Question

score (_/_)
area of polygons
name: ______
date: ______
period: ____
directions: determine the area of the sandcastle by calculating the area of the polygons.
do not round answers!
area of a square:
a = ______ ft²
area of a triangle:
a = ______ ft²
area of a trapezoid:
a = ______ ft²
area of a parallelogram:
a = ______ ft²
area of the sandcastle:
how many of the same figure?
a_square = ____ ft² x _ = ______ ft²
a_triangle = ____ ft² x _ = ______ ft²
a_trapezoid = ____ ft² x _ = ______ ft²
a_parallelogram = ____ ft² x _ = ______ ft²
a_sandcastle = ______ ft²
© 2020 math in demand

Explanation:

Step1: Area of a square

The side length of the square is \( 4.5 \) ft. The formula for the area of a square is \( A = s^2 \), where \( s \) is the side length. So, \( A = 4.5^2 = 20.25 \) \( \text{ft}^2 \).

Step2: Area of a triangle

The base and height of the triangle are both \( 4.5 \) ft. The formula for the area of a triangle is \( A=\frac{1}{2}bh \), where \( b \) is the base and \( h \) is the height. So, \( A=\frac{1}{2}\times4.5\times4.5 = 10.125 \) \( \text{ft}^2 \).

Step3: Area of a trapezoid

Assume the trapezoid has bases \( b_1 = 4.5 \) ft, \( b_2 = 9 \) ft (since two squares side - by - side would have a length of \( 4.5\times2 = 9 \) ft) and height \( h = 4.5 \) ft. The formula for the area of a trapezoid is \( A=\frac{(b_1 + b_2)}{2}h \). So, \( A=\frac{(4.5+9)}{2}\times4.5=\frac{13.5}{2}\times4.5 = 30.375 \) \( \text{ft}^2 \).

Step4: Area of a parallelogram

The base of the parallelogram is \( 4.5 \) ft and the height is \( 4.5 \) ft. The formula for the area of a parallelogram is \( A = bh \). So, \( A=4.5\times4.5 = 20.25 \) \( \text{ft}^2 \).

Step5: Number of each figure and their total areas

• Squares: From the diagram, we count 7 squares. \( A_{\text{square total}}=20.25\times7 = 141.75 \) \( \text{ft}^2 \)
• Triangles: We count 6 triangles. \( A_{\text{triangle total}}=10.125\times6 = 60.75 \) \( \text{ft}^2 \)
• Trapezoids: We count 3 trapezoids. \( A_{\text{trapezoid total}}=30.375\times3 = 91.125 \) \( \text{ft}^2 \)
• Parallelograms: We count 2 parallelograms. \( A_{\text{parallelogram total}}=20.25\times2 = 40.5 \) \( \text{ft}^2 \)

Step6: Area of the sandcastle

To find the area of the sandcastle, we sum up the total areas of all the figures: \( A_{\text{sandcastle}}=141.75 + 60.75+91.125 + 40.5=334.125 \) \( \text{ft}^2 \)

Answer:

• Area of square: \( 20.25 \)
• Area of triangle: \( 10.125 \)
• Area of trapezoid: \( 30.375 \)
• Area of parallelogram: \( 20.25 \)
• Square total: \( 141.75 \) (with 7 squares)
• Triangle total: \( 60.75 \) (with 6 triangles)
• Trapezoid total: \( 91.125 \) (with 3 trapezoids)
• Parallelogram total: \( 40.5 \) (with 2 parallelograms)
• Sandcastle area: \( 334.125 \)