Area of Composite Figures Worksheets - Math Monks - Free Printable
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Step-by-step solution for: Area of Composite Figures Worksheets - Math Monks
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Step-by-step solution for: Area of Composite Figures Worksheets - Math Monks
Let’s solve each problem step by step. The worksheet asks for the area of regular polygons and composite figures.
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Actually, looking at the diagram: it shows a square (since all sides look equal) with an apothem (distance from center to side) of 7.5 m, and half the side length is 10.9 m? Wait — that doesn’t make sense for a square.
Wait — actually, this looks like a regular polygon, but the label says “7.5 m” as the apothem and “10.9 m” as the side length? But 10.9 m is labeled along the bottom — likely the side length.
Actually, looking again — this is likely a regular pentagon or hexagon? No — the shape has 4 sides? It looks like a square, but the apothem is drawn from center perpendicular to a side.
But if it’s a square, then:
- Apothem = distance from center to side = half the side length.
- So if apothem = 7.5 m → side = 15 m → area = 15² = 225 m².
BUT the base is labeled 10.9 m — which contradicts.
Wait — perhaps it’s a regular polygon with side length = 10.9 m and apothem = 7.5 m.
That makes sense! For any regular polygon:
> Area = (1/2) × Perimeter × Apothem
So if it’s a regular polygon with side length 10.9 m and apothem 7.5 m, we need to know how many sides.
Looking at the shape — it has 4 sides? Actually, no — the diagram shows a 4-sided figure, but the apothem is drawn from center to one side — that suggests it's a regular quadrilateral = square.
But in a square, apothem = side/2 → so if apothem = 7.5, side = 15 → but labeled side is 10.9 → contradiction.
Alternatively — maybe the 10.9 m is the length of the entire base, and the apothem is 7.5 m — so perhaps it’s a rectangle? But then why draw apothem?
Wait — I think there’s a mistake in interpretation.
Actually, looking at the image again — the first shape is a square, and the 7.5 m is the height from center to side (apothem), and 10.9 m is the side length? That can’t be — because in a square, apothem = side/2.
Unless it’s not a square — maybe it’s a regular polygon with 5 or 6 sides? But the shape drawn has 4 sides.
This is confusing.
Wait — perhaps the 10.9 m is the side length, and 7.5 m is the apothem, and it’s a regular polygon with n sides — but n is not given.
Actually — looking at the diagram carefully — the shape has 4 sides, and the apothem is drawn from center perpendicular to one side — so it must be a regular quadrilateral = square.
Then, if apothem = 7.5 m → side = 15 m → area = 225 m².
But the base is labeled 10.9 m — which is inconsistent.
Alternatively — maybe the 10.9 m is the length of the side, and 7.5 m is the apothem, and it’s a regular polygon — but how many sides?
Actually, let’s assume it’s a regular polygon with side length s = 10.9 m and apothem a = 7.5 m.
Then area = (1/2) * perimeter * apothem.
But we don’t know the number of sides.
Wait — perhaps the shape is a rectangle? But then apothem doesn’t apply.
I think there might be a mislabeling.
Alternatively — perhaps the 7.5 m is the height of a triangle inside the square? No.
Let me check the other problems for pattern.
Actually, looking at problem 2: pentagon with side 13.2 yd and apothem 3.5 yd — that makes sense.
Problem 3: hexagon with side 12 cm and apothem 5√3 cm — also makes sense.
So for problem 1 — likely it’s a regular polygon with side length 10.9 m and apothem 7.5 m — but how many sides?
The shape drawn has 4 sides — so perhaps it’s a square, and the 10.9 m is a mistake? Or perhaps it’s a rhombus? But not regular unless square.
I think the best assumption is that it’s a regular polygon with side length 10.9 m and apothem 7.5 m, and we use the formula:
> Area = (1/2) × Perimeter × Apothem
But we need the number of sides.
Looking at the shape — it has 4 sides — so n=4.
So perimeter = 4 × 10.9 = 43.6 m
Apothem = 7.5 m
Area = (1/2) × 43.6 × 7.5 = (1/2) × 327 = 163.5 m²
But in a square, apothem should be half the side — 10.9/2 = 5.45, not 7.5 — so it’s not a square.
Perhaps it’s a regular octagon or something else — but the diagram shows 4 sides.
I think there’s an error in the diagram or my interpretation.
Alternatively — perhaps the 10.9 m is the length of the base, and the 7.5 m is the height of the polygon — but for a regular polygon, height is not typically used.
Another idea: perhaps it’s a triangle? But no, it has 4 sides.
I think for the sake of progress, I’ll assume it’s a regular polygon with side length 10.9 m and apothem 7.5 m, and since the shape has 4 sides, n=4.
So area = (1/2) * 4 * 10.9 * 7.5 = 2 * 10.9 * 7.5 = 21.8 * 7.5 = 163.5 m²
Let’s go with that.
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Given:
- Side length = 13.2 yd
- Apothem = 3.5 yd
- Number of sides, n = 5
Perimeter = 5 × 13.2 = 66 yd
Area = (1/2) × Perimeter × Apothem = (1/2) × 66 × 3.5 = 33 × 3.5 = 115.5 yd²
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Given:
- Side length = 12 cm
- Apothem = 5√3 cm
For a regular hexagon, apothem = (s√3)/2, where s is side length.
Check: (12√3)/2 = 6√3 ≈ 10.392, but given apothem is 5√3 ≈ 8.66 — not matching.
So either it’s not regular, or the apothem is given as 5√3.
We’ll use the given values.
n = 6
Perimeter = 6 × 12 = 72 cm
Area = (1/2) × 72 × 5√3 = 36 × 5√3 = 180√3 cm²
We can leave it as 180√3 cm², or approximate: √3 ≈ 1.732 → 180×1.732 = 311.76 cm²
But since the apothem is given as 5√3, likely they want exact form.
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Base = 22 in, Height = 16 in
Area = (1/2) × base × height = (1/2) × 22 × 16 = 11 × 16 = 176 in²
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Dimensions:
- Rectangle: width = 18 ft, length = 32 ft
- Semicircle on the right end — diameter = 18 ft (same as width)
Area of rectangle = 32 × 18 = 576 ft²
Area of semicircle = (1/2) × π × r², r = 9 ft
= (1/2) × π × 81 = 40.5π ft²
Total area = 576 + 40.5π ft²
Approximate: 40.5 × 3.1416 ≈ 127.23, so total ≈ 576 + 127.23 = 703.23 ft²
But likely they want exact form.
---
---
The figure is a triangle on top of a semicircle.
Given:
- Total height = 12 mm
- Diameter of semicircle = 9 mm → radius = 4.5 mm
- So height of triangle = 12 - 4.5 = 7.5 mm? Wait — no.
Actually, the semicircle is at the bottom, and the triangle is on top.
The diameter of the semicircle is 9 mm, so the base of the triangle is also 9 mm (since it sits on the semicircle).
The total height is 12 mm — this includes the height of the triangle plus the radius of the semicircle? No — the semicircle has radius 4.5 mm, but the height from base to top of semicircle is 4.5 mm, and the triangle goes up to 12 mm total, so the height of the triangle is 12 - 4.5 = 7.5 mm.
Is that correct? Let me see.
The semicircle is below, so its highest point is at 4.5 mm above the base. The triangle starts from the diameter (base) and goes up to 12 mm, so yes, height of triangle = 12 - 4.5 = 7.5 mm.
Area of triangle = (1/2) × base × height = (1/2) × 9 × 7.5 = 4.5 × 7.5 = 33.75 mm²
Area of semicircle = (1/2) × π × r² = (1/2) × π × (4.5)² = (1/2) × π × 20.25 = 10.125π mm²
Total area = 33.75 + 10.125π mm²
Approximate: 10.125 × 3.1416 ≈ 31.81, so total ≈ 33.75 + 31.81 = 65.56 mm²
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---
Rectangle: length = 10 km, width = 6 km → area = 10 × 6 = 60 km²
Attached to the right is a triangle — the base is the same as the width of the rectangle, 6 km? No — looking at the diagram, the triangle is attached to the right side, and the height is given as 4.8 km, and it's a right triangle? The dashed line is the height, perpendicular to the base.
Actually, the triangle has base = 6 km (same as rectangle height) and height = 4.8 km.
Area of triangle = (1/2) × 6 × 4.8 = 3 × 4.8 = 14.4 km²
Total area = 60 + 14.4 = 74.4 km²
---
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Square: side = 16 cm → area = 16 × 16 = 256 cm²
Below the square is a triangle — the base is the same as the square, 16 cm? No — looking at the diagram, the triangle is attached to the bottom, and it's an isosceles triangle with two equal sides of 11.5 cm, and the base is 16 cm? But the height is given as 8.5 cm.
Actually, the height is drawn from the apex to the base, and it's 8.5 cm, and the base is 16 cm (same as square).
Area of triangle = (1/2) × base × height = (1/2) × 16 × 8.5 = 8 × 8.5 = 68 cm²
Total area = 256 + 68 = 324 cm²
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## Final Answers:
1. 163.5 m²
2. 115.5 yd²
3. 180√3 cm²
4. 176 in²
5. 576 + 40.5π ft² (or approx 703.23 ft²)
6. 33.75 + 10.125π mm² (or approx 65.56 mm²)
7. 74.4 km²
8. 324 cm²
Note: For problems 5 and 6, if decimal approximation is required, use π ≈ 3.14 or as specified. Otherwise, exact form is acceptable.
For problem 1, if it's intended to be a square with side 15 m (since apothem 7.5 m implies side 15 m), then area would be 225 m², but the diagram labels 10.9 m, so I stuck with the given numbers and assumed n=4. If it's a different polygon, the answer may vary.
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Problem 1: Square (or rectangle?) with apothem?
Actually, looking at the diagram: it shows a square (since all sides look equal) with an apothem (distance from center to side) of 7.5 m, and half the side length is 10.9 m? Wait — that doesn’t make sense for a square.
Wait — actually, this looks like a regular polygon, but the label says “7.5 m” as the apothem and “10.9 m” as the side length? But 10.9 m is labeled along the bottom — likely the side length.
Actually, looking again — this is likely a regular pentagon or hexagon? No — the shape has 4 sides? It looks like a square, but the apothem is drawn from center perpendicular to a side.
But if it’s a square, then:
- Apothem = distance from center to side = half the side length.
- So if apothem = 7.5 m → side = 15 m → area = 15² = 225 m².
BUT the base is labeled 10.9 m — which contradicts.
Wait — perhaps it’s a regular polygon with side length = 10.9 m and apothem = 7.5 m.
That makes sense! For any regular polygon:
> Area = (1/2) × Perimeter × Apothem
So if it’s a regular polygon with side length 10.9 m and apothem 7.5 m, we need to know how many sides.
Looking at the shape — it has 4 sides? Actually, no — the diagram shows a 4-sided figure, but the apothem is drawn from center to one side — that suggests it's a regular quadrilateral = square.
But in a square, apothem = side/2 → so if apothem = 7.5, side = 15 → but labeled side is 10.9 → contradiction.
Alternatively — maybe the 10.9 m is the length of the entire base, and the apothem is 7.5 m — so perhaps it’s a rectangle? But then why draw apothem?
Wait — I think there’s a mistake in interpretation.
Actually, looking at the image again — the first shape is a square, and the 7.5 m is the height from center to side (apothem), and 10.9 m is the side length? That can’t be — because in a square, apothem = side/2.
Unless it’s not a square — maybe it’s a regular polygon with 5 or 6 sides? But the shape drawn has 4 sides.
This is confusing.
Wait — perhaps the 10.9 m is the side length, and 7.5 m is the apothem, and it’s a regular polygon with n sides — but n is not given.
Actually — looking at the diagram carefully — the shape has 4 sides, and the apothem is drawn from center perpendicular to one side — so it must be a regular quadrilateral = square.
Then, if apothem = 7.5 m → side = 15 m → area = 225 m².
But the base is labeled 10.9 m — which is inconsistent.
Alternatively — maybe the 10.9 m is the length of the side, and 7.5 m is the apothem, and it’s a regular polygon — but how many sides?
Actually, let’s assume it’s a regular polygon with side length s = 10.9 m and apothem a = 7.5 m.
Then area = (1/2) * perimeter * apothem.
But we don’t know the number of sides.
Wait — perhaps the shape is a rectangle? But then apothem doesn’t apply.
I think there might be a mislabeling.
Alternatively — perhaps the 7.5 m is the height of a triangle inside the square? No.
Let me check the other problems for pattern.
Actually, looking at problem 2: pentagon with side 13.2 yd and apothem 3.5 yd — that makes sense.
Problem 3: hexagon with side 12 cm and apothem 5√3 cm — also makes sense.
So for problem 1 — likely it’s a regular polygon with side length 10.9 m and apothem 7.5 m — but how many sides?
The shape drawn has 4 sides — so perhaps it’s a square, and the 10.9 m is a mistake? Or perhaps it’s a rhombus? But not regular unless square.
I think the best assumption is that it’s a regular polygon with side length 10.9 m and apothem 7.5 m, and we use the formula:
> Area = (1/2) × Perimeter × Apothem
But we need the number of sides.
Looking at the shape — it has 4 sides — so n=4.
So perimeter = 4 × 10.9 = 43.6 m
Apothem = 7.5 m
Area = (1/2) × 43.6 × 7.5 = (1/2) × 327 = 163.5 m²
But in a square, apothem should be half the side — 10.9/2 = 5.45, not 7.5 — so it’s not a square.
Perhaps it’s a regular octagon or something else — but the diagram shows 4 sides.
I think there’s an error in the diagram or my interpretation.
Alternatively — perhaps the 10.9 m is the length of the base, and the 7.5 m is the height of the polygon — but for a regular polygon, height is not typically used.
Another idea: perhaps it’s a triangle? But no, it has 4 sides.
I think for the sake of progress, I’ll assume it’s a regular polygon with side length 10.9 m and apothem 7.5 m, and since the shape has 4 sides, n=4.
So area = (1/2) * 4 * 10.9 * 7.5 = 2 * 10.9 * 7.5 = 21.8 * 7.5 = 163.5 m²
Let’s go with that.
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Problem 1 Answer: 163.5 m²
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Problem 2: Regular Pentagon
Given:
- Side length = 13.2 yd
- Apothem = 3.5 yd
- Number of sides, n = 5
Perimeter = 5 × 13.2 = 66 yd
Area = (1/2) × Perimeter × Apothem = (1/2) × 66 × 3.5 = 33 × 3.5 = 115.5 yd²
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Problem 2 Answer: 115.5 yd²
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Problem 3: Regular Hexagon
Given:
- Side length = 12 cm
- Apothem = 5√3 cm
For a regular hexagon, apothem = (s√3)/2, where s is side length.
Check: (12√3)/2 = 6√3 ≈ 10.392, but given apothem is 5√3 ≈ 8.66 — not matching.
So either it’s not regular, or the apothem is given as 5√3.
We’ll use the given values.
n = 6
Perimeter = 6 × 12 = 72 cm
Area = (1/2) × 72 × 5√3 = 36 × 5√3 = 180√3 cm²
We can leave it as 180√3 cm², or approximate: √3 ≈ 1.732 → 180×1.732 = 311.76 cm²
But since the apothem is given as 5√3, likely they want exact form.
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Problem 3 Answer: 180√3 cm²
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Problem 4: Triangle
Base = 22 in, Height = 16 in
Area = (1/2) × base × height = (1/2) × 22 × 16 = 11 × 16 = 176 in²
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Problem 4 Answer: 176 in²
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Problem 5: Composite Figure — Rectangle + Semicircle
Dimensions:
- Rectangle: width = 18 ft, length = 32 ft
- Semicircle on the right end — diameter = 18 ft (same as width)
Area of rectangle = 32 × 18 = 576 ft²
Area of semicircle = (1/2) × π × r², r = 9 ft
= (1/2) × π × 81 = 40.5π ft²
Total area = 576 + 40.5π ft²
Approximate: 40.5 × 3.1416 ≈ 127.23, so total ≈ 576 + 127.23 = 703.23 ft²
But likely they want exact form.
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Problem 5 Answer: 576 + 40.5π ft² or approximately 703.23 ft²
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Problem 6: Composite Figure — Triangle + Semicircle
The figure is a triangle on top of a semicircle.
Given:
- Total height = 12 mm
- Diameter of semicircle = 9 mm → radius = 4.5 mm
- So height of triangle = 12 - 4.5 = 7.5 mm? Wait — no.
Actually, the semicircle is at the bottom, and the triangle is on top.
The diameter of the semicircle is 9 mm, so the base of the triangle is also 9 mm (since it sits on the semicircle).
The total height is 12 mm — this includes the height of the triangle plus the radius of the semicircle? No — the semicircle has radius 4.5 mm, but the height from base to top of semicircle is 4.5 mm, and the triangle goes up to 12 mm total, so the height of the triangle is 12 - 4.5 = 7.5 mm.
Is that correct? Let me see.
The semicircle is below, so its highest point is at 4.5 mm above the base. The triangle starts from the diameter (base) and goes up to 12 mm, so yes, height of triangle = 12 - 4.5 = 7.5 mm.
Area of triangle = (1/2) × base × height = (1/2) × 9 × 7.5 = 4.5 × 7.5 = 33.75 mm²
Area of semicircle = (1/2) × π × r² = (1/2) × π × (4.5)² = (1/2) × π × 20.25 = 10.125π mm²
Total area = 33.75 + 10.125π mm²
Approximate: 10.125 × 3.1416 ≈ 31.81, so total ≈ 33.75 + 31.81 = 65.56 mm²
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Problem 6 Answer: 33.75 + 10.125π mm² or approximately 65.56 mm²
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Problem 7: Composite Figure — Rectangle + Triangle
Rectangle: length = 10 km, width = 6 km → area = 10 × 6 = 60 km²
Attached to the right is a triangle — the base is the same as the width of the rectangle, 6 km? No — looking at the diagram, the triangle is attached to the right side, and the height is given as 4.8 km, and it's a right triangle? The dashed line is the height, perpendicular to the base.
Actually, the triangle has base = 6 km (same as rectangle height) and height = 4.8 km.
Area of triangle = (1/2) × 6 × 4.8 = 3 × 4.8 = 14.4 km²
Total area = 60 + 14.4 = 74.4 km²
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Problem 7 Answer: 74.4 km²
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Problem 8: Composite Figure — Square + Triangle
Square: side = 16 cm → area = 16 × 16 = 256 cm²
Below the square is a triangle — the base is the same as the square, 16 cm? No — looking at the diagram, the triangle is attached to the bottom, and it's an isosceles triangle with two equal sides of 11.5 cm, and the base is 16 cm? But the height is given as 8.5 cm.
Actually, the height is drawn from the apex to the base, and it's 8.5 cm, and the base is 16 cm (same as square).
Area of triangle = (1/2) × base × height = (1/2) × 16 × 8.5 = 8 × 8.5 = 68 cm²
Total area = 256 + 68 = 324 cm²
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Problem 8 Answer: 324 cm²
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## Final Answers:
1. 163.5 m²
2. 115.5 yd²
3. 180√3 cm²
4. 176 in²
5. 576 + 40.5π ft² (or approx 703.23 ft²)
6. 33.75 + 10.125π mm² (or approx 65.56 mm²)
7. 74.4 km²
8. 324 cm²
Note: For problems 5 and 6, if decimal approximation is required, use π ≈ 3.14 or as specified. Otherwise, exact form is acceptable.
For problem 1, if it's intended to be a square with side 15 m (since apothem 7.5 m implies side 15 m), then area would be 225 m², but the diagram labels 10.9 m, so I stuck with the given numbers and assumed n=4. If it's a different polygon, the answer may vary.
Parent Tip: Review the logic above to help your child master the concept of area of composite figures worksheet 6th grade.
