Area Of Irregular Shapes Worksheet for 4th - 8th Grade | Lesson Planet - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use $\pi \approx 3.14$ as instructed.

1. L-shaped Figure
* Split the shape: Divide the "L" into two rectangles.
* Vertical rectangle: $2 \text{ m} \times 6 \text{ m} = 12 \text{ m}^2$.
* Horizontal rectangle (bottom part): The total width is $8 \text{ m}$ and the top part is $2 \text{ m}$, so the bottom width is $8 - 2 = 6 \text{ m}$. The height is $2 \text{ m}$. Area $= 6 \text{ m} \times 2 \text{ m} = 12 \text{ m}^2$.
* Total Area: $12 + 12 = 24 \text{ m}^2$.

2. House Shape
* Split the shape: A rectangle on the bottom and a triangle on top.
* Rectangle: $12 \text{ ft} \times 6 \text{ ft} = 72 \text{ ft}^2$.
* Triangle: Base $= 12 \text{ ft}$, Height $= 4 \text{ ft}$. Area $= \frac{1}{2} \times 12 \times 4 = 24 \text{ ft}^2$.
* Total Area: $72 + 24 = 96 \text{ ft}^2$.

3. Arrow Shape
* Split the shape: A large triangle on top and a rectangle on the bottom.
* Triangle: Base $= 10 \text{ in}$, Height $= 5 \text{ in}$. Area $= \frac{1}{2} \times 10 \times 5 = 25 \text{ in}^2$.
* Rectangle: Width $= 2 \text{ in}$, Height $= 6 \text{ in}$. Area $= 2 \times 6 = 12 \text{ in}^2$.
* Total Area: $25 + 12 = 37 \text{ in}^2$.

4. Stepped Shape
* Split the shape: Two rectangles side-by-side.
* Left rectangle: $3 \text{ cm} \times 5 \text{ cm} = 15 \text{ cm}^2$.
* Right rectangle: Width is $4 \text{ cm}$. Height is $2 \text{ cm}$ (since total height is $5$ and left side is $3$, wait—looking at diagram, right side height is labeled $2$). Let's assume the vertical line labeled $2$ is the height of the right block. Area $= 4 \text{ cm} \times 2 \text{ cm} = 8 \text{ cm}^2$.
* Total Area: $15 + 8 = 23 \text{ cm}^2$.

5. Semicircle attached to Rectangle
* Split the shape: A rectangle and a semicircle.
* Rectangle: $10 \text{ mm} \times 6 \text{ mm} = 60 \text{ mm}^2$.
* Semicircle: Diameter $= 6 \text{ mm}$, so Radius $= 3 \text{ mm}$.
* Area of full circle $= \pi r^2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26 \text{ mm}^2$.
* Area of semicircle $= 28.26 / 2 = 14.13 \text{ mm}^2$.
* Total Area: $60 + 14.13 = 74.13 \text{ mm}^2$.

6. Tombstone Shape
* Split the shape: A rectangle and a semicircle on top.
* Rectangle: Width $= 4 \text{ m}$, Height $= 3 \text{ m}$. Area $= 4 \times 3 = 12 \text{ m}^2$.
* Semicircle: Diameter $= 4 \text{ m}$, so Radius $= 2 \text{ m}$.
* Area of full circle $= 3.14 \times 2^2 = 3.14 \times 4 = 12.56 \text{ m}^2$.
* Area of semicircle $= 12.56 / 2 = 6.28 \text{ m}^2$.
* Total Area: $12 + 6.28 = 18.28 \text{ m}^2$.

7. Chevron/Arrowhead Shape
* Method: Calculate the area of the large outer rectangle and subtract the empty triangle at the bottom.
* Large Rectangle: Width $= 10 \text{ yd}$, Height $= 6 \text{ yd}$. Area $= 10 \times 6 = 60 \text{ yd}^2$.
* Empty Triangle: Base $= 10 \text{ yd}$, Height $= 3 \text{ yd}$. Area $= \frac{1}{2} \times 10 \times 3 = 15 \text{ yd}^2$.
* Total Area: $60 - 15 = 45 \text{ yd}^2$.

8. Capsule Shape
* Split the shape: A rectangle in the middle and two semicircles on the ends. The two semicircles make one full circle.
* Rectangle: Length $= 10 \text{ ft}$, Width (Diameter) $= 4 \text{ ft}$. Area $= 10 \times 4 = 40 \text{ ft}^2$.
* Circle: Diameter $= 4 \text{ ft}$, so Radius $= 2 \text{ ft}$.
* Area $= \pi r^2 = 3.14 \times 2^2 = 3.14 \times 4 = 12.56 \text{ ft}^2$.
* Total Area: $40 + 12.56 = 52.56 \text{ ft}^2$.

9. Square with Semicircles
* Analyze the shape: It looks like a square with four semicircles attached to the sides. However, usually, these problems imply the circles are *inside* or the shape is composed of them. Looking closely at the shading, it appears to be a central square with four semicircles on the outside.
* Central Square: Side $= 10 \text{ in}$. Area $= 10 \times 10 = 100 \text{ in}^2$.
* Four Semicircles: Diameter $= 10 \text{ in}$, so Radius $= 5 \text{ in}$. Four semicircles equal two full circles.
* Area of one circle $= 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ in}^2$.
* Area of two circles $= 78.5 \times 2 = 157 \text{ in}^2$.
* Total Area: $100 + 157 = 257 \text{ in}^2$.
*(Note: If the shape is interpreted as a square with corners cut out by circles, the calculation would differ, but based on standard "irregular shape" additions, this additive method is standard for this visual).*

10. Window Shape
* Split the shape: A rectangle and a semicircle on top.
* Rectangle: Width $= 6 \text{ ft}$, Height $= 8 \text{ ft}$. Area $= 6 \times 8 = 48 \text{ ft}^2$.
* Semicircle: Diameter $= 6 \text{ ft}$, so Radius $= 3 \text{ ft}$.
* Area of full circle $= 3.14 \times 3^2 = 28.26 \text{ ft}^2$.
* Area of semicircle $= 28.26 / 2 = 14.13 \text{ ft}^2$.
* Total Area: $48 + 14.13 = 62.13 \text{ ft}^2$.

11. Track/Capsule Shape
* Split the shape: A rectangle and two semicircles (which make one full circle).
* Rectangle: Length $= 12 \text{ m}$, Width $= 4 \text{ m}$. Area $= 12 \times 4 = 48 \text{ m}^2$.
* Circle: Diameter $= 4 \text{ m}$, so Radius $= 2 \text{ m}$.
* Area $= 3.14 \times 2^2 = 12.56 \text{ m}^2$.
* Total Area: $48 + 12.56 = 60.56 \text{ m}^2$.

12. Flag/Pennant Shape
* Split the shape: A rectangle and a triangle.
* Rectangle: Width $= 10 \text{ ft}$, Height $= 6 \text{ ft}$. Area $= 10 \times 6 = 60 \text{ ft}^2$.
* Triangle: Base $= 6 \text{ ft}$ (same as rectangle height), Height $= 4 \text{ ft}$. Area $= \frac{1}{2} \times 6 \times 4 = 12 \text{ ft}^2$.
* Total Area: $60 + 12 = 72 \text{ ft}^2$.

13. Geography: Nevada
* Split the shape: A trapezoid on the left and a triangle on the right.
* Trapezoid: Parallel sides are $300 \text{ mi}$ and $200 \text{ mi}$. Height (horizontal distance) $= 200 \text{ mi}$.
* Area $= \frac{(300 + 200)}{2} \times 200 = 250 \times 200 = 50,000 \text{ mi}^2$.
* Triangle: Base $= 200 \text{ mi}$ (vertical side shared with trapezoid? No, looking at diagram: The triangle is on the right. Its vertical base is part of the right side of the trapezoid? Actually, let's look at the dimensions provided).
* Let's re-evaluate based on standard decomposition:
* Left part (Trapezoid): Height $= 200$, Base1 $= 300$, Base2 $= 200$. Area $= 50,000$.
* Right part (Triangle): The diagram shows a triangle attached to the right. The vertical side of the triangle corresponds to the shorter parallel side of the trapezoid ($200$). The horizontal height of the triangle is $100$.
* Area of Triangle $= \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 200 \times 100 = 10,000 \text{ mi}^2$.
* Total Area: $50,000 + 10,000 = 60,000 \text{ mi}^2$.

Final Answer:
1. 24 m²
2. 96 ft²
3. 37 in²
4. 23 cm²
5. 74.13 mm²
6. 18.28 m²
7. 45 yd²
8. 52.56 ft²
9. 257 in²
10. 62.13 ft²
11. 60.56 m²
12. 72 ft²
13. 60,000 mi²