Worksheet for calculating the areas of regular polygons and composite figures, featuring eight problems with geometric shapes and dimensions.

Let’s solve each problem one by one. We’ll calculate the area for each figure using the right formulas and given measurements.

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Problem 1: Rectangle with a triangle inside?

Wait — actually, looking at the diagram: it shows a rectangle with a diagonal line from top-left to bottom-right, and a perpendicular height of 7.5 m drawn from the center point to the base (10.9 m). But that doesn’t make sense for a rectangle unless it’s showing half the rectangle?

Actually, re-examining: The figure is a rectangle, but they’ve drawn a triangle inside it? Or maybe it’s just showing the dimensions of the rectangle? Wait — no, the 7.5 m is labeled as the height from the center to the base, and there’s a right angle mark. That suggests this might be a triangle? But the outer shape looks like a rectangle.

Hold on — let me think again.

Looking closely: It’s a rectangle, but they’re giving you the base = 10.9 m, and the “height” from the center to the base is 7.5 m? That doesn’t help directly.

Wait — perhaps it’s a parallelogram? No, it has right angles.

Actually, I think there’s a mistake in interpretation. Let me look at the labels:

- Base = 10.9 m
- Height = 7.5 m (with a right angle symbol)

And it’s drawn as a rectangle with a diagonal, but the height is measured from the midpoint? Hmm.

Alternatively — maybe it’s a triangle? But the shape is rectangular.

Wait — perhaps the figure is a rectangle, and they want the area of the whole rectangle? Then why show the 7.5 m from center?

Another idea: Maybe it’s a rhombus or parallelogram? But it has right angles.

I think I need to reinterpret.

Looking at Problem 2 and 3 — those are regular polygons with apothems given. So maybe Problem 1 is also meant to be solved using apothem?

But Problem 1 is a rectangle — which is not a regular polygon unless it’s a square.

Wait — perhaps the figure is a square? But 10.9 ≠ 7.5.

Alternatively — maybe the 7.5 m is the height of a triangle that makes up half the rectangle? If so, then area of triangle = (1/2)*base*height = (1/2)*10.9*7.5, and since there are two such triangles, total area = 10.9 * 7.5.

That makes sense! Because if you draw a diagonal in a rectangle, you get two congruent triangles, each with area (1/2)*base*height. So total area = base * height.

So for Problem 1: Area = 10.9 m × 7.5 m

Calculate:
10.9 × 7.5
= (10 + 0.9) × 7.5
= 10×7.5 + 0.9×7.5
= 75 + 6.75
= 81.75 m²

✔ So Area = 81.75 m²

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Problem 2: Regular Pentagon

Given:
- Side length = 13.2 yd
- Apothem = 3.5 yd

Formula for area of regular polygon:
Area = (1/2) × perimeter × apothem

Perimeter = number of sides × side length = 5 × 13.2 = 66 yd

Area = (1/2) × 66 × 3.5
= 33 × 3.5
= 33 × 3 + 33 × 0.5
= 99 + 16.5
= 115.5 yd²

✔ Area = 115.5 yd²

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Problem 3: Regular Hexagon

Given:
- Side length = 12 cm
- Apothem = 5√3 cm

Perimeter = 6 × 12 = 72 cm

Area = (1/2) × perimeter × apothem
= (1/2) × 72 × 5√3
= 36 × 5√3
= 180√3 cm²

We can leave it exact, or approximate? Since the problem didn’t specify, and apothem is given as 5√3, probably exact form is fine.

But let’s check: sometimes they expect decimal. However, since √3 is irrational, and the input uses radical, we’ll keep it as 180√3.

But wait — let me verify if 5√3 is correct for a hexagon with side 12.

In a regular hexagon, apothem = (side × √3)/2 = (12 × √3)/2 = 6√3 ≈ 10.392, but here it’s given as 5√3 ≈ 8.66 — which is less. So maybe it’s not standard? Or perhaps it’s a different figure?

Wait — the diagram shows a hexagon with side 12 cm and apothem 5√3 cm. So we must use the given values, even if they don’t match standard geometry. Maybe it’s not regular? But the title says "regular polygons".

Hmm — contradiction. In a regular hexagon, apothem should be (s√3)/2 = 6√3 for s=12. But here it’s given as 5√3. So either it’s a typo, or we proceed with given numbers.

Since the worksheet gives these values, we’ll use them as is.

So Area = (1/2) × 72 × 5√3 = 180√3 cm²

If we want decimal: √3 ≈ 1.732, so 180 × 1.732 = let’s compute:

180 × 1.7 = 306
180 × 0.032 = 5.76
Total ≈ 311.76 cm²

But since the problem gave apothem as 5√3, likely expects exact answer.

However, looking at other problems, they might expect numerical answers. Let me see Problem 4,5,etc — they have integers or decimals.

Perhaps for consistency, we should compute numerically.

But 5√3 is approximately 8.660, so:

Area = 0.5 × 72 × 8.660 = 36 × 8.660

36 × 8 = 288
36 × 0.66 = 23.76
Total = 311.76 cm²

But let's do exact calculation:

5√3 = 5 * 1.7320508 ≈ 8.660254

36 * 8.660254 = let's compute:

30 * 8.660254 = 259.80762
6 * 8.660254 = 51.961524
Sum = 311.769144 ≈ 311.77 cm²

But perhaps the worksheet expects us to use the formula with given numbers without questioning.

I think for accuracy, since it's a math worksheet, and apothem is given as 5√3, we can leave as 180√3, but I suspect they want numerical value.

Looking back at Problem 1, we got decimal. Problem 2 decimal. So probably decimal expected.

Moreover, in Problem 3, if it were regular hexagon, apothem should be 6√3, but it's given as 5√3, so maybe it's not regular? But the title says "regular polygons". This is confusing.

Perhaps it's a mistake, but we have to go with given data.

I'll compute numerically: 180 * √3 ≈ 180 * 1.732 = 311.76 cm²

Round to two decimals? 311.76 cm²

But let's confirm calculation:

√3 ≈ 1.73205080757

180 * 1.73205080757 = 311.7691453626 ≈ 311.77 cm²

I'll use 311.77 cm²

But to be precise, let's see if the problem intends exact value. Since other problems have decimals, I'll go with decimal.

✔ Area = 311.77 cm² (approximately)

But wait — perhaps I should use the exact value as per given. Let me check online or standard practice.

In many worksheets, if radical is given, they accept radical form. But here, since it's "find the area", and others are numerical, I'll provide both, but for final answer, I'll use the calculated decimal.

Actually, let's move on and come back.

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Problem 4: Triangle

Base = 22 in, height = 16 in

Area of triangle = (1/2) × base × height = (1/2) × 22 × 16

= 11 × 16 = 176 in²

✔ Area = 176 in²

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Problem 5: Composite Figure - Rectangle + Semicircle

Rectangle: width = 18 ft, length = 32 ft? Wait, the arrow shows 32 ft for the straight part, and the semicircle is attached to the end.

The figure is a rectangle with a semicircle on one end. The diameter of the semicircle is equal to the width of the rectangle, which is 18 ft.

So:

Area of rectangle = length × width = 32 ft × 18 ft = 576 ft²

Area of semicircle = (1/2) × π × r²

Radius r = diameter / 2 = 18 / 2 = 9 ft

So area = (1/2) × π × 81 = (81/2)π = 40.5π ft²

Using π ≈ 3.1416, 40.5 × 3.1416 ≈ ?

40 × 3.1416 = 125.664
0.5 × 3.1416 = 1.5708
Sum = 127.2348 ft²

Total area = rectangle + semicircle = 576 + 127.2348 = 703.2348 ft² ≈ 703.23 ft²

But let's be precise.

Sometimes they use π = 3.14

40.5 × 3.14 = 40.5 × 3 = 121.5, 40.5 × 0.14 = 5.67, total 127.17

Then 576 + 127.17 = 703.17 ft²

I think 703.23 is fine, but let's use more accurate.

Actually, in school, often π=3.14 is used.

So I'll use π=3.14

Semicircle area = 0.5 * 3.14 * 81 = 0.5 * 254.34 = 127.17 ft²? Wait:

πr² = 3.14 * 81 = let's compute: 3*81=243, 0.14*81=11.34, total 254.34

Half of that is 127.17 ft²

Rectangle: 32*18=576 ft²

Total: 576 + 127.17 = 703.17 ft²

✔ Area = 703.17 ft²

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Problem 6: Composite Figure - Triangle + Semicircle

The figure is an isosceles triangle on top of a semicircle.

Given:
- Height of triangle = 12 mm
- Diameter of semicircle = 9 mm (since the dashed line is 9 mm, and it's the base of the triangle and diameter of semicircle)

First, area of triangle: base = 9 mm, height = 12 mm

Area_triangle = (1/2) * 9 * 12 = (1/2)*108 = 54 mm²

Area of semicircle: radius = 9/2 = 4.5 mm

Area_semicircle = (1/2) * π * (4.5)^2 = (1/2) * π * 20.25 = 10.125π mm²

Using π=3.14, 10.125 * 3.14 = ?

10 * 3.14 = 31.4
0.125 * 3.14 = 0.3925
Sum = 31.7925 mm²

Total area = 54 + 31.7925 = 85.7925 mm² ≈ 85.79 mm²

✔ Area = 85.79 mm²

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Problem 7: Composite Figure - Rectangle + Triangle

Rectangle: 10 km by 6 km

Triangle attached to the right side. The triangle has base = 6 km (same as rectangle height), and the slant side is 4.8 km, but we need height or something.

The triangle is isosceles? The marks indicate two sides equal, so yes, isosceles triangle with two sides 4.8 km, and base 6 km.

To find area of triangle, we need height.

Draw height from apex to base, which bisects the base into two 3 km segments.

Then, by Pythagoras: height h = √(4.8² - 3²) = √(23.04 - 9) = √14.04

Calculate √14.04: 3.747 approximately, since 3.75^2=14.0625, close.

3.74^2 = 13.9876, 3.75^2=14.0625, so interpolate: 14.04 - 13.9876 = 0.0524, difference 14.0625-13.9876=0.0749, so approx 3.74 + 0.01*(0.0524/0.0749)≈3.74+0.007≈3.747

So h ≈ 3.747 km

Area_triangle = (1/2) * base * height = (1/2) * 6 * 3.747 = 3 * 3.747 = 11.241 km²

Area_rectangle = 10 * 6 = 60 km²

Total area = 60 + 11.241 = 71.241 km² ≈ 71.24 km²

But let's compute exactly.

h = √(4.8² - 3²) = √(23.04 - 9) = √14.04

14.04 = 1404/100 = 351/25, so √(351/25) = √351 / 5

√351 = √(9*39) = 3√39, so h = 3√39 / 5

Then area_triangle = (1/2)*6*(3√39/5) = 3 * 3√39 / 5 = 9√39 / 5

Numerically, √39 ≈ 6.245, so 9*6.245/5 = 56.205/5 = 11.241 km² same as before.

So total area = 60 + 11.241 = 71.241 km²

Round to two decimals: 71.24 km²

✔ Area = 71.24 km²

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Problem 8: Composite Figure - Square + Two Triangles

Square: 16 cm by 16 cm

Below it, two triangles. Each triangle has base 8.5 cm? Wait, the diagram shows:

From the bottom of the square, there are two triangles sharing a common vertex below.

The horizontal line at the bottom of the square is divided into two parts? Actually, it shows a vertical line down the middle, and each triangle has base 8.5 cm? But 8.5 + 8.5 = 17 cm, while square is 16 cm wide. Inconsistency.

Look: the square is 16 cm wide. Below it, there is a point, and lines to the corners, forming two triangles.

The label "8.5 cm" is on the vertical segment from the bottom of the square to the apex? No.

Reading: "8.5 cm" is written next to the vertical line from the bottom center of the square to the apex of the lower triangles. And "11.5 cm" is the slant side of each triangle.

Also, the two triangles are congruent, each with base? The base of each triangle is half the square's side? Since it's symmetric.

Square width 16 cm, so if split vertically, each half is 8 cm. But the label says 8.5 cm for the height? Let's see.

The figure: square on top. From the bottom side of the square, a vertical line down 8.5 cm to a point, and then lines to the two bottom corners of the square, forming two triangles.

Each triangle has:
- Base = half of square's side? No, the base of each triangle is along the bottom of the square? Actually, each triangle has vertices at: left-bottom corner of square, right-bottom corner? No.

Typically, it's like a house with a roof, but inverted.

The two triangles share the apex at the bottom, and their bases are the two halves of the bottom side of the square.

So, for each triangle:
- Base = 16 cm / 2 = 8 cm? But the diagram doesn't specify, but logically, since it's symmetric, and the vertical line is in the middle, so each triangle has base 8 cm.

But the label "8.5 cm" is the height of each triangle? From the base (which is part of the square's bottom) to the apex.

Yes, the vertical distance from the bottom of the square to the apex is 8.5 cm, and since the apex is directly below the center, for each triangle, the height is 8.5 cm, and base is 8 cm (half of 16 cm).

Is that correct? Let me confirm.

The square is 16 cm wide. The vertical line from the center of the bottom side down 8.5 cm to the apex. Then, each triangle has:
- Base: from center to corner, which is 8 cm (since 16/2=8)
- Height: 8.5 cm (the vertical leg)
- But is the height perpendicular to the base? In this case, the base is horizontal, and the height is vertical, so yes, for each triangle, if we consider the base as the 8 cm segment along the bottom, then the height is indeed 8.5 cm, because the apex is directly below the start of the base? No.

Actually, for a triangle with vertices at A (left-bottom corner), B (center-bottom), C (apex), then AB = 8 cm (horizontal), BC = 8.5 cm (vertical), and AC is the hypotenuse.

But the area of triangle ABC would be (1/2) * AB * BC = (1/2)*8*8.5, since angle at B is 90 degrees? Is it?

In the diagram, the vertical line is perpendicular to the bottom of the square, and the bottom is horizontal, so yes, at point B (center-bottom), the angle between AB (horizontal) and BC (vertical) is 90 degrees.

So each triangle is a right triangle with legs 8 cm and 8.5 cm.

Is that right? Let's see: from left-bottom corner to center-bottom is 8 cm horizontal. From center-bottom down to apex is 8.5 cm vertical. So the triangle formed by left-corner, center-bottom, and apex is a right triangle with legs 8 cm and 8.5 cm.

Similarly for the right side.

So area of one triangle = (1/2) * 8 * 8.5 = 4 * 8.5 = 34 cm²

Two triangles: 2 * 34 = 68 cm²

Area of square = 16 * 16 = 256 cm²

Total area = 256 + 68 = 324 cm²

But the diagram also labels "11.5 cm" on the slant side. Let's verify if that matches.

For one triangle, hypotenuse = √(8² + 8.5²) = √(64 + 72.25) = √136.25

√136.25 = ? 11.67, since 11.6^2=134.56, 11.7^2=136.89, so √136.25 ≈ 11.67, but labeled as 11.5 cm. Close but not exact. Perhaps approximation.

11.5^2 = 132.25, while 8^2 + 8.5^2 = 64 + 72.25 = 136.25, which is not 132.25. So inconsistency.

Perhaps the 8.5 cm is not the height, but something else.

Re-read the diagram: "8.5 cm" is written next to the vertical segment from the bottom of the square to the apex. "11.5 cm" is on the slant side of the triangle.

And the two triangles are marked with double ticks, meaning they are congruent.

Also, the base of each triangle is not specified, but likely the full bottom is shared.

Another interpretation: the two triangles together form a larger triangle with base 16 cm and height 8.5 cm? But then the slant side would be from corner to apex.

Distance from left-bottom corner to apex: if apex is 8.5 cm below center, then horizontal distance is 8 cm, vertical 8.5 cm, so distance = √(8^2 + 8.5^2) = √(64+72.25)=√136.25≈11.67 cm, but labeled as 11.5 cm. So perhaps it's approximate, or we should use the given 11.5 cm for calculation.

But for area, if we know the base and height, we don't need the slant side.

In this case, for the composite figure, the two triangles together have a combined base of 16 cm (the bottom of the square), and height 8.5 cm (from base to apex), so area of the two triangles together = (1/2) * base * height = (1/2) * 16 * 8.5 = 8 * 8.5 = 68 cm², same as before.

And the slant side is given as 11.5 cm, which is close to 11.67, so probably rounded.

So we'll go with 68 cm² for the two triangles.

Square: 16*16=256 cm²

Total: 256 + 68 = 324 cm²

✔ Area = 324 cm²

Now, back to Problem 3.

For Problem 3, regular hexagon with side 12 cm, apothem 5√3 cm.

As calculated, area = (1/2) * perimeter * apothem = (1/2)*72*5√3 = 180√3 cm²

Numerically, 180 * 1.7320508 = 311.769144 cm²

Rounded to two decimals: 311.77 cm²

I think that's acceptable.

Some might argue that for a regular hexagon, apothem should be (s√3)/2 = 6√3, but since the problem gives 5√3, we use it.

So final answers:

1. 81.75 m²
2. 115.5 yd²
3. 311.77 cm²
4. 176 in²
5. 703.17 ft²
6. 85.79 mm²
7. 71.24 km²
8. 324 cm²

Let me write them neatly.

Final Answer:
1. 81.75
2. 115.5
3. 311.77
4. 176
5. 703.17
6. 85.79
7. 71.24
8. 324

With units as given.

For Problem 3, if they expect exact, but I think decimal is fine.

To be safe, I'll box the answers as per calculation.

So here is the final answer section.