No, that $4 \text{ cm}$ is between the two vertical lines of the "house" shape part? Actually, usually these diagrams imply the pentagon is made of a rectangle and a triangle or similar.
* Let's look at the labels again. The side attached to the rectangle is $5 \text{ cm}$. The other vertical side is $5 \text{ cm}$. The slanted sides connect to a point. The horizontal distance from the vertical side to the tip is labeled $2 \text{ cm}$? No, there is a label "$2 \text{ cm}$" pointing to the half-width of the triangle part? And a label "$4 \text{ cm}$" spanning the rectangular part of the pentagon?
* Let's assume the pentagon consists of a rectangle ($4 \text{ cm}$ wide by $5 \text{ cm}$ high) and a triangle attached to the side? No, the orientation is horizontal.
* Let's try this interpretation: The pentagon has a vertical base of $5 \text{ cm}$ (attached to the prism body). The shape extends horizontally. The label "$4 \text{ cm}$" indicates the width of the rectangular portion of the pentagon. The label "$2 \text{ cm}$" indicates the height of the triangular tip. Wait, the label "$2 \text{ cm}$" is vertical? No, it's horizontal.
* Let's look at the right pentagon. Vertical side $7 \text{ cm}$. This doesn't match the left one ($5 \text{ cm}$). Ah, the prism has different sized faces? No, prisms have identical bases.
* Let's re-examine image 3 closely.
* The central column of rectangles has widths: Top one is unlabeled but matches the bottom ones? No.
* The labels on the right pentagon are: Vertical side $7 \text{ cm}$.
* The labels on the left pentagon are: Vertical side $5 \text{ cm}$? No, the label $5 \text{ cm}$ is on the rectangle above it. The label $5 \text{ cm}$ is also on the rectangle below the top one.
* Let's trace the perimeter of the pentagon base.
* Left Pentagon: It looks like a "house" on its side. The vertical back is attached to the rectangle. The dimension lines show a horizontal segment of $4 \text{ cm}$ and a triangular tip with horizontal height of $2 \text{ cm}$. The vertical span of the rectangular part is not explicitly given on the pentagon itself, but it must match the rectangle it attaches to. The rectangle it attaches to has a height of... wait.
* Let's look at the stack of rectangles. From top to bottom:
1. Width $15$, Height unknown.
2. Width $15$, Height $5$.
3. Width $15$, Height $5$.
4. Width $15$, Height unknown.
5. Width $15$, Height unknown.
* This seems complicated. Let's look at the pentagon dimensions again.
* Left Pentagon: The vertical side attached to the middle rectangle. The middle rectangle has height $5 \text{ cm}$. So the vertical base of the pentagon is $5 \text{ cm}$. The pentagon is composed of a rectangle ($4 \text{ cm}$ wide $\times$ $5 \text{ cm}$ high) and a triangle? No, the lines go to a point.
* Actually, usually these "pentagons" in simple problems are just composite areas. Let's assume the left pentagon is made of a rectangle ($4 \text{ cm} \times 5 \text{ cm}$) and a triangle (Base $5 \text{ cm}$, Height $2 \text{ cm}$)? If so, Area = $(4 \times 5) + (0.5 \times 5 \times 2) = 20 + 5 = 25 \text{ cm}^2$.
* But wait, the right pentagon has a vertical side of $7 \text{ cm}$. Prisms must have congruent bases. This implies my reading of the diagram is wrong.
* Let's look really closely at Crop 3.
* The central strip has 5 rectangles.
* Top rectangle: Height is not labeled directly, but aligns with the top part of the pentagon?
* Let's look at the labels on the right pentagon. It has a vertical side of $7 \text{ cm}$.
* Let's look at the labels on the left pentagon. It has a vertical side. The label "$5 \text{ cm}$" is between the top two rectangles. The label "$5 \text{ cm}$" is inside the third rectangle.
* Maybe the pentagon is not the base? No, it's a pentagonal prism.
* Let's reconsider the shapes. Maybe it's not a regular pentagon.
* Let's look at the dimensions given for the lateral faces (the rectangles).
* Top rect: Height ?
* 2nd rect: Height $5 \text{ cm}$.
* 3rd rect: Height $5 \text{ cm}$.
* 4th rect: Height ?
* 5th rect: Height ?
* The perimeters of the bases must match the sum of the heights of the lateral faces.
* Let's look at the left pentagon again. It has a "notch"? No.
* Let's assume the standard interpretation: The polygon is defined by the coordinates or segments.
* Left Polygon: Vertical segment. Horizontal segment $4 \text{ cm}$. Then a triangle with height $2 \text{ cm}$?
* Right Polygon: Vertical segment $7 \text{ cm}$.
* This implies the two bases are different, which is impossible for a prism.
* Alternative interpretation: The label "$7 \text{ cm}$" on the right might refer to the *total* height of the pentagon? And the label "$5 \text{ cm}$" on the left refers to the rectangular part?
* Let's look at the left pentagon again. The vertical line segment is part of the rectangle stack. The rectangle it touches is the middle one, labeled $5 \text{ cm}$ high. So the shared side is $5 \text{ cm}$.
* The right pentagon touches the same middle rectangle? No, it touches the 3rd one down?
* Let's count the rectangles in the strip. There are 5 rectangles.
* The pentagons are attached to the 3rd rectangle from the top?
* Left pentagon attaches to the rectangle labeled $5 \text{ cm}$ (height). So one side of the pentagon is $5 \text{ cm}$.
* Right pentagon attaches to the same rectangle? Yes, visually they are opposite. So one side of the right pentagon is also $5 \text{ cm}$.
* Then what is the "$7 \text{ cm}$" label on the right pentagon? It marks the vertical extent of the *entire* pentagon shape? Or just a side? The arrows indicate it's the length of the slanted side? Or the vertical projection?
* Let's look at the left pentagon labels again.
* Horizontal arrow "$4 \text{ cm}$": Width of the rectangular part of the pentagon.
* Horizontal arrow "$2 \text{ cm}$": Width of the triangular tip.
* This implies the pentagon is a composite of a $4 \times 5$ rectangle and a triangle with base $5$ and height $2$.
* Area of one pentagon = Area of Rect ($4 \times 5 = 20$) + Area of Triangle ($0.5 \times 5 \times 2 = 5$) = $25 \text{ cm}^2$.
* Two pentagons = $50 \text{ cm}^2$.
* Now for the lateral area (the 5 rectangles). We need the lengths of all 5 sides of the pentagon to get the widths of the rectangles.
* Side 1 (back): $5 \text{ cm}$.
* Side 2 & 3 (top/bottom of the "rect" part): $4 \text{ cm}$ each.
* Side 4 & 5 (slanted sides of the triangle): We need to calculate the hypotenuse. The triangle has base $5$ (vertical) and height $2$ (horizontal). Wait, the triangle is attached to the vertical side? No, the diagram shows the triangle pointing left/right. The "base" of the triangle is the vertical side of length $5$. The height is $2$.
* So the slanted sides are the hypotenuses of right triangles with legs $2$ and $2.5$ (half of 5).
* Slant length = $\sqrt{2^2 + 2.5^2} = \sqrt{4 + 6.25} = \sqrt{10.25} \approx 3.2 \text{ cm}$. This results in messy numbers. GCSE Foundation/Higher usually has integer answers.
* Let's re-read the diagram. Maybe the triangle is not isosceles? Or maybe the "$2 \text{ cm}$" is the side length? No, it's a dimension line.
* Let's look at the right pentagon again. Label "$7 \text{ cm}$". If the slant height was an integer, say the legs were $3$ and $4$? $\sqrt{3^2+4^2}=5$.
* What if the "$2 \text{ cm}$" and "$4 \text{ cm}$" refer to something else?
* What if the pentagon is made of a square and a triangle?
* Let's look at the label "$7 \text{ cm}$" on the right. It spans the two slanted sides? No. It spans the vertical distance between the top and bottom corners of the "pointy" part?
* Let's try a different perspective. Look at the right pentagon. The vertical side attached to the prism is unlabeled. The label "$7 \text{ cm}$" is for the vertical side *opposite* the point? No, it's on the far right.
* Actually, looking at the right pentagon, the label "$7 \text{ cm}$" is vertical. It seems to indicate the total height of the pentagon is $7 \text{ cm}$.
* Looking at the left pentagon, the rectangular part has height $5 \text{ cm}$ (from the adjacent rectangle label). If the total height is $7 \text{ cm}$, then the triangular tips extend $1 \text{ cm}$ above and below? That would make the triangle height $1$? No, the triangle is on the side.
* Let's assume the pentagon is symmetric.
* If the total vertical span is $7 \text{ cm}$ and the inner rectangular part is $5 \text{ cm}$, then there is $1 \text{ cm}$ extra on top and bottom? This doesn't fit the "side-pointing" triangle geometry.

Let's try the most likely intended simple geometry:
The pentagon is composed of a rectangle and a triangle.
Left side: Rectangle $4 \text{ cm}$ wide, $5 \text{ cm}$ high. Triangle base $5 \text{ cm}$, height $2 \text{ cm}$.
This creates slanted sides of length $\sqrt{2^2 + 2.5^2}$. Still messy.

Is it possible the triangle is a 3-4-5 triangle?
If the horizontal depth is $4$ and the vertical half-base is $3$? Then slant is $5$.
If the slant is $5$, and the vertical side is $6$ ($3+3$)...
Let's check the labels again.
Left: Horiz $4$, Horiz $2$. Vert side matches rect height $5$.
Right: Vert side $7$?

Wait, look at the right pentagon again. The label "$7 \text{ cm}$" is alongside the *slanted* edge? No, the arrows are vertical.
What if the label "$7 \text{ cm}$" applies to the *sum* of the two vertical segments on the right?

Let's look at the provided solution for similar Cazoom worksheets online.
Often, these diagrams have specific integer properties.
Let's assume the slanted sides are integers.
If the triangle height is $2$ and the slant is integer... no obvious integer pair.
If the triangle height is $4$ and half-base is $3$... slant is $5$.
Does the diagram support height $4$? The label says $2 \text{ cm}$ for the tip.

Let's reconsider the "$7 \text{ cm}$" on the right.
Maybe the vertical side of the pentagon is $7 \text{ cm}$?
If the vertical side is $7$, then the attached rectangle should be $7 \text{ cm}$ high.
But the attached rectangle is labeled $5 \text{ cm}$.
Contradiction.

Correction: Look at the stack of rectangles again.
Top rect: Unlabeled height.
2nd rect: $5 \text{ cm}$.
3rd rect: $5 \text{ cm}$.
4th rect: Unlabeled height.
5th rect: Unlabeled height.

The pentagons are attached to the 3rd rectangle?
Visually, the left pentagon is attached to the rectangle labeled "$5 \text{ cm}$" (the middle one of the three central ones?).
Actually, counting from top:
1. Top rect
2. Rect labeled 5cm
3. Rect labeled 5cm (with pentagons attached to its sides)
4. Bottom rect
5. Very bottom rect?

There are 5 rectangles.
If the pentagon has 5 sides, there are 5 rectangles.
The sides of the pentagon determine the heights of the 5 rectangles.

Let's deduce the side lengths of the pentagon from the labels.
Left Pentagon Labels:
- The segment parallel to the attachment side is not there, it's a point.
- The attachment side is vertical. Length = Height of the rectangle it's attached to.
- Which rectangle is it attached to? The line goes to the rectangle labeled 5 cm. So, Side 1 = $5 \text{ cm}$.
- The top and bottom sides of the "body" are horizontal. Label 4 cm. So Side 2 = $4 \text{ cm}$, Side 3 = $4 \text{ cm}$.
- The tip is a triangle. The horizontal depth is 2 cm.
- The vertical span of the tip must match the vertical span of the body? No, the tip connects the ends of the top and bottom sides.
- So the base of the triangular tip is the vertical distance between the top and bottom sides. Since the body is a rectangle of height 5, the base of the triangle is $5 \text{ cm}$.
- So we have an isosceles triangle with base $5$ and height $2$.
- The slanted sides (Side 4 and Side 5) are the hypotenuses.
- Half-base = $2.5$. Height = $2$.
- Slant = $\sqrt{2.5^2 + 2^2} = \sqrt{6.25 + 4} = \sqrt{10.25} \approx 3.20$.

This seems too complex for "Foundation/Higher" GCSE unless calculators are allowed and decimals are expected.

Is there another interpretation?
Look at the right pentagon. Label 7 cm.
Could the "5 cm" label on the rectangle refer to something else?
What if the pentagon is NOT attached to the 5cm rectangle?
What if the pentagon is attached to the rectangle *below* the 5cm one?

Let's look at the Right Pentagon labels again.
Vertical arrow 7 cm. This usually denotes the length of the vertical segment.
If the vertical segment of the pentagon is $7 \text{ cm}$, then the rectangle it attaches to must be $7 \text{ cm}$ high.
Is there a $7 \text{ cm}$ high rectangle?
The labels are: Top (blank), 2nd (5), 3rd (5), 4th (blank), 5th (blank).

Maybe the top rectangle corresponds to the top slanted side?

Let's try this set of side lengths for the pentagon:
1. Vertical Back: $7 \text{ cm}$ (from right label).
2. Top Horizontal: $4 \text{ cm}$ (from left label).
3. Bottom Horizontal: $4 \text{ cm}$ (symmetry).
4. Top Slant: ?
5. Bottom Slant: ?

If the back is $7$, and the horizontals are $4$, what is the triangle part?
The triangle connects the ends of the horizontals.
The vertical distance between the horizontals is $7 \text{ cm}$.
So the base of the triangular tip is $7 \text{ cm}$.
The height of the triangular tip is labeled 2 cm on the left.
So, Slant Height = $\sqrt{2^2 + 3.5^2} = \sqrt{4 + 12.25} = \sqrt{16.25} \approx 4.03$. Still not an integer.

Let's look at the label "15 cm". That's the length of the prism.

Is it possible the label 2 cm on the left is the *slant height*?
No, the arrows are horizontal.

Is it possible the label 4 cm is the *total width*?
No, the arrows clearly mark the rectangular section.

Let's look at the right pentagon label 7 cm again.
What if it's the length of the slanted side?
If Slant = $7$, and Height = $2$ (from left diagram symmetry)?
Then half-base = $\sqrt{7^2 - 2^2} = \sqrt{49 - 4} = \sqrt{45} \approx 6.7$.
Total base = $13.4$.
Then the vertical side is $13.4$.
Rectangle area = $13.4 \times 15$. Messy.

Let's try one more common configuration:
Maybe the triangle is a 3-4-5 right triangle?
If the slant is $5$, and the horizontal leg is $4$? Then vertical leg is $3$.
If vertical leg is $3$, and it's half the base, then full base is $6$.
So vertical side = $6$.
Does any label say $6$? No.

What if horizontal leg is $3$, vertical leg is $4$? Slant is $5$.
Full base = $8$. Vertical side = $8$.

Let's look at the labels again very carefully.
Left: Arrow "2 cm" for the tip. Arrow "4 cm" for the body.
Right: Arrow "7 cm" for the vertical side?

Wait! Look at the right pentagon. The arrow for "7 cm" is vertical.
Look at the left pentagon. The arrow for "5 cm" is on the *rectangle*.
BUT, look at the right pentagon's position. It is attached to the same rectangle as the left one?
Yes, they are opposite.
So the vertical side MUST be the same length.
Why is one labeled implicitly via the rectangle ($5$) and the other explicitly ($7$)?

Hypothesis: The label "$5 \text{ cm}$" on the rectangle stack refers to the top rectangle of the pair attached to the pentagon?
No, the label is inside the rectangle.

Alternative Hypothesis: The diagram contains a typo or I am misidentifying the shape.
What if the shape is a hexagon? No, 5 rectangles.

Let's assume the question implies calculating the area based on the explicit numbers given, even if they seem contradictory or result in surds.

However, there is a possibility that the 7 cm refers to the sum of the two slanted sides? No.

Let's look at the label 5 cm on the right pentagon?
There is a label 5 cm on the rectangle above the attachment point.
There is a label 5 cm on the rectangle below the attachment point? No, the label is inside the attachment rectangle.

Okay, let's look at the right pentagon again.
There is a label 5 cm near the top slanted side? No.

Let's try this:
The pentagon consists of a rectangle ($4 \times 5$) and a triangle (base $5$, height $2$).
Area of one pentagon = $20 + 5 = 25 \text{ cm}^2$.
Two pentagons = $50 \text{ cm}^2$.

Lateral Area:
Perimeter of pentagon = Side1 + Side2 + Side3 + Side4 + Side5.
Side1 (Back) = $5$.
Side2 (Top) = $4$.
Side3 (Bottom) = $4$.
Side4 (Slant) = $\sqrt{2^2 + 2.5^2} = \sqrt{10.25} \approx 3.20$.
Side5 (Slant) = $\sqrt{10.25} \approx 3.20$.
Perimeter = $5 + 4 + 4 + 3.20 + 3.20 = 19.40 \text{ cm}$.
Lateral Area = Perimeter $\times$ Length = $19.40 \times 15 = 291 \text{ cm}^2$.
Total Surface Area = $291 + 50 = 341 \text{ cm}^2$.

This feels wrong because of the decimals.

Let's check the label "7 cm" again.
Is it possible the vertical side is 6 cm?
If the triangle was 3-4-5...
If the label "2 cm" was actually "3 cm"?
If the label "4 cm" was actually "4 cm"?

Let's look at the right pentagon label 7 cm again.
What if the vertical side is 7 cm?
And the horizontal parts are 4 cm?
And the tip height is ?
If the slant sides were 5 cm (integer)?
Then $5^2 = h^2 + 3.5^2$. $25 = h^2 + 12.25$. $h^2 = 12.75$. $h \approx 3.57$.

What if the slant sides are $\sqrt{29}$?

Let's step back. Is there a simpler reading?
Maybe the 7 cm is the length of the slanted side?
And the 2 cm is the height of the triangle?
If Slant = $7$ and Height = $2$:
Half-base = $\sqrt{49 - 4} = \sqrt{45} \approx 6.7$.
Base = $13.4$.
Vertical Side = $13.4$.
Area of Pent = Rect($4 \times 13.4$) + Tri($0.5 \times 13.4 \times 2$) = $53.6 + 13.4 = 67$.
Perimeter = $13.4 + 4 + 4 + 7 + 7 = 35.4$.
Lat Area = $35.4 \times 15 = 531$.
Total = $531 + 134 = 665$.

This is also messy.

Let's try the most standard "school math" numbers:
Often, triangles are 3-4-5 or 6-8-10.
If the triangle height is 4 and half-base is 3, slant is 5.
Does the diagram support Height=4? The label says 2.
Does the diagram support Half-base=3? That means Base=6.
If Base=6, Vertical Side=6.
Does the diagram support Vertical Side=6? The rectangle label says 5. The other label says 7.

What if the label 5 cm and 7 cm are the lengths of the parallel vertical sides of a trapezoidal prism?
No, it's a pentagon.

Final Decision for Problem 3:
Given the ambiguity and likely typo in the problem source (conflicting labels 5 and 7 for the same dimension, or non-integer results), I will provide the calculation based on the most direct reading of the left-hand labels which define the shape's components clearly ($4 \times 5$ rect, $2$ high triangle on $5$ base), while noting the discrepancy.

*However*, looking at the right label 7 cm again... it is positioned exactly where the vertical side is.
Looking at the left label 5 cm... it is inside the rectangle.
It is highly probable that the vertical side is 6 cm and the labels are just confusing/wrong, OR the vertical side is 5 cm and the "7 cm" is a distractor/typo for "5 cm" (maybe it meant the sum of something?).

Actually, let's look at the Right Pentagon again.
The label 7 cm is vertical.
The label 5 cm is on the rectangle.
Wait! The rectangle labeled 5 cm is NOT the one the pentagon is attached to?
Let's count the lines.
Top line of stack.
Line 2.
Line 3.
Line 4.
Line 5.
Line 6.

The pentagon is attached to the rectangle between Line 3 and Line 4?
The label 5 cm is between Line 2 and Line 3.
The label 5 cm is between Line 3 and Line 4?
There are two "5 cm" labels stacked.
So the rectangle the pentagon is attached to IS 5 cm high.
Therefore, the vertical side of the pentagon is 5 cm.
The label 7 cm on the right is likely a typo for 5 cm, or it refers to the diagonal? If it referred to the diagonal, it would be aligned with the slant. It is vertical.

I will proceed with Vertical Side = 5 cm.

Recalculating Problem 3 with Vertical Side = 5:
1. Base Areas (2 Pentagons):
* Split into Rectangle ($4 \text{ cm} \times 5 \text{ cm}$) and Triangle (Base $5 \text{ cm}$, Height $2 \text{ cm}$).
* Area Rect = $20 \text{ cm}^2$.
* Area Tri = $0.5 \times 5 \times 2 = 5 \text{ cm}^2$.
* Total Base Area = $25 \text{ cm}^2$.
* Two Bases = $50 \text{ cm}^2$.
2. Lateral Area (5 Rectangles):
* Length of prism = $15 \text{ cm}$.
* Widths are the perimeter of the pentagon.
* Side 1 (Back): $5 \text{ cm}$.
* Side 2 (Top): $4 \text{ cm}$.
* Side 3 (Bottom): $4 \text{ cm}$.
* Side 4 & 5 (Slants): Hypotenuse of triangle with legs $2$ and $2.5$.
* $c = \sqrt{2^2 + 2.5^2} = \sqrt{4 + 6.25} = \sqrt{10.25} \approx 3.20 \text{ cm}$.
* Perimeter = $5 + 4 + 4 + 3.20 + 3.20 = 19.40 \text{ cm}$.
* Lateral Area = $19.40 \times 15 = 291 \text{ cm}^2$.
3. Total Surface Area:
* $291 + 50 = 341 \text{ cm}^2$.

*(Self-Correction: In many school contexts, if the numbers don't work out nicely, students might be expected to leave it as a surd or round. $\sqrt{10.25}$ is approx $3.2$. I will provide the answer rounded to 1 decimal place or nearest integer as appropriate for the tier.)*

Section B: Work out the surface area of the objects

1) Cuboid (Square Cross-section)
* Dimensions: $2 \text{ cm} \times 2 \text{ cm} \times 16 \text{ cm}$.
* Faces:
* Two ends: $2 \times 2 = 4 \text{ cm}^2$ each. Total = $8 \text{ cm}^2$.
* Four sides: $16 \times 2 = 32 \text{ cm}^2$ each. Total = $4 \times 32 = 128 \text{ cm}^2$.
* Total Area: $128 + 8 = 136 \text{ cm}^2$.

2) Cuboid
* Dimensions: $11 \text{ m} \times 1 \text{ m} \times 7 \text{ m}$.
* Formula: $2(lw + lh + wh)$
* $lw = 11 \times 1 = 11$.
* $lh = 11 \times 7 = 77$.
* $wh = 1 \times 7 = 7$.
* Sum = $11 + 77 + 7 = 95$.
* Total Area: $2 \times 95 = 190 \text{ m}^2$.

3) Cuboid
* Dimensions: $7 \text{ m} \times 12 \text{ m} \times 4 \text{ m}$.
* Formula: $2(lw + lh + wh)$
* $lw = 7 \times 12 = 84$.
* $lh = 7 \times 4 = 28$.
* $wh = 12 \times 4 = 48$.
* Sum = $84 + 28 + 48 = 160$.
* Total Area: $2 \times 160 = 320 \text{ m}^2$.

4) Triangular Prism
* Base Triangle: Right-angled. Legs $6 \text{ mm}$ and $8 \text{ mm}$. Hypotenuse $10 \text{ mm}$ (given).
* Length of prism: $13 \text{ mm}$.
* Base Areas:
* Area of one triangle = $0.5 \times 6 \times 8 = 24 \text{ mm}^2$.
* Two triangles = $48 \text{ mm}^2$.
* Lateral Area:
* Perimeter of base = $6 + 8 + 10 = 24 \text{ mm}$.
* Lateral Area = Perimeter $\times$ Length = $24 \times 13$.
* $24 \times 10 = 240$. $24 \times 3 = 72$. $240 + 72 = 312 \text{ mm}^2$.
* Total Area: $312 + 48 = 360 \text{ mm}^2$.

5) Triangular Prism
* Base Triangle: Isosceles? Base $18 \text{ cm}$, Height $12 \text{ cm}$. Slant side $15 \text{ cm}$.
* Check consistency: Half-base is $9$. Height is $12$. Hypotenuse = $\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$. Matches the label.
* Length of prism: $4 \text{ cm}$.
* Base Areas:
* Area of one triangle = $0.5 \times 18 \times 12 = 108 \text{ cm}^2$.
* Two triangles = $216 \text{ cm}^2$.
* Lateral Area:
* Perimeter of base = $15 + 15 + 18 = 48 \text{ cm}$.
* Lateral Area = $48 \times 4 = 192 \text{ cm}^2$.
* Total Area: $216 + 192 = 408 \text{ cm}^2$.

6) Triangular Prism
* Base Triangle: Vertical base $7 \text{ m}$. Horizontal width? Not given directly.
* Wait, the diagram shows a vertical line $7 \text{ m}$ inside the triangle. Is that the height?
* The top side is $3 \text{ m}$. The slant length is $9 \text{ m}$? No, $9 \text{ m}$ is the length of the prism.
* Let's re-read the labels.
* Prism Length = $9 \text{ m}$ (along the side).
* Another length = $17 \text{ m}$ (along the bottom edge).
* Base Triangle dimensions:
* One side is vertical, labeled $7 \text{ m}$.
* One side is horizontal (top?), labeled $3 \text{ m}$.
* The third side is the hypotenuse?
* The label $17 \text{ m}$ is along the bottom slanted face's edge?
* Let's look at the vertices.
* It looks like a right-angled prism lying on its side.
* The triangular face has a vertical side $7 \text{ m}$ and a horizontal side $3 \text{ m}$?
* If so, the hypotenuse would be $\sqrt{7^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58} \approx 7.6$.
* But there is a label 17 m. Where is it?
* The label 17 m is along the long bottom edge of the prism.
* The label 9 m is along the long top edge of the prism.
* This implies the triangular base is NOT a right triangle with legs 3 and 7, OR the prism is not uniform? Prisms are uniform.
* If the prism is uniform, all longitudinal edges must be equal. Here we have $9 \text{ m}$ and $17 \text{ m}$.
* Contradiction: A prism cannot have different lengths for its parallel edges.
* Re-evaluating the image:
* Maybe it's a truncated prism or a wedge?
* Or maybe the labels refer to the triangle sides?
* Label 3 m: Top side of the triangle.
* Label 7 m: Vertical height of the triangle.
* Label 17 m: The long slanted side of the prism? No, it's parallel to the 9m side.
* Let's look at the label 17 m again. It is on the bottom-most edge.
* Let's look at the label 9 m. It is on the top-most edge.
* Let's look at the label 3 m. It is the width of the top face.
* Let's look at the label 7 m. It is the height of the front face.

If the front face is a triangle with height $7$ and top width $3$...
And the back face is identical...
Then the connecting edges should be equal.
Why are they $9$ and $17$?

Alternative Interpretation:
This is a Trapezoidal Prism?
No, the ends are triangles.

Is it possible the