Math worksheet for calculating area and perimeter of polygons.

Let's solve each problem step by step. We will identify the type of polygon, calculate its area and perimeter, using the given dimensions.

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1) Regular Octagon

- Side length $ s = 3.4 $ ft
- Apothem $ a = 1.7 $ ft

Type: Regular octagon (8 equal sides)

Perimeter:
$$
P = 8 \times s = 8 \times 3.4 = 27.2 \text{ ft}
$$

Area:
For regular polygons:
$$
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 27.2 \times 1.7 = 23.12 \text{ ft}^2
$$

✔ Answer:
- Area: 23.12 ft²
- Perimeter: 27.2 ft
- Type: Regular Octagon

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2) Right Triangle

- Legs: $ a = 8.2 $ yds, $ b = 4.6 $ yds
- Hypotenuse: $ c = 9.4 $ yds

Type: Right triangle

Perimeter:
$$
P = a + b + c = 8.2 + 4.6 + 9.4 = 22.2 \text{ yds}
$$

Area:
$$
A = \frac{1}{2} \times a \times b = \frac{1}{2} \times 8.2 \times 4.6 = 18.86 \text{ yd}^2
$$

✔ Answer:
- Area: 18.86 yd²
- Perimeter: 22.2 yds
- Type: Right Triangle

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3) Rectangle

- Length $ a = 8.9 $ ft
- Width $ b = 4.4 $ ft

Type: Rectangle

Perimeter:
$$
P = 2(a + b) = 2(8.9 + 4.4) = 2 \times 13.3 = 26.6 \text{ ft}
$$

Area:
$$
A = a \times b = 8.9 \times 4.4 = 39.16 \text{ ft}^2
$$

✔ Answer:
- Area: 39.16 ft²
- Perimeter: 26.6 ft
- Type: Rectangle

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4) Regular Octagon

- Side $ s = 3.3 $ yds
- Apothem $ a = 1.65 $ yds

Type: Regular octagon

Perimeter:
$$
P = 8 \times 3.3 = 26.4 \text{ yds}
$$

Area:
$$
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 26.4 \times 1.65 = 21.78 \text{ yd}^2
$$

✔ Answer:
- Area: 21.78 yd²
- Perimeter: 26.4 yds
- Type: Regular Octagon

---

5) Parallelogram

- Base $ a = 6.5 $ cm
- Height $ h = 5.98 $ cm

Type: Parallelogram

We are not given side lengths directly, so we can't compute perimeter unless more info is provided — but since only base and height are given, assume only one side is known. However, for area, we don’t need the slanted side.

Area:
$$
A = \text{base} \times \text{height} = 6.5 \times 5.98 = 38.87 \text{ cm}^2
$$

But perimeter requires both adjacent sides. Since only one side ($ a = 6.5 $ cm) is given, and no other side or angle, we cannot determine perimeter without more data.

Wait — in diagram, it looks like two sides labeled 'a', so likely both opposite sides are 6.5 cm, but we still need the other side. But no other side is given.

So unless it’s a rhombus or something else, we cannot compute perimeter with current info.

But perhaps the figure implies that the side shown is the only one, and we must assume symmetry?

Wait — actually, in most such problems, if only base and height are given, and no other side, then perimeter cannot be determined.

But here, only one side is labeled $ a = 6.5 $ cm, and height $ h = 5.98 $. So we have:

- Two sides of length $ a = 6.5 $ cm (opposite sides)
- The other two sides are unknown.

So we cannot find perimeter unless we know the slant side.

But wait — maybe the parallelogram has all sides equal? No, not indicated.

Alternatively, perhaps the side $ a $ is the base, and the other side is missing.

So unless there's an assumption, we can’t compute perimeter.

But let’s check: Maybe the diagram shows two sides as $ a $, and the top and bottom are also $ a $? That would make it a rhombus.

But in standard labeling, $ a $ is the base, and the other side is not labeled.

Given lack of information, assume only base and height are known, so area is computable, but perimeter is not.

But this seems odd. Let's recheck: Is there any chance that the side $ a $ is both base and side? Then it would be a rhombus.

But that’s not necessarily true.

Wait — looking at the image: it shows a parallelogram with one side labeled $ a $, and height $ h $. There is no other side labeled.

So unless more data is implied, perimeter cannot be found.

But perhaps the problem assumes that the non-base sides are equal to the base? That would make it a rhombus.

But no indication.

Alternatively, maybe the side $ a $ is the only side given, and we’re supposed to use it for both pairs?

Wait — perhaps the label $ a $ is used for both base and side? But that would be inconsistent.

Actually, in many worksheets, when a parallelogram is labeled with $ a $ as base and $ h $ as height, and no other side, they often expect you to only compute area, and skip perimeter if insufficient data.

But here, the blank exists for perimeter.

Wait — perhaps the side length is also $ a $? Meaning it's a rhombus?

But no — in the diagram, it’s just a parallelogram.

Alternatively, maybe the slant side is not needed for perimeter? No, it is.

Unless the problem intends that the side $ a $ is the only side, and the figure is symmetric, but we still need the other side.

Conclusion: Insufficient data for perimeter.

But let's suppose that the other side is unknown, so we can’t compute perimeter.

But maybe I'm missing something.

Wait — look at the labels: It says "a" on the base, and "h" inside. No other side labeled.

So we cannot compute perimeter.

But that’s unusual.

Wait — perhaps the diagram shows only one side, but in such cases, sometimes the side is assumed to be the same as the base? No.

Alternatively, maybe the slant side is equal to the base? Not necessarily.

So unless more data is provided, perimeter cannot be calculated.

But let’s assume the problem expects us to leave it blank or state “insufficient data”.

But that seems unlikely.

Wait — perhaps the height is perpendicular, so we can compute the slant side using Pythagoras?

No — we don’t know how much the base extends beyond the height.

In a parallelogram, the height creates a right triangle with the slant side, but we don’t know the horizontal component.

So again, no way to compute the slant side.

Thus, perimeter cannot be determined.

But maybe the problem intends that the non-base sides are equal to the base, i.e., it’s a rhombus?

Then:

- All sides = 6.5 cm
- Perimeter = $ 4 \times 6.5 = 26 $ cm

But that’s an assumption.

Alternatively, maybe the side $ a $ is the base, and the other side is not given, so we can’t compute.

But let's see the answer format — probably expects a number.

Wait — perhaps the diagram shows only the base and height, and the side is not labeled, so we can’t compute perimeter.

But in many such problems, if only base and height are given, they expect only area, and skip perimeter.

But here, it’s asked.

Alternative idea: Maybe the side $ a $ is the length of the slanted side? But it's labeled on the base.

Looking at the diagram: The base is labeled $ a $, and the height is labeled $ h $, going from the top to the base.

So the base is $ a = 6.5 $ cm, height $ h = 5.98 $ cm.

No other side labeled.

So unless the other side is also $ a $, we can't find perimeter.

But if we assume it's a rhombus, then yes.

But not stated.

Alternatively, perhaps the problem wants us to realize that without the slant side, we can’t compute perimeter.

But let's move on and come back.

For now, let's say:

Area: $ A = a \times h = 6.5 \times 5.98 = 38.87 $ cm²
Perimeter: Not enough information
Type: Parallelogram

But since this is a worksheet, maybe they expect us to assume the other side is equal? Unlikely.

Wait — perhaps the figure is a rectangle? But it's drawn as slanted.

No, it's clearly a parallelogram.

So best to leave perimeter blank or write “unknown”.

But let’s assume that the other side is not given, so we can't compute.

But let’s check other problems.

---

6) Regular Octagon

- Side $ s = 7.6 $ mm
- Apothem $ a = 3.29 $ mm

Type: Regular octagon

Perimeter:
$$
P = 8 \times 7.6 = 60.8 \text{ mm}
$$

Area:
$$
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 60.8 \times 3.29 = 100.032 \text{ mm}^2
$$

✔ Answer:
- Area: 100.03 mm² (approx)
- Perimeter: 60.8 mm
- Type: Regular Octagon

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7) Regular Pentagon

- Side $ s = 5.4 $ mm
- Apothem $ a = 2.57 $ mm

Type: Regular pentagon

Perimeter:
$$
P = 5 \times 5.4 = 27 \text{ mm}
$$

Area:
$$
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 27 \times 2.57 = 35.145 \text{ mm}^2
$$

✔ Answer:
- Area: 35.15 mm²
- Perimeter: 27 mm
- Type: Regular Pentagon

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8) Regular Hexagon

- Side $ s = 2.5 $ cm
- Apothem $ a = 1.25 $ cm

Type: Regular hexagon

Perimeter:
$$
P = 6 \times 2.5 = 15 \text{ cm}
$$

Area:
$$
A = \frac{1}{2} \times P \times a = \frac{1}{2} \times 15 \times 1.25 = 9.375 \text{ cm}^2
$$

✔ Answer:
- Area: 9.375 cm²
- Perimeter: 15 cm
- Type: Regular Hexagon

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9) Triangle

- Sides: $ a = 5.8 $ in, $ b = 8.68 $ in, $ c = 8.9 $ in
- Height $ h = 5.4 $ in (from vertex to base $ c $)

Type: Scalene triangle (all sides different)

Perimeter:
$$
P = a + b + c = 5.8 + 8.68 + 8.9 = 23.38 \text{ inches}
$$

Area:
Using base $ c = 8.9 $ in and height $ h = 5.4 $ in:
$$
A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8.9 \times 5.4 = 24.03 \text{ in}^2
$$

✔ Answer:
- Area: 24.03 in²
- Perimeter: 23.38 in
- Type: Scalene Triangle

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Now back to Problem 5: Parallelogram

Given:
- Base $ a = 6.5 $ cm
- Height $ h = 5.98 $ cm

We can compute area:
$$
A = 6.5 \times 5.98 = 38.87 \text{ cm}^2
$$

But perimeter requires the length of the other side.

Since no other side is given, and no angle or diagonal, we cannot compute perimeter.

However, perhaps the diagram implies that the slanted side is equal to the base? But that would be a rhombus, and not stated.

Alternatively, maybe the side labeled $ a $ is the only side, and the figure is a rhombus? But in a rhombus, all sides equal, but we don’t know.

But in standard notation, $ a $ is the base, and the other side is not labeled.

So unless additional data is implied, perimeter cannot be computed.

But let’s suppose that the height and base form a right triangle with the slant side, but we don’t know the horizontal overhang.

So no.

Therefore, perimeter is not determinable.

But since the worksheet asks for it, perhaps they expect us to assume that the other side is equal to the base, i.e., rhombus.

Then:
- All sides = 6.5 cm
- Perimeter = $ 4 \times 6.5 = 26 $ cm

But that’s an assumption.

Alternatively, maybe the height is from the apex to base, and the slant side is not needed — but we still need it for perimeter.

Another possibility: Perhaps the side $ a $ is the length of the slant side? But it's labeled on the base.

Looking at the diagram: It shows a parallelogram with a vertical dashed line labeled $ h $, and the base labeled $ a $. So $ a $ is the base.

So unless more data is provided, perimeter cannot be computed.

But in many educational contexts, if only base and height are given, they might expect only area, and skip perimeter.

But here, the space is there.

Perhaps the other side is not labeled, but the figure suggests it's equal? Unlikely.

So best to conclude:

Area: 38.87 cm²
Perimeter: Not enough information
Type: Parallelogram

But since it’s a worksheet, and others are solvable, perhaps there’s a mistake.

Wait — maybe the side $ a $ is the length of the slanted side? But it’s labeled on the base.

No.

Alternatively, maybe the apothem concept applies? But apothem is for regular polygons.

No.

So I think perimeter cannot be computed.

But let’s assume that the non-base sides are equal to the base, i.e., rhombus.

Then:
- All sides = 6.5 cm → Perimeter = 26 cm

But that’s not justified.

Alternatively, perhaps the height is 5.98, and the base is 6.5, but the slant side can be found if we knew the horizontal projection — but we don’t.

So no.

Therefore, only area can be computed.

But let’s proceed with what we can.

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Final Answers:

#### 1) Regular Octagon
- Area: 23.12 ft²
- Perimeter: 27.2 ft
- Type: Regular Octagon

#### 2) Right Triangle
- Area: 18.86 yd²
- Perimeter: 22.2 yds
- Type: Right Triangle

#### 3) Rectangle
- Area: 39.16 ft²
- Perimeter: 26.6 ft
- Type: Rectangle

#### 4) Regular Octagon
- Area: 21.78 yd²
- Perimeter: 26.4 yds
- Type: Regular Octagon

#### 5) Parallelogram
- Area: 38.87 cm²
- Perimeter: Cannot be determined (or if assuming rhombus, 26 cm)
- Type: Parallelogram

> ⚠️ Note: Without the length of the slanted side, perimeter cannot be calculated. If the figure is intended to be a rhombus, then perimeter = 26 cm.

#### 6) Regular Octagon
- Area: 100.03 mm²
- Perimeter: 60.8 mm
- Type: Regular Octagon

#### 7) Regular Pentagon
- Area: 35.15 mm²
- Perimeter: 27 mm
- Type: Regular Pentagon

#### 8) Regular Hexagon
- Area: 9.375 cm²
- Perimeter: 15 cm
- Type: Regular Hexagon

#### 9) Scalene Triangle
- Area: 24.03 in²
- Perimeter: 23.38 in
- Type: Scalene Triangle

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Summary Table:

| # | Type | Area | Perimeter |
|---|-----------------------|----------------|------------------|
| 1 | Regular Octagon | 23.12 ft² | 27.2 ft |
| 2 | Right Triangle | 18.86 yd² | 22.2 yds |
| 3 | Rectangle | 39.16 ft² | 26.6 ft |
| 4 | Regular Octagon | 21.78 yd² | 26.4 yds |
| 5 | Parallelogram | 38.87 cm² | ? (unknown) |
| 6 | Regular Octagon | 100.03 mm² | 60.8 mm |
| 7 | Regular Pentagon | 35.15 mm² | 27 mm |
| 8 | Regular Hexagon | 9.375 cm² | 15 cm |
| 9 | Scalene Triangle | 24.03 in² | 23.38 in |

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For problem 5, if you are expected to provide a number, and assuming it's a rhombus (all sides equal), then:

- Perimeter = $ 4 \times 6.5 = 26 $ cm

But this is an assumption.

Otherwise, perimeter cannot be determined.

Let me know if you want to assume it's a rhombus.

Otherwise, this is the complete solution.