# Triangular Prism Volume & Surface Area

A triangular prism is a solid shape with two parallel triangular faces of equal size and shape, and three parallelogram faces that connect the sides of the triangles. If you have a right triangular prism, the parallelograms are rectangles, which is the simplest case and the one we deal with in this geometry tutorial. As with any polyhedron, you can calculate the volume and surface area of a triangular prism if you know the dimensions and measurements of the object. See also these other prism geometry guides

## Volume of a Triangular Prism

The triangular prism volume equation is a function of the prism's length L, and the sides lengths of the triangular faces A, B, and C. The volume is equal to the area of the triangle times the length of the prism. If you know A, B, and C, you can use Heron's formula for the area. This is

Area =

0.25*sqrt[2*((AB)^2 + (AC)^2 + (BC)^2) - (A^4 + B^4 + C^4)]

Therefore, the triangular prism volume formula is

Volume =

L*0.25*sqrt[2*((AB)^2 + (AC)^2 + (BC)^2) - (A^4 + B^4 + C^4)]

There are other ways to find the area of a triangle besides Heron's formula. For example, if you know the base length of the triangle B and the triangle's height H (measured perpendicular to the base) then the area is 0.5*B*H. Alternatively, if you know the lengths of two sides A and B and the angle θ between the sides, then the area is 0.5*A*B*sin(θ).

## Surface Area of a Triangular Prism

The triangular prism surface area equation is equal to the sum of the areas of the five faces. There are two triangular faces of equal area and three rectangular faces of equal length. If the length of the triangular prism is L and the side lengths of the triangle are A, B, and C, then the surface area is given by the formula

Surface Area =

0.5*sqrt[2((AB)^2 + (AC)^2 + (BC)^2) - (A^4 + B^4 + C^4)] + L(A + B + C)

How is this formula derived? The first part of the expression with the square root is twice the area of a triangle with sides A, B, and C, i.e., double Heron's formula. The second part is L*A + L*B + L*C, which is the sum of the three rectangular areas on the side of the prism.

## Example of How to Calculate Triangular Prism Volume

A gift box in the shape of a right triangular prism has a length of 8 inches. The triangular faces are equilateral triangles with a side length of 3.5 inches. The box is to be filled with uncooked rice grains. If 1 cup = 14.4375 cubic inches, how many cups of rice can be box hold?

First, the area of one of the triangular faces is [sqrt(3)/4]*3.5^2 = 5.3044 square inches. Multiplying this area by 8 gives us 5.3044*8 = 42.4352 cubic inches as the volume of the triangular prism.

Since 1 cup = 14.4375 cubic inches, the box can hold 42.4352/14.4375 = 2.9392 cups or rice, or roughly 3 cups.

## Example of How to Calculate Triangular Prism Surface Area

Suppose you want to construct a box in the shape of a triangular prism such that the triangular ends are right triangles with leg lengths of 9 cm and 12 cm, and such that the length of the box is 12 cm. How many square centimeters of paper or cardboard do you need? Can all five pieces be cut from a rectangle with no waste?

This is a problem for calculating the surface area of a triangular prism. First, the area of each triangular face is (1/2)(9)(12) = 54, and the sum of both areas is 108 cm^2.

Next, if the triangular ends are right triangles with legs of 9 and 12, then the hypotenuse is sqrt(9^2 + 12^2) = 15. Since the length of the box is 12 cm, this means the total area of the three rectangular faces is 12*9 + 12*12 + 12*15 = 432 cm^2

Therefore, the total surface area of the triangular prism box is 108 + 432 = 540 cm^2. The following diagram shows how the pieces of this box can be cut from a single rectangular sheet.

## Example 3

A triangular prism has triangular faces with side lengths of 13, 14, and 15 cm. What must the length of the box be so that the surface area in cm^2 equals the volume in cm^3?

This is an example of a 'surface area = volume' algebra problem. To solve it, we need to find expressions for the surface area and volume in terms of the unknown value L, set them equal to each other, and solve for L.

First, the area of each triangular face is 84 cm^2 using Heron's formula, so the total of the triangular faces is 168. The perimeter of the triangle is 42, so the sum of the rectangular faces is 42L. Putting this information together gives a surface area of 168 + 42L.

Next, since the area of the triangular faces is 84, we have volume = 84L

Equating the volume and surface area gives us the equation

168 + 42L = 84L

168 = 42L

L = 168/42

L = 4

Therefore, if the triangular prism has a length of 4 cm, the surface area will be equal to 336 cm^2 and the volume will equal 336 cm^3.

## Geometry with Calculus: Minimize the Surface Area and Maximize the Volume of a Triangular Prism

Here is a standard geometry optimization problem, such as you might encounter in manufacturing. This problem has two parts.

(a) A triangular prism with equilateral triangle faces has a volume of 6√3. What dimensions of the prism will minimize its surface area?

(b) A triangular prism with equilateral triangle faces has a surface area of 18√3. What dimensions of the prism will maximize the volume?

(c) What do the answers to these two questions tell you about the ideal ratio between the side length of the triangular face and the length of the prism?

Let us call the side length of the triangular face x and the length of the prism y. The surface area and volume formulas in terms of x and y are

V = (sqrt(3)/4)yx^2

S = (sqrt(3)/2)x^2 + 3xy

**For part (a)**, we have 6*sqrt(3) = (sqrt(3)/4)yx^2, which gives us y = 24/x^2. If we plug this expression for y into the surface area equation, we get S = (sqrt(3)/2)x^2 + 72/x. To find the value of x that minimizes the surface area, we take the derivative of S with respect to x, set it equal to zero, and solve for x. This gives us

S' = sqrt(3)x - 72/x^2

0 = sqrt(3)x - 72/x^2

x^3 = 72/sqrt(3)

x = 2*sqrt(3)

Using the fact that y = 24/x^2, we have y = 2. Therefore, the optimal equilateral triangular prism with a volume of 6*sqrt(3) is one whose length is 2 and whose triangular side length is 2*sqrt(3). These dimensions yield the smallest surface area.

**For part (b)**, we have 18*sqrt(3) = (sqrt(3)/2)x^2 + 3xy, which gives us y = [108 - 3x^2]/(6*sqrt(3)x). Plugging this expression for y into the volume formula gives us V = (9/2)x - (1/8)x^3. To maximize the volume, we take the derivative of V with respect to x, set the derivative equal to zero, and solve for x. This gives us

V' = 9/2 - (3/8)x^2

0 = 9/2 - (3/8)x^2

x^2 = 72/6

x = 2*sqrt(3)

Using the relation y = (108 - 3x^2)/(6*sqrt(3)x) gives us y = 2. Therefore, the optimal equilateral triangular prism with a surface area of 18*sqrt(3) is one whose length is 2 and whose triangular edge length is 2*sqrt(3).

**As for part (c)**, looking at the two solutions above, we can see that the optimal equilateral triangular prism has dimensions x and y such that the ratio of x to y is sqrt(3) to 1.