Volume of a triangular prism – what is it? How to understand how much space this figure takes up? It’s simple: this is the total volume occupied by the prism in three dimensions.
Everyone knows that the volume of any prism can be found by multiplying the area of the base by the height of the prism. And what do we know about the triangular prism? It has two triangular bases. Area of a triangle? Easy: Half the length of the base multiplied by the height.
So, in this post, we will consider how to calculate the volume of a triangular prism using formulas. And, of course, we will solve several practical problems. Ready to learn more? Let’s get started!
How to Calculate the Volume of a Triangular Prism: Practical Formulas
Alright, so you want to find the volume of a triangular prism? You do it by multiplying the area of its base by its height. Since the base is a triangle, we first need to figure out the area of that triangle.
We know the area of a triangle is calculated by multiplying the base length by the height and then dividing by two. So, the formula for finding the volume of a triangular prism looks like this:
What If We Don’t Know the Height of the Base?
Sometimes, you might not know the height of the triangle’s base. But don’t worry! You can still calculate the volume of a triangular prism. Let’s explore a few methods depending on the type of triangle and the information you have.
- Equilateral Triangle: If the base is an equilateral triangle, use this formula: SABC=(√3⋅AB2)/4;
- Isosceles Triangle: If the base is isosceles (where two sides are equal), use: SABC=(AC/4) ⋅√(4⋅AB2-AC2);
- General Triangle: If it’s any triangle and you know all three sides, Heron’s formula comes in handy: SABC=√(s⋅(s-AB)⋅(s-BC)⋅(s-AC)), where s is the semi-perimeter s=(AB+BC+AC)/2;
- Two Sides and Included Angle: If you know two sides and the angle between them: SABC=(1/2)⋅AB⋅AC⋅sin(α).
Once you have the area of the base, you can find the volume by multiplying the area by the height of the prism.
Note: if we denote the sides of the base, the height of the base, and the height of the triangular prism with the letters a, b, c, h, and l, respectively, the volume formulas will be rewritten in a more familiar letter form:
Calculation of the Volume of a Triangular Prism: Examples and Answers
Let’s solve some examples to put these formulas to work. Each example has a solution, but try to solve them yourself before looking at the answers!
Example 1: A triangular prism has a height of 4 cm, and its triangular base has a side of 3 cm and a height drawn to it of 5 cm. What is the volume of the triangular prism?
So, using the formula for the volume with the values h=5, c=3 and l=4, we get:
The volume of a triangular prism is 30 cm3.
Example 2: Find the volume of a triangular prism with a base height of 4 cm, a side of 3 cm and a prism height of 6 cm
In this case, the height of the prism is 6 cm, the side of the base is 3 cm, and the height drawn to it is 4 cm. Substituting these values into the volume formula, we get:
Thus, the volume of a triangular prism is 36 cm3.
Example 3: Determine the volume of a triangular prism in which the base of the triangle is 7 cm, the height is 10 cm, and the height of the prism is 5 cm
By convention, we have h=10, c=7 and l=5. Using the formula for the volume of a triangular prism with this value, we get:
Thus, the volume of a triangular prism is 175 cm3.
Example 4: The volume of a triangular prism is 510 cm3, and its triangular base has a side of 3 cm and a height drawn to it of 4 cm. What is the length of the height of the prism?
In this case, knowing the volume, we must find the height of the prism. So, using the same formula, we substitute the given values and find l:
Hence, the height of the prism is 85 cm.
Example 5: Two sides of the base of a triangular prism and the angle between them are equal to 4 cm, 5 cm and 30 degrees, respectively. What is the volume of the prism if its height is 10 cm?
So, by convention, we have that a=4, c=5, α=30° and l=10. Therefore, using the prism volume formula with this value, we get:
Thus, the volume of a triangular prism is 50 cm3.
Deeper into Triangular Prism Geometry: All the Important Aspects!
Curious to learn more? Here are some additional topics you might find interesting:
- What is a Triangular Prism? – Everything you need to know about triangular prisms.
- Surface Area of a Triangular Prism – Discover how to calculate the surface area using different formulas.
Programming the Volume of a Triangular Prism: A Fun Project
If you’re into programming, why not write a program to calculate the volume of a triangular prism? Use a flowchart as your guide and combine your coding skills with geometry knowledge. It’s a great way to apply what you’ve learned in a practical and fun way!
Ready to give it a try? Start coding and see how you can automate these calculations!