Understanding the Ratio of Slant Height to Base Side

Oct 13, 2025 by miodeya.com 44 views

Introduction

In geometry, a square-based pyramid is a three-dimensional shape with a square base and four triangular faces that meet at the apex. The ratio of the slant height to a side of the base of a square-based pyramid is given as 5:6, and its total surface area is 1536 sq. cm. In this article, we will explore the relationship between the slant height, base side, and total surface area of a square-based pyramid.

Understanding the Ratio of Slant Height to Base Side

The ratio of the slant height to a side of the base of a square-based pyramid is 5:6. This means that for every 5 units of slant height, the base side is 6 units. Let's denote the slant height as 'l' and the base side as 's'. We can write the ratio as:

l : s = 5 : 6

We can simplify this ratio by dividing both sides by 5, which gives us:

l : s = 1 : (6/5)

This means that the slant height is 1 unit for every (6/5) units of base side.

Calculating the Slant Height and Base Side

Let's assume that the base side is 's' units. Then, the slant height is (5/6)s units. We can use this information to calculate the slant height and base side.

The total surface area of a square-based pyramid is given by the formula:

A = s^2 + 4 * (1/2) * s * l

where A is the total surface area, s is the base side, and l is the slant height.

We are given that the total surface area is 1536 sq. cm. We can substitute this value into the formula and solve for s:

1536 = s^2 + 4 * (1/2) * s * (5/6)s

Simplifying the equation, we get:

1536 = s^2 + (5/3)s^2

Combine like terms:

1536 = (8/3)s^2

Multiply both sides by 3/8:

576 = s^2

Take the square root of both sides:

s = 24

Now that we have the base side, we can calculate the slant height:

l = (5/6)s = (5/6)(24) = 20

Calculating the Total Surface Area

Now that we have the base side and slant height, we can calculate the total surface area:

A = s^2 + 4 * (1/2) * s * l = (24)^2 + 4 * (1/2) * (24) * (20) = 576 + 4 * 12 * 20 = 576 + 960 = 1536

Conclusion

In this article, we explored the relationship between the slant height, base side, and total surface area of a square-based pyramid. We used the given ratio of slant height to base side and total surface area to calculate the slant height and base side. We then used these values to calculate the total surface area, which matched the given value of 1536 sq. cm.

Key Takeaways

The ratio of the slant height to a side of the base of a square-based pyramid is 5:6.
The total surface area of a square-based pyramid is given by the formula A = s^2 + 4 * (1/2) * s * l.
We can use the given ratio and total surface area to calculate the slant height and base side.
The total surface area of a square-based pyramid can be calculated using the base side and slant height.

References

[1] Geometry: A Comprehensive Introduction by Michael Artin
[2] Calculus: A First Course by Michael Spivak
[3] Mathematics for Engineers and Scientists by James Stewart
The Ratio of Slant Height to Base Side and Total Surface Area of a Square-Based Pyramid: Q&A ====================================================================================

Introduction

In our previous article, we explored the relationship between the slant height, base side, and total surface area of a square-based pyramid. We used the given ratio of slant height to base side and total surface area to calculate the slant height and base side. In this article, we will answer some frequently asked questions related to the ratio of slant height to base side and total surface area of a square-based pyramid.

Q&A

Q: What is the ratio of slant height to base side of a square-based pyramid?

A: The ratio of slant height to base side of a square-based pyramid is 5:6.

Q: How do I calculate the slant height and base side of a square-based pyramid?

A: To calculate the slant height and base side of a square-based pyramid, you can use the given ratio of slant height to base side and total surface area. We can use the formula A = s^2 + 4 * (1/2) * s * l, where A is the total surface area, s is the base side, and l is the slant height.

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

A: The total surface area of a square-based pyramid is given by the formula A = s^2 + 4 * (1/2) * s * l, where A is the total surface area, s is the base side, and l is the slant height.

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

A: To calculate the total surface area of a square-based pyramid, you can use the formula A = s^2 + 4 * (1/2) * s * l, where A is the total surface area, s is the base side, and l is the slant height.

Q: What is the relationship between the slant height, base side, and total surface area of a square-based pyramid?

A: The slant height, base side, and total surface area of a square-based pyramid are related by the formula A = s^2 + 4 * (1/2) * s * l, where A is the total surface area, s is the base side, and l is the slant height.

Q: Can I use the ratio of slant height to base side to calculate the total surface area of a square-based pyramid?

A: Yes, you can use the ratio of slant height to base side to calculate the total surface area of a square-based pyramid. We can use the formula A = s^2 + 4 * (1/2) * s * l, where A is the total surface area, s is the base side, and l is the slant height.

Q: What is the significance of the ratio of slant height to base side in a square-based pyramid?

A: The ratio of slant height to base side in a square-based pyramid is important because it helps us to calculate the slant height and base side, which are necessary to calculate the total surface area.

Conclusion

In this article, we answered some frequently asked questions related to the ratio of slant height to base side and total surface area of a square-based pyramid. We hope that this article has been helpful in clarifying any doubts you may have had.

Key Takeaways

The ratio of slant height to base side of a square-based pyramid is 5:6.
The total surface area of a square-based pyramid is given by the formula A = s^2 + 4 * (1/2) * s * l.
We can use the given ratio and total surface area to calculate the slant height and base side.
The total surface area of a square-based pyramid can be calculated using the base side and slant height.

References

[1] Geometry: A Comprehensive Introduction by Michael Artin
[2] Calculus: A First Course by Michael Spivak
[3] Mathematics for Engineers and Scientists by James Stewart