Vikram Is Studying The Square Pyramid Below.To Find The Surface Area Of The Pyramid, In Square Inches, Vikram Wrote $(33.2)(34.2) + 4\left(\frac{1}{2}(34.2)(28.4)\right$\]. What Error Did Vikram Make?A. He Used The Same Expression For The Area

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Introduction

When studying the properties of a square pyramid, it's essential to understand the different components that contribute to its overall surface area. The surface area of a pyramid is the sum of the areas of its base and its four triangular faces. In this article, we will explore the correct formula for calculating the surface area of a square pyramid and identify the error made by Vikram in his calculation.

The Formula for Surface Area

The surface area of a square pyramid can be calculated using the following formula:

A = B + 4T

where A is the total surface area, B is the area of the base, and T is the area of one triangular face.

Calculating the Area of the Base

The base of the pyramid is a square with a side length of 34.2 inches. To calculate the area of the base, we use the formula:

B = s^2

where s is the side length of the square.

B = (34.2)^2 B = 1168.64

Calculating the Area of One Triangular Face

Each triangular face of the pyramid is an isosceles triangle with a base length of 34.2 inches and a height of 28.4 inches. To calculate the area of one triangular face, we use the formula:

T = (1/2)bh

where b is the base length and h is the height.

T = (1/2)(34.2)(28.4) T = 486.528

Calculating the Total Surface Area

To calculate the total surface area of the pyramid, we add the area of the base to four times the area of one triangular face:

A = B + 4T A = 1168.64 + 4(486.528) A = 1168.64 + 1946.112 A = 3114.752

Identifying Vikram's Error

Vikram's calculation for the surface area of the pyramid is:

(33.2)(34.2) + 4\left(\frac{1}{2}(34.2)(28.4)\right)

However, this expression is incorrect because it uses the same expression for the area of the base and the area of one triangular face. The correct expression for the area of the base is s^2, and the correct expression for the area of one triangular face is (1/2)bh.

Conclusion

In conclusion, the surface area of a square pyramid can be calculated using the formula A = B + 4T, where B is the area of the base and T is the area of one triangular face. Vikram's error was using the same expression for the area of the base and the area of one triangular face. By using the correct formulas and expressions, we can accurately calculate the surface area of a square pyramid.

Discussion

What do you think is the most common mistake students make when calculating the surface area of a pyramid? How can we help students avoid this mistake and ensure they understand the correct formulas and expressions?

Related Topics

  • Surface Area of a Cube: A cube is a three-dimensional solid object with six square faces. The surface area of a cube can be calculated using the formula A = 6s^2, where s is the side length of the cube.
  • Surface Area of a Rectangular Prism: A rectangular prism is a three-dimensional solid object with six rectangular faces. The surface area of a rectangular prism can be calculated using the formula A = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height of the prism.
  • Surface Area of a Sphere: A sphere is a three-dimensional solid object with a curved surface. The surface area of a sphere can be calculated using the formula A = 4πr^2, where r is the radius of the sphere.
    Frequently Asked Questions (FAQs) about the Surface Area of a Square Pyramid ================================================================================

Q: What is the surface area of a square pyramid?

A: The surface area of a square pyramid is the sum of the areas of its base and its four triangular faces.

Q: How do I calculate the surface area of a square pyramid?

A: To calculate the surface area of a square pyramid, you need to calculate the area of the base and the area of one triangular face, and then add them together. The formula for the surface area of a square pyramid is A = B + 4T, where B is the area of the base and T is the area of one triangular face.

Q: What is the formula for the area of the base of a square pyramid?

A: The formula for the area of the base of a square pyramid is B = s^2, where s is the side length of the square base.

Q: What is the formula for the area of one triangular face of a square pyramid?

A: The formula for the area of one triangular face of a square pyramid is T = (1/2)bh, where b is the base length and h is the height of the triangle.

Q: How do I calculate the surface area of a square pyramid with a square base of side length 34.2 inches and triangular faces with a base length of 34.2 inches and a height of 28.4 inches?

A: To calculate the surface area of a square pyramid with a square base of side length 34.2 inches and triangular faces with a base length of 34.2 inches and a height of 28.4 inches, you need to calculate the area of the base and the area of one triangular face, and then add them together. The area of the base is B = (34.2)^2 = 1168.64, and the area of one triangular face is T = (1/2)(34.2)(28.4) = 486.528. The surface area of the pyramid is A = B + 4T = 1168.64 + 4(486.528) = 3114.752.

Q: What is the most common mistake students make when calculating the surface area of a pyramid?

A: The most common mistake students make when calculating the surface area of a pyramid is using the same expression for the area of the base and the area of one triangular face. This can lead to incorrect calculations and answers.

Q: How can I avoid making this mistake?

A: To avoid making this mistake, make sure to use the correct formulas and expressions for the area of the base and the area of one triangular face. Double-check your calculations and answers to ensure they are correct.

Q: What are some real-world applications of the surface area of a square pyramid?

A: The surface area of a square pyramid has many real-world applications, including:

  • Architecture: The surface area of a square pyramid is used in the design of buildings and monuments, such as the Great Pyramid of Giza.
  • Engineering: The surface area of a square pyramid is used in the design of bridges and other structures.
  • Mathematics: The surface area of a square pyramid is used in mathematical problems and puzzles.

Q: How can I practice calculating the surface area of a square pyramid?

A: You can practice calculating the surface area of a square pyramid by working through examples and problems, such as the one described above. You can also use online resources and tools, such as calculators and software, to help you with your calculations.

Q: What are some tips for calculating the surface area of a square pyramid?

A: Here are some tips for calculating the surface area of a square pyramid:

  • Read the problem carefully: Make sure you understand what is being asked and what information is given.
  • Use the correct formulas and expressions: Use the correct formulas and expressions for the area of the base and the area of one triangular face.
  • Double-check your calculations: Double-check your calculations and answers to ensure they are correct.
  • Practice, practice, practice: Practice calculating the surface area of a square pyramid by working through examples and problems.