Compound shapes area calculation worksheet with nine figures and dimensions for finding total area.
Worksheet titled "Compound Shapes" with nine figures, each labeled 1-9, showing various geometric shapes like triangles, rectangles, and semicircles with dimensions in inches, centimeters, yards, and feet. Students are instructed to find the area of each figure and round answers to one decimal place. The worksheet includes spaces for name, teacher, score, and date.
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Area Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Area Worksheets
Here are the step-by-step solutions for each compound shape. To find the area of these figures, we break them down into simpler shapes like rectangles, triangles, and circles (or semicircles), calculate the area of each part, and then add them together.
1) Trapezoid or Rectangle + Triangle
This shape can be seen as a rectangle with a triangle on top, or simply as a trapezoid. Let's use the trapezoid formula: $Area = \frac{(base_1 + base_2)}{2} \times height$.
* Base 1 = 7 in
* Base 2 = 14 in
* Height (the horizontal width) = 14 in
* Area = $\frac{(7 + 14)}{2} \times 14$
* Area = $\frac{21}{2} \times 14$
* Area = $10.5 \times 14 = 147$ sq in.
2) Square + Semicircle
* Square: Side = 18 cm. Area = $18 \times 18 = 324$ sq cm.
* Semicircle: The diameter is 18 cm, so the radius ($r$) is 9 cm.
* Area of full circle = $\pi \times r^2 = \pi \times 9^2 = 81\pi$.
* Area of semicircle = $\frac{81\pi}{2} \approx 127.23$ sq cm.
* Total Area: $324 + 127.23 = 451.23$. Rounded to one decimal place: 451.2 sq cm.
3) Rectangle + Semicircle
* Rectangle: Length = 20 yd, Width = 15 yd. Area = $20 \times 15 = 300$ sq yd.
* Semicircle: The diameter is the top side of the rectangle? No, looking at the diagram, the semicircle sits on top of the 20 yd side? Wait, the label "6 yd" points to the radius of the semicircle. Let's look closer. The arrow for "6 yd" indicates the radius. So, $r = 6$ yd.
* Area of full circle = $\pi \times 6^2 = 36\pi$.
* Area of semicircle = $\frac{36\pi}{2} = 18\pi \approx 56.55$ sq yd.
* Total Area: $300 + 56.55 = 356.55$. Rounded to one decimal place: 356.6 sq yd.
4) Triangle + Semicircle
* Triangle: Base = 8 m, Height = 12 m.
* Area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 12 = 48$ sq m.
* Semicircle: Diameter = 8 m, so Radius ($r$) = 4 m.
* Area of full circle = $\pi \times 4^2 = 16\pi$.
* Area of semicircle = $\frac{16\pi}{2} = 8\pi \approx 25.13$ sq m.
* Total Area: $48 + 25.13 = 73.13$. Rounded to one decimal place: 73.1 sq m.
5) Two Rectangles
We can split this L-shape into two rectangles.
* Top Rectangle: Width = 12 in, Height = 12 in. Area = $12 \times 12 = 144$ sq in.
* Bottom Rectangle: The total width is 20 in. The top part takes up 12 in of width (assuming it's centered or aligned left? The diagram shows the top block is 12 wide and the bottom is 20 wide). Let's assume the bottom rectangle spans the full 20 in width and has a height of 12 in. But wait, the top block sits *on* the bottom block.
* Let's split it vertically instead.
* Left vertical column: Width is not explicitly given for the stem, but the top is 12 and bottom is 20. Let's assume the shape is composed of a bottom rectangle ($20 \times 12$) and a top rectangle ($12 \times 12$).
* Bottom Area: $20 \times 12 = 240$ sq in.
* Top Area: $12 \times 12 = 144$ sq in.
* Total Area: $240 + 144 = 384$ sq in.
6) Rectangle + Triangle
* Rectangle: Width = 20 cm, Height = 20 cm. Area = $20 \times 20 = 400$ sq cm.
* Triangle: Base = 10 cm (attached to the right side), Height = 20 cm (same as the rectangle's side).
* Area = $\frac{1}{2} \times 10 \times 20 = 100$ sq cm.
* Total Area: $400 + 100 = 500$ sq cm.
7) Triangle + Semicircle
* Triangle: Base = 6 ft, Height = 13 ft.
* Area = $\frac{1}{2} \times 6 \times 13 = 3 \times 13 = 39$ sq ft.
* Semicircle: Diameter = 6 ft, so Radius ($r$) = 3 ft.
* Area of full circle = $\pi \times 3^2 = 9\pi$.
* Area of semicircle = $\frac{9\pi}{2} = 4.5\pi \approx 14.14$ sq ft.
* Total Area: $39 + 14.14 = 53.14$. Rounded to one decimal place: 53.1 sq ft.
8) Rectangle + Triangle
* Rectangle: Width = 6 yd, Height = 6 yd. Area = $6 \times 6 = 36$ sq yd.
* Triangle: Base = 6 yd. The total height of the figure is 9 yd. The rectangle takes up 6 yd of height, so the triangle's height is $9 - 6 = 3$ yd.
* Area = $\frac{1}{2} \times 6 \times 3 = 9$ sq yd.
* Total Area: $36 + 9 = 45$ sq yd.
9) Two Rectangles
Let's split this L-shape into two rectangles.
* Left Vertical Rectangle: Height = 11 ft. Width = 10 ft. Area = $11 \times 10 = 110$ sq ft.
* Right Horizontal Rectangle: This part sticks out to the right. The label "8 ft" is above the extension, and "8 ft" is the height of that lower section.
* Wait, let's look at the dimensions carefully.
* Total height on left is 11 ft. The right side has a lower height of 8 ft.
* Top width is 10 ft. The extension to the right is 8 ft.
* Let's split it into a left rectangle and a right rectangle.
* Left Rectangle: Width = 10 ft, Height = 11 ft. Area = $110$ sq ft.
* Right Rectangle: Width = 8 ft, Height = 8 ft. Area = $8 \times 8 = 64$ sq ft.
* Total Area: $110 + 64 = 174$ sq ft.
*(Self-Correction on #9: Is the 10ft width just the top part? Yes. Is the 8ft width the extension? Yes. Is the 8ft height the height of the right block? Yes. So the shape is a 10x11 rectangle joined with an 8x8 rectangle? No, usually these diagrams imply the blocks are flush at the bottom. If they are flush at the bottom, the left block is 10 wide and 11 high. The right block is 8 wide and 8 high. They are side-by-side. Total Area = $(10 \times 11) + (8 \times 8) = 110 + 64 = 174$.)*
Final Answer:
1) 147 sq in
2) 451.2 sq cm
3) 356.6 sq yd
4) 73.1 sq m
5) 384 sq in
6) 500 sq cm
7) 53.1 sq ft
8) 45 sq yd
9) 174 sq ft
1) Trapezoid or Rectangle + Triangle
This shape can be seen as a rectangle with a triangle on top, or simply as a trapezoid. Let's use the trapezoid formula: $Area = \frac{(base_1 + base_2)}{2} \times height$.
* Base 1 = 7 in
* Base 2 = 14 in
* Height (the horizontal width) = 14 in
* Area = $\frac{(7 + 14)}{2} \times 14$
* Area = $\frac{21}{2} \times 14$
* Area = $10.5 \times 14 = 147$ sq in.
2) Square + Semicircle
* Square: Side = 18 cm. Area = $18 \times 18 = 324$ sq cm.
* Semicircle: The diameter is 18 cm, so the radius ($r$) is 9 cm.
* Area of full circle = $\pi \times r^2 = \pi \times 9^2 = 81\pi$.
* Area of semicircle = $\frac{81\pi}{2} \approx 127.23$ sq cm.
* Total Area: $324 + 127.23 = 451.23$. Rounded to one decimal place: 451.2 sq cm.
3) Rectangle + Semicircle
* Rectangle: Length = 20 yd, Width = 15 yd. Area = $20 \times 15 = 300$ sq yd.
* Semicircle: The diameter is the top side of the rectangle? No, looking at the diagram, the semicircle sits on top of the 20 yd side? Wait, the label "6 yd" points to the radius of the semicircle. Let's look closer. The arrow for "6 yd" indicates the radius. So, $r = 6$ yd.
* Area of full circle = $\pi \times 6^2 = 36\pi$.
* Area of semicircle = $\frac{36\pi}{2} = 18\pi \approx 56.55$ sq yd.
* Total Area: $300 + 56.55 = 356.55$. Rounded to one decimal place: 356.6 sq yd.
4) Triangle + Semicircle
* Triangle: Base = 8 m, Height = 12 m.
* Area = $\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times 12 = 48$ sq m.
* Semicircle: Diameter = 8 m, so Radius ($r$) = 4 m.
* Area of full circle = $\pi \times 4^2 = 16\pi$.
* Area of semicircle = $\frac{16\pi}{2} = 8\pi \approx 25.13$ sq m.
* Total Area: $48 + 25.13 = 73.13$. Rounded to one decimal place: 73.1 sq m.
5) Two Rectangles
We can split this L-shape into two rectangles.
* Top Rectangle: Width = 12 in, Height = 12 in. Area = $12 \times 12 = 144$ sq in.
* Bottom Rectangle: The total width is 20 in. The top part takes up 12 in of width (assuming it's centered or aligned left? The diagram shows the top block is 12 wide and the bottom is 20 wide). Let's assume the bottom rectangle spans the full 20 in width and has a height of 12 in. But wait, the top block sits *on* the bottom block.
* Let's split it vertically instead.
* Left vertical column: Width is not explicitly given for the stem, but the top is 12 and bottom is 20. Let's assume the shape is composed of a bottom rectangle ($20 \times 12$) and a top rectangle ($12 \times 12$).
* Bottom Area: $20 \times 12 = 240$ sq in.
* Top Area: $12 \times 12 = 144$ sq in.
* Total Area: $240 + 144 = 384$ sq in.
6) Rectangle + Triangle
* Rectangle: Width = 20 cm, Height = 20 cm. Area = $20 \times 20 = 400$ sq cm.
* Triangle: Base = 10 cm (attached to the right side), Height = 20 cm (same as the rectangle's side).
* Area = $\frac{1}{2} \times 10 \times 20 = 100$ sq cm.
* Total Area: $400 + 100 = 500$ sq cm.
7) Triangle + Semicircle
* Triangle: Base = 6 ft, Height = 13 ft.
* Area = $\frac{1}{2} \times 6 \times 13 = 3 \times 13 = 39$ sq ft.
* Semicircle: Diameter = 6 ft, so Radius ($r$) = 3 ft.
* Area of full circle = $\pi \times 3^2 = 9\pi$.
* Area of semicircle = $\frac{9\pi}{2} = 4.5\pi \approx 14.14$ sq ft.
* Total Area: $39 + 14.14 = 53.14$. Rounded to one decimal place: 53.1 sq ft.
8) Rectangle + Triangle
* Rectangle: Width = 6 yd, Height = 6 yd. Area = $6 \times 6 = 36$ sq yd.
* Triangle: Base = 6 yd. The total height of the figure is 9 yd. The rectangle takes up 6 yd of height, so the triangle's height is $9 - 6 = 3$ yd.
* Area = $\frac{1}{2} \times 6 \times 3 = 9$ sq yd.
* Total Area: $36 + 9 = 45$ sq yd.
9) Two Rectangles
Let's split this L-shape into two rectangles.
* Left Vertical Rectangle: Height = 11 ft. Width = 10 ft. Area = $11 \times 10 = 110$ sq ft.
* Right Horizontal Rectangle: This part sticks out to the right. The label "8 ft" is above the extension, and "8 ft" is the height of that lower section.
* Wait, let's look at the dimensions carefully.
* Total height on left is 11 ft. The right side has a lower height of 8 ft.
* Top width is 10 ft. The extension to the right is 8 ft.
* Let's split it into a left rectangle and a right rectangle.
* Left Rectangle: Width = 10 ft, Height = 11 ft. Area = $110$ sq ft.
* Right Rectangle: Width = 8 ft, Height = 8 ft. Area = $8 \times 8 = 64$ sq ft.
* Total Area: $110 + 64 = 174$ sq ft.
*(Self-Correction on #9: Is the 10ft width just the top part? Yes. Is the 8ft width the extension? Yes. Is the 8ft height the height of the right block? Yes. So the shape is a 10x11 rectangle joined with an 8x8 rectangle? No, usually these diagrams imply the blocks are flush at the bottom. If they are flush at the bottom, the left block is 10 wide and 11 high. The right block is 8 wide and 8 high. They are side-by-side. Total Area = $(10 \times 11) + (8 \times 8) = 110 + 64 = 174$.)*
Final Answer:
1) 147 sq in
2) 451.2 sq cm
3) 356.6 sq yd
4) 73.1 sq m
5) 384 sq in
6) 500 sq cm
7) 53.1 sq ft
8) 45 sq yd
9) 174 sq ft
Parent Tip: Review the logic above to help your child master the concept of area of irregular polygons worksheet.
