Truncated Prisms

Summary

The video explains the process of calculating the volume of a truncated triangular prism with varying heights. It demonstrates solving two problems: first, finding the side of the base of a truncated prism with an equilateral triangle base, given the volume and heights. The second problem involves calculating the volume of a truncated triangular prism using Heron's formula to find the area of the triangular base and averaging the heights. The video walks through the formulas step by step, emphasizing the importance of correct calculations in geometry.

Takeaways

• 📐 A truncated prism is discussed in terms of its base, right section, and mean height calculation.
• 📏 The mean height of a truncated prism is the average of multiple height measurements.
• 🔢 Example provided: calculating mean height as (h1 + h2 + h3)/3 for three heights.
• 📊 The formula for the volume of a truncated prism is presented as V = Rm, where R is the area of the base and m is the mean height.
• 🔺 A problem involves finding the side length of an equilateral triangle base for a truncated prism with a volume of 3600 cubic centimeters and varying vertical edges.
• 📐 Using the formula for the area of an equilateral triangle (√3/4 * e^2), the side length is calculated to be approximately 37.22 centimeters.
• 📦 Another problem involves calculating the volume of a right truncated triangular prism with given triangular side lengths and varying heights.
• 📏 The area of the triangular base is calculated using Heron’s formula, with the semi-perimeter s = (x + y + z)/2.
• 💡 The area of the base triangle is calculated to be approximately 4403.9 square centimeters.
• 🧮 The volume of the truncated triangular prism is then calculated by multiplying the area by the mean height, resulting in a volume of 311208.93 cubic centimeters.

Q & A

What is a truncated prism?

A truncated prism is a prism that has had part of its upper section cut off, resulting in a non-parallel top and base. The vertical edges of the prism are no longer equal in length.

How do you find the volume of a truncated prism?

The volume (V) of a truncated prism is calculated by multiplying the area of the base (R) by the average height (M) of the vertical edges. The formula is V = R * M.

What is the average height used in the volume formula?

The average height is calculated by summing the vertical heights (h1, h2, h3, etc.) and dividing by the total number of heights. For example, if there are three heights, M = (h1 + h2 + h3) / 3.

What formula is used to find the area of the base for a triangular prism with an equilateral triangle?

For an equilateral triangle, the area (A) is calculated using the formula A = (sqrt(3)/4) * e², where 'e' is the length of one side of the equilateral triangle.

In the problem where the volume of the truncated prism is 3600 cm³, what is the side length of the triangular base?

The side length of the equilateral triangular base is 37.22 cm, found by solving the area equation with the given volume.

What is Heron’s formula and how is it used here?

Heron’s formula is used to calculate the area of a triangle when the lengths of all three sides are known. The formula is: A = sqrt(s * (s - x) * (s - y) * (s - z)), where 's' is the semi-perimeter and x, y, z are the sides of the triangle.

How do you calculate the semi-perimeter in Heron’s formula?

The semi-perimeter 's' is calculated by summing the lengths of all three sides of the triangle and dividing by two. For example, for a triangle with sides 100, 90, and 120 cm, s = (100 + 90 + 120) / 2 = 155 cm.

What is the area of the triangular base with sides 100 cm, 90 cm, and 120 cm?

Using Heron’s formula, the area of the triangular base is approximately 4403.9 cm².

How is the volume of a right truncated triangular prism calculated in the second problem?

The volume is calculated using the formula V = R * M, where R is the area of the triangular base (4403.9 cm²), and M is the average of the heights 55 cm, 71 cm, and 86 cm. The calculated volume is approximately 311208.93 cm³.

What are the three vertical heights used in the second problem?

The three vertical heights used in the second problem are 55 cm, 71 cm, and 86 cm.