How do i find the volume of a triangular prism

Triangular Prism Shape

a = side length a
b = side length b = bottom triangle base b
c = side length c
h = height of prism
H = height of bottom triangle
V = volume
Atot = total surface area = all sides
Alat = lateral surface area = all rectangular sides
Atop = top surface area = top triangle
Abot = bottom surface area = bottom triangle

A triangular prism is a geometric solid shape with a triangle as its base. It's a three-sided prism where the base and top are equal triangles and the remaining 3 sides are rectangles.

Calculator Use

This calculator finds the volume, surface area and height of a triangular prism. Surface area calculations include top, bottom, lateral sides and total surface area. Height is calculated from known volume or lateral surface area.

Units: Units are shown for convenience but do not affect calculations. Answers will be the same whether in feet, ft2, ft3, or meters, m2, m3, or any other unit measure.

Significant Figures: Choose the number of significant figures or leave on auto to let the calculator determine number precision.

Triangular Prism Formulas in terms of height and triangle side lengths a, b and c:

Volume of a Triangular Prism Formula

Finds the 3-dimensional space occupied by a triangular prism.

\[ V = \dfrac{1}{4}h \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]

\[ V = \dfrac{1}{4}h \sqrt{(c+a-b)(a+b-c)} \\\times \sqrt{(a+b+c)(b+c-a)} \]

Top Surface Area of a Triangular Prism Formula

Finds the area contained by the triangular surface at the top of the prism. This is the same area as the bottom surface area.

\[ A_{top} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]

\[ A_{top} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]

Bottom Surface Area of a Triangular Prism Formula

Finds the area contained by the triangular surface at the bottom of the prism. This is the same area as the top surface area.

\[ A_{bot} = \dfrac{1}{4} \sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)} \]

\[ A_{bot} = \dfrac{1}{4} \sqrt{\begin{aligned}(a+&b+c)(b+c-a)\\&\times(c+a-b)(a+b-c)\end{aligned}} \]

Lateral Surface Area of a Triangular Prism Formula

Finds the total area contained by the three rectangular sides of the prism. You can think of the lateral surface area as the total surface area of the prism minus the two triangular areas at the top and bottom of the prism.

\[ A_{lat} = h (a+b+c) \]

Total Surface Area of a Triangular Prism Formula

Finds the total area of all sides of a triangular prism. Total surface area of a prism includes the area of the top and bottom triangle sides of the prism, plus the area of all 3 rectangular sides.

\[ A_{tot} = A_{top} + A_{bot} + A_{lat} \]

Height of a Triangular Prism Formula in Terms of Volume

Finds the height of a triangular prism by solving the Volume Formula for height. Height, h, is calculated from volume, V, and side lengths a, b and c.

\[ h = \dfrac{4V}{\sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}} \]

\[ h = 4V \div \left[ \, \sqrt{(c+a-b)(a+b-c)} \\\times \sqrt{(a+b+c)(b+c-a)} \, \right] \]

Height of a Triangular Prism Formula in Terms of Lateral Surface Area

Finds the height of a triangular prism by solving the Lateral Surface Area Formula for height. Height, h, is calculated from lateral surface area, Alat, and side lengths a, b and c.

\[ h = \dfrac{A_{lat}}{(a+b+c)} \]

Reference

Weisstein, Eric W. "Triangle Area." From MathWorld--A Wolfram Web Resource, Triangle Area.

Video Transcript

Determine the volume of the given triangular prism.

The volume of a triangular prism can be found by multiplying the base times the height, where the shaded pink portion represents the base. The volume is then the area of the base multiplied by the height. And the green portion represents the height. It’s the distance from one base to the other.

And how do we go about finding the area of the base? Well, like any triangle, we multiply one-half, the base of that triangle, times the height of that triangle. So we see, in our case, the base of our triangle is 10 feet and the height of our triangle is six feet. And the height of our triangular prism is 17 feet.

Let’s start plugging things into our formula. For our base, our triangle, one-half times six times 10. Then we multiply the area of our base by the height of our prism, 17 feet. Now we simply multiply. six times 10 times one-half equals 30. And 30 times 17 equals 510.

But we’re not finished here because we need to decide what to do with our units. When we multiply feet by feet by feet and when we’re discussing volume, we know that our units will be cubed. That’s our final answer: The volume of the given triangular prism equals 510 feet cubed.

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