Area Of A Trapezoid Word Problems

The trapezoid, a quadrilateral with at least one pair of parallel sides, often presents intriguing challenges in geometry. Mastering the concept of a trapezoid's area and its application through word problems is vital for students and enthusiasts alike. This article delves into the area of a trapezoid, offering clear explanations, step-by-step solutions to various word problems, and insights to enhance your understanding.

Understanding the Trapezoid and Its Area

A trapezoid (also known as a trapezium) is a four-sided figure with at least one pair of parallel sides. These parallel sides are called bases, and the non-parallel sides are called legs. The height of a trapezoid is the perpendicular distance between the two bases.

The formula for the area of a trapezoid is:

Area = (1/2) * (base1 + base2) * height

Where:

• base1 and base2 are the lengths of the parallel sides.
• height is the perpendicular distance between the bases.

Solving Area of a Trapezoid Word Problems: A Step-by-Step Guide

Word problems involving the area of a trapezoid can seem daunting at first. However, by breaking them down into manageable steps, you can easily solve them. Here's a general approach:

1. Read the problem carefully: Understand what information is given and what you are asked to find.
2. Identify the bases and the height: Look for the lengths of the parallel sides (bases) and the perpendicular distance between them (height). Sometimes, the height might be indirectly given and require some extra calculations using other geometric principles (like the Pythagorean theorem).
3. Apply the formula: Substitute the values of the bases and the height into the formula: Area = (1/2) * (base1 + base2) * height
4. Calculate the area: Perform the arithmetic operations to find the numerical value of the area.
5. Include the units: Remember to include the appropriate units (e.g., square inches, square meters) in your final answer.
6. Check your answer: Does the answer make sense in the context of the problem? Estimate the area to ensure your calculated answer is reasonable.

Example Word Problems and Solutions

Let's tackle several word problems to illustrate the application of the trapezoid area formula.

Problem 1: A garden is shaped like a trapezoid. The lengths of the parallel sides are 15 meters and 11 meters, and the distance between them is 8 meters. What is the area of the garden?

Solution:

1. Identify the given information:
   • base1 = 15 meters
   • base2 = 11 meters
   • height = 8 meters
2. Apply the formula:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (15 + 11) * 8
3. Calculate the area:
   • Area = (1/2) * (26) * 8
   • Area = 13 * 8
   • Area = 104 square meters
4. Answer: The area of the garden is 104 square meters.

Problem 2: A farmer has a field in the shape of a trapezoid. One base is 240 feet long, and the other base is 180 feet long. The height of the trapezoid is 90 feet. If the farmer wants to fertilize the field and each bag of fertilizer covers 500 square feet, how many bags of fertilizer does the farmer need?

Solution:

1. Identify the given information:
   • base1 = 240 feet
   • base2 = 180 feet
   • height = 90 feet
2. Calculate the area of the field:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (240 + 180) * 90
   • Area = (1/2) * (420) * 90
   • Area = 210 * 90
   • Area = 18900 square feet
3. Determine the number of fertilizer bags needed:
   • Bags = Total Area / Area per bag
   • Bags = 18900 / 500
   • Bags = 37.8
4. Round up to the nearest whole number: Since the farmer can't buy a fraction of a bag, they need to buy 38 bags.
5. Answer: The farmer needs 38 bags of fertilizer.

Problem 3: The area of a trapezoid is 56 square inches. The lengths of the bases are 6 inches and 8 inches. Find the height of the trapezoid.

Solution:

1. Identify the given information:
   • Area = 56 square inches
   • base1 = 6 inches
   • base2 = 8 inches
   • height = ? (unknown)
2. Apply the formula and solve for height:
   • Area = (1/2) * (base1 + base2) * height
   • 56 = (1/2) * (6 + 8) * height
   • 56 = (1/2) * (14) * height
   • 56 = 7 * height
   • height = 56 / 7
   • height = 8 inches
3. Answer: The height of the trapezoid is 8 inches.

Problem 4: A window is shaped like an isosceles trapezoid. The lengths of the parallel sides are 3 feet and 5 feet. The length of each of the non-parallel sides is 2 feet. Find the area of the window.

Solution:

1. Identify the given information:
   • base1 = 3 feet
   • base2 = 5 feet
   • leg length = 2 feet
2. Determine the height: This problem requires an extra step to find the height. Since it's an isosceles trapezoid, we can drop perpendiculars from the shorter base to the longer base, creating two congruent right triangles. The base of each triangle is (5 - 3) / 2 = 1 foot.
3. Use the Pythagorean theorem: to find the height: height^2 + base^2 = leg^2
   • height^2 + 1^2 = 2^2
   • height^2 + 1 = 4
   • height^2 = 3
   • height = √3 feet
4. Apply the area formula:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (3 + 5) * √3
   • Area = (1/2) * (8) * √3
   • Area = 4√3 square feet
5. Approximate the area:
   • Area ≈ 4 * 1.732
   • Area ≈ 6.928 square feet
6. Answer: The area of the window is approximately 6.928 square feet.

Problem 5: A table top is in the shape of a trapezoid with bases of length 4 feet and 6 feet and a height of 3 feet. If the material costs $5 per square foot, what is the cost of the material for the table top?

Solution:

1. Identify the given information:
   • base1 = 4 feet
   • base2 = 6 feet
   • height = 3 feet
   • cost per square foot = $5
2. Calculate the area of the table top:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (4 + 6) * 3
   • Area = (1/2) * (10) * 3
   • Area = 5 * 3
   • Area = 15 square feet
3. Calculate the total cost:
   • Total Cost = Area * Cost per square foot
   • Total Cost = 15 * $5
   • Total Cost = $75
4. Answer: The cost of the material for the table top is $75.

Problem 6: A banner is being designed in the shape of a trapezoid. The top edge is 18 inches long, and the bottom edge is 24 inches long. The area of the banner needs to be 252 square inches. What should the height of the banner be?

Solution:

1. Identify the given information:
   • base1 = 18 inches
   • base2 = 24 inches
   • Area = 252 square inches
   • height = ? (unknown)
2. Apply the formula and solve for height:
   • Area = (1/2) * (base1 + base2) * height
   • 252 = (1/2) * (18 + 24) * height
   • 252 = (1/2) * (42) * height
   • 252 = 21 * height
   • height = 252 / 21
   • height = 12 inches
3. Answer: The height of the banner should be 12 inches.

Problem 7: A swimming pool is shaped like a trapezoid. The parallel sides are 30 feet and 50 feet. The pool is 10 feet deep throughout. How many cubic feet of water are needed to fill the pool?

Solution:

1. Identify the given information:
   • base1 = 30 feet
   • base2 = 50 feet
   • depth (height of the trapezoid) = 10 feet
   • The depth of the pool itself is the "height" of the prism.
2. Calculate the area of the trapezoidal base:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (30 + 50) * 10
   • Area = (1/2) * (80) * 10
   • Area = 40 * 10
   • Area = 400 square feet
3. Calculate the volume of the pool: Since the pool has a constant depth, we're calculating the volume of a trapezoidal prism. The volume is the area of the trapezoidal base times the depth of the pool.
   • Volume = Area * Depth
   • Volume = 400 * 10
   • Volume = 4000 cubic feet
4. Answer: 4000 cubic feet of water are needed to fill the pool.

Problem 8: A piece of land is shaped like a trapezoid. The lengths of the parallel sides are 200 meters and 300 meters. The distance between the parallel sides is 150 meters. If the land is sold for $50 per square meter, what is the total selling price of the land?

Solution:

1. Identify the given information:
   • base1 = 200 meters
   • base2 = 300 meters
   • height = 150 meters
   • price per square meter = $50
2. Calculate the area of the land:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (200 + 300) * 150
   • Area = (1/2) * (500) * 150
   • Area = 250 * 150
   • Area = 37500 square meters
3. Calculate the total selling price:
   • Total Price = Area * Price per square meter
   • Total Price = 37500 * $50
   • Total Price = $1,875,000
4. Answer: The total selling price of the land is $1,875,000.

Problem 9: A kite is shaped like a trapezoid joined with a triangle. The trapezoid has bases of 8 inches and 12 inches, with a height of 5 inches. The triangle has a base of 12 inches and a height of 4 inches. Find the total area of the kite.

Solution:

1. Identify the given information:
   • Trapezoid: base1 = 8 inches, base2 = 12 inches, height = 5 inches
   • Triangle: base = 12 inches, height = 4 inches
2. Calculate the area of the trapezoid:
   • Area_trapezoid = (1/2) * (base1 + base2) * height
   • Area_trapezoid = (1/2) * (8 + 12) * 5
   • Area_trapezoid = (1/2) * (20) * 5
   • Area_trapezoid = 10 * 5
   • Area_trapezoid = 50 square inches
3. Calculate the area of the triangle:
   • Area_triangle = (1/2) * base * height
   • Area_triangle = (1/2) * 12 * 4
   • Area_triangle = 6 * 4
   • Area_triangle = 24 square inches
4. Calculate the total area of the kite:
   • Total_Area = Area_trapezoid + Area_triangle
   • Total_Area = 50 + 24
   • Total_Area = 74 square inches
5. Answer: The total area of the kite is 74 square inches.

Problem 10: A cross-section of a dam is in the shape of a trapezoid. The top of the dam is 20 meters wide, and the base of the dam is 60 meters wide. The dam is 30 meters high. What is the area of the cross-section?

Solution:

1. Identify the given information:
   • base1 = 20 meters
   • base2 = 60 meters
   • height = 30 meters
2. Apply the area formula:
   • Area = (1/2) * (base1 + base2) * height
   • Area = (1/2) * (20 + 60) * 30
3. Calculate the area:
   • Area = (1/2) * (80) * 30
   • Area = 40 * 30
   • Area = 1200 square meters
4. Answer: The area of the cross-section of the dam is 1200 square meters.

Common Mistakes to Avoid

• Confusing bases and legs: Always remember that the bases are the parallel sides, and the legs are the non-parallel sides.
• Incorrectly identifying the height: The height is the perpendicular distance between the bases. It's not necessarily the length of one of the legs.
• Forgetting units: Always include the appropriate units (square units) in your final answer.
• Misinterpreting the problem: Read the problem carefully and make sure you understand what you are being asked to find. Draw a diagram if necessary.
• Arithmetic errors: Double-check your calculations to avoid careless mistakes.

Tips for Success

• Draw diagrams: Visualizing the problem can often make it easier to understand.
• Label the parts: Label the bases, height, and any other relevant information on your diagram.
• Break down complex problems: If the problem seems overwhelming, try breaking it down into smaller, more manageable steps.
• Practice, practice, practice: The more problems you solve, the more comfortable you will become with the concept of the area of a trapezoid.
• Check your work: Always take a few minutes to review your work and make sure your answer is reasonable.

Conclusion

Understanding the area of a trapezoid is an essential skill in geometry. By mastering the formula, following a systematic approach to problem-solving, and avoiding common mistakes, you can confidently tackle any word problem involving trapezoids. With practice and perseverance, you'll be able to apply this knowledge to real-world scenarios and excel in your mathematical endeavors. Remember to carefully read the problem, identify the bases and height, apply the formula, and double-check your work. Good luck!