Prism Surface Area Calculator

What shape is the prism's base?

Three base edges and the length. Uses Heron's formula for the triangle. Leave the lateral surface area blank to compute it, or type it in to solve for a missing dimension.

First base edge (a)

Second base edge (b)

Third base edge (c)

Length of the prism (L)

Lateral surface area (LSA)

Total surface area (TSA)

Please input values into the fields

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This Prism Surface Area Calculator can calculate the lateral surface area and total surface area of prisms with common base shapes. It supports triangles, rectangles, squares, pentagons, hexagons, and octagons. If a prism doesn't fit those shapes, you can also compute arbitrary shapes based on the base area and perimeter of the base. After selecting the base shape and entering the dimensions, the tool will show the area of each side combined. It also supports calculating in reverse and shows volume as well.

What is a Prism?

A prism is a geometric solid with two identical, parallel bases connected by flat sides. The two identical bases are called the ends of the prism and the name of the prism depends on the shape of its base. A triangular prism has triangles as its ends while a hexagonal prism has hexagons as its ends. The faces that connect the sides to both ends are called lateral faces. In a right prism, all the lateral faces are rectangles.

Tents, Toblerone chocolate bars, steel I-beam sections, flat pools and standard cartons are all examples of prisms. The common thing is that the cross section is constant. No matter where along its length it is cut, the shape remains the same.

Surface area and total surface area

Since there are two types of surface area to calculate for a prism, you must determine which one you need. The lateral surface area (LSA) is the sum of the areas of the sides ("lateral" means "side") that connect the base(s) of a prism; in other words, the LSA doesn't add in the area of the bases themselves. Imagine if you were trying to wrap a can with a label - only the sides of the can would be covered by the label, not the top and bottom of the can.

The total surface area (TSA) is the sum of all the areas on the outer side of a shape. It can be calculated by adding up the areas of different surfaces. The TSA is used when an object needs to be completely covered, painted or coated. If both ends are open or not included in the calculation then the lateral surface area (LSA) is used instead.

A general formula for any prism:

Regardless of the shape of the base surface, two basic rules apply to all prisms. The lateral area is the product of the perimeter of the base and the length of the prism. This is because the side face can be unfolded into a rectangle. The height of this rectangle corresponds to the length of the prism, while the width corresponds to the perimeter of the base surface.

\text{LSA} = P \times L

To calculate the total area, the areas of two bases are added in addition. Each base has an area of B.

\text{TSA} = P \times L + 2B

Here P is the perimeter of the base, L is the length of the prism and B is the area of a base. For the different shapes described below, just substitute the correct values for the perimeter and area of the base into these two rules. The volume is calculated in a similar way by multiplying the area of the base by the length.

V = B \times L

Formulas for surface area depending on shape of base.

The calculator knows the perimeter and area formulas for various common shapes so you only need to input the length of each side. The formulas used are as follows: where a is the length of a side, L is the length of the prism, and b or h is the base and height of the triangle respectively.

Triangular prism.

For a triangle with sides of length a, b and c the surface area is the product of the sum of the three sides and the length. As the base areas of two triangles can be calculated using Heron's formula, there is no need to measure the height of the triangle separately.

\text{LSA} = (a + b + c)\,L

\text{TSA} = (a + b + c)\,L + \tfrac{1}{2}\sqrt{(a+b+c)(b+c-a)(a+c-b)(a+b-c)}

Rectangular Prism.

A cuboid is a box-shaped figure whose length is l, width is w and the length of the prism is L.

\text{LSA} = 2(l + w)\,L \qquad \text{TSA} = 2(lw + wL + lL)

Square Prism.

A square prism has a square base with side length a. When two of the three sides are equal in length, you get this prismatic shape.

\text{LSA} = 4aL \qquad \text{TSA} = 4aL + 2a^2

Pentagonal Prism, Hexagonal Prism, Octagonal Prism

The area of regular polygons follows the same pattern. The perimeter is the product of the number of sides and the side length, while the area is calculated using the formula for regular polygons.

\text{Pentagon: }\ \text{TSA} = 5aL + \tfrac{1}{2}\sqrt{5(5 + 2\sqrt{5})}\,a^2

\text{Hexagon: }\ \text{TSA} = 6aL + 3\sqrt{3}\,a^2

\text{Octagon: }\ \text{TSA} = 8aL + 4(1 + \sqrt{2})\,a^2

Any Prism

If the base is an irregular polygon shape, a trapezoid or any other measurable shape use the arbitrary prism mode. Enter the area of the base and the perimeter of the base manually and then the calculator will apply general rules directly. This is the core of the tool. The other modes are just shortcuts to arrive at those two values.

A full example

Let's take a rectangular prism as an example. The length of the base is 6 cm, the width is 5 cm and the height of the prism is 7 cm. The lateral surface area is the product of the perimeter of the base and the height.

\text{LSA} = 2(6 + 5)\times 7 = 154\ \text{cm}^2

In addition to the total area, there are two base areas, each of which has an area that is the product of 6 and 5.

\text{TSA} = 2(6\cdot 5 + 5\cdot 7 + 6\cdot 7) = 214\ \text{cm}^2

So the sum of the lateral areas is 154 square centimeters while the sum of the areas of the entire outer surface of the body is 214 square centimeters. This means that the sum of the bases of both ends is 60 square centimeters. The volume is the product of the base area and length, so it's the product of 30 and 7 which leads to 210 cubic centimeters.

Backwards calculation:

Since the lateral area is the product of the perimeter and length, it can also be calculated backwards. If the lateral area and dimensions of the base are known and one side length or the length of the prism is missing, a calculator will calculate that missing dimension for you. Manufacturers of tents who have set the surface area of the canvas they use can use this to figure out how long their tent can be. Manufacturers with sheets of metal of certain dimensions can use it to figure out what length of tubing they can cover with it. Enter in the lateral area and all known dimensions except one, and read off the missing dimension directly from the display.

Summary of all formulas for prisms.

This table summarizes the formulas used by a calculation tool. Each row represents one formula. The letter "P" stands for perimeter of the base, "B" is the area of the base and "L" is the length of the prism.

Quantity	Formula
Lateral surface area	P x L
Total surface area	P x L + 2B
Volume	B x L
Triangular base area (Heron)	one quarter times the square root of (a+b+c)(b+c-a)(a+c-b)(a+b-c)
Square base area	a squared
Regular pentagon base area	one quarter times the square root of 5(5 + 2 root 5) times a squared
Regular hexagon base area	three root three over two, times a squared
Regular octagon base area	2(1 + root 2) times a squared

Some examples of prisms are:

These examples show the surface area and total area for various shapes. All measurements are small and given in centimeters which makes it easy to check by hand calculations.

Shape	Dimensions	LSA	TSA
Triangular	sides 3, 4, 5; length 10	120 sq cm	132 sq cm
Square	side 5; length 7	140 sq cm	190 sq cm
Rectangular	6 by 5; length 7	154 sq cm	214 sq cm
Hexagonal	side 4; length 10	240 sq cm	323.1 sq cm

Applications of the surface of a prism:

The surface area of a prism is used when an object has a constant cross section and needs to be covered or enclosed. Packaging engineers calculate the cost of cardboard required, builders estimate the amount of plaster or lining material needed for walls, and metalworkers determine the dimensions of sheeting required for ventilation ducts or funnels. For labels or wall coverings only the lateral surface area is needed, while for all surfaces exposed to view the total surface area must be calculated.

The surface area also affects the rate of heat transfer, as warm solids lose heat through their outer layer. The performance of heaters, coolers and ice cubes all depend on the size of their outside surface area. For this reason, the ratio of surface area to volume is very useful in many areas of engineering and biology.

Frequently asked questions

What is the formula for surface area of a prism?

All prisms have two common rules. The lateral surface area is the product of the perimeter of the base and the length of the prism, so LSA=P*L. The entire surface area is obtained by adding both base areas in addition. This means that TSA = (P*L) + 2B, where B is the area of a base. For any named shape, from triangular to octagonal prisms, all you need to do is plug the correct perimeter and base area into these two rules.

What is the difference between surface area and total surface area?

The lateral surface area is the area of all the sides of a prism not including the bases. The total surface area is the sum of the lateral surface area and the areas of both bases, or the area of the entire outside of the figure. Lateral surface area is used when you need to find the surface area of an object without the top and bottom surfaces, such as a wrap-around label or an open-ended channel. Total surface area is used when you need to cover the whole object.

How do I find the surface area of a triangular prism?

If you take the sum of the three sides of the base and multiply it by the length, you get the surface area, so the result of (a + b + c) multiplied by L. To the total area then two triangles are added as end faces. The calculation tool determines the area of the triangles using Heron's formula from the three sides, so that there is no need to measure the height of the triangles. You simply select the triangular prism mode and enter the three sides and the length.

Can this calculation tool also start from the surface?

Yes, it is possible. As the surface area has a linear relationship with the perimeter times length, you can convert between them. If the surface area and all dimensions except one are known, this tool will calculate the missing dimension. Enter the surface area and leave either of the side lengths or the length of the prism blank, then the tool will automatically fill in the value.

What to do if the base of the prism is not a regular shape?

Use the general prism mode. Do not select any of the preset shapes but directly enter the base area and perimeter of the base. The calculation tool will then apply general rules. The lateral surface is the product of the circumference and length, and the total surface is the result plus twice the base area. This applies to prisms with trapezoidal bases, prisms with irregular polygons or any other cross-section shape that can be measured.

What units are used to express the surface area of a prism?

The unit for surface area is always a square unit and the unit for volume is always a cubic unit. If you input values in centimeters, your results will be in square centimeters for surface area and cubic centimeters for volume. Switching to inches will change the results accordingly to square inches and cubic inches. The unit for each value can be changed and the calculator will ensure that all units remain consistent.

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Cube Surface Area CalculatorEnter any one of a cube's edge, total or lateral surface area, volume, or space diagonal and get all the others instantly, with full unit conversion.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Prism (geometry)
Definition of a prism, its bases and lateral faces, and right versus oblique prisms.
Math is Fun: Prisms
Plain-language explanation of prism surface area and volume with diagrams.
Wikipedia: Surface area
General treatment of surface area for solids.