Pyramid Surface Area Calculator

Base edge (a)

Height (h)

Slant height (s)

Pick a base shape, then enter any two of the base edge, height, or slant height. The calculator fills in the third and works out the lateral, base, and total surface area.

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This straight cone surface area calculator allows you to calculate the lateral surface area, base surface area and total surface area. The shape of the base can be chosen between a triangle and an octagon, and two out of three given dimensions must be entered: side length, height and slant height. The tool will complete the missing dimension and in addition to each surface area it will give you the volume, distance from center to side, perimeter, edges and all angles of the cone.

What is the surface area of a cone?

The base of a pyramid is a polygon, which is a flat shape. Lines that run from each side of the base to an apex meet at one point and form a series of triangular faces. These triangles are called lateral faces.

The sum of all the lateral faces is called the lateral surface area (LSA). The total surface area is the sum of the LSA and the base area. It represents the amount of material required to cover the entire closed cone.

Since any surface is two-dimensional, the surface of a cone, even though it is a three-dimensional object, is measured in square units such as square centimeters or square feet.

How to use this calculator.

First select the shape of the base. When people think of a pyramid they often mean a tetrahedron like the pyramids at Giza. So this tool starts with a square base but you can change to a triangular, pentagonal, hexagonal or octagonal base at any time.

Enter any two of the following values: side length, height or slant height. Set the unit used. If you enter two values, the tool will instantly calculate the third value and in addition to showing the lateral surface area, base area and total area also show the volume and geometric data for the base.

You do not have to specify which combination of values you entered. If when measuring a pyramid, you can only measure the length of one side and the height, enter those two values. If you have a drawing with the length of one side and the slant line for each face, enter those two values. You can also check the slope of each face as well as the angles at the corners of the triangular sides by activating the "angles" option.

Formula for surface area of a pyramid:

All regular pyramids follow the same two rules regardless of how many sides the base has. The lateral area is the product of the perimeter of the base and half the slant height, while the total surface area is this plus the area of the base.

\text{LSA} = \tfrac{1}{2}\,P\,s \qquad \text{TSA} = \tfrac{1}{2}\,P\,s + B

In this formula, P is the perimeter of the base, s is the slant height and B is the area of the base. If the side length is a and it is a regular n-sided polygon, then the perimeter is the product of n and a. To calculate the area of the base, use the distance from center to side (m). This is the distance from the center of the base to any point in the middle of a side.

P = n\,a \qquad m = \frac{a}{2\tan(\pi/n)} \qquad B = \tfrac{1}{2}\,P\,m

The hypotenuse is an important length that connects these two measures together. The hypotenuse is the longest side of a right triangle, with the other two sides being the distance from the center to one edge and the vertical height. Thus, these three lengths are related by the Pythagorean theorem.

s = \sqrt{m^2 + h^2}

Because of this relationship, the tool can calculate a missing dimension (base, height or hypotenuse) if the other two values are known. In addition to that, it also displays the volume as a result. However, for calculating the volume, the vertical height is used and not the hypotenuse.

V = \tfrac{1}{3}\,B\,h

Surface according to basic shape:

When the number of edges is substituted into the general formula, a simple equation results for regular pyramids. In each row, "a" represents the base length, "h" represents the height and "s" represents the hypotenuse.

Base	Sides	Base area B	Total surface area
Triangle	3	(square root of 3 over 4) a squared	B + 1.5 a s
Square	4	a squared	a squared + 2 a s
Pentagon	5	about 1.720 a squared	B + 2.5 a s
Hexagon	6	(3 square root of 3 over 2) a squared	B + 3 a s
Octagon	8	about 4.828 a squared	B + 4 a s

Since the regular square pyramid is much more often searched for than other pyramids, it makes sense to state its formula separately. The total area of a regular square pyramid with base side 'a' and height 'h' is:

A = a^2 + a\sqrt{a^2 + 4h^2}

Hypotenuse, height, side edge: three different lengths.

A common mistake when looking at pyramids is to confuse the slant height with the vertical height. The vertical height runs along a straight line inside and extends from the center of the base to the top. The slant height goes outside, lies on the midline of the triangular sides, and extends from the top down to the middle of the base side.

Since the surface covers the area we use the slant height for the lateral faces. Since volume is a measure contained inside we use the vertical height. The slant height is always longer than the vertical height and even if only one height is known tools will usually calculate the slant height so there should be no confusion.

Another important line is the side edge. This is the slanted edge where two triangular faces meet and extends from the tip to one of the base corners. Because it connects the center of the side with a corner, it's longer than the slant height, and tools also calculate the side edges.

What is the distance to the center of the edge? Why is this important?

The distance from the center of an edge to the base is the distance from the center of any given edge of the polygon to its center. For a square base this is half the length of a side, while for pentagons or hexagons it requires calculation with trigonometric functions. This is why the general formula contains a tangent term.

The distance to the center of edge is an important geometrical value for pyramids. The distance to the center of edge determines the base area and in combination with the height also the slant height. If the distance to the center of edge is correct, all surfaces can be calculated without problems.

How steep is a pyramid's slope?

When the Show Angle option is enabled, it displays the dihedral angle between the faces and the base. For a shallow pyramid this will be small, while for a tall pointy one it will approach 90 degrees.

The face angles of the Great Pyramid at Giza are about 51.8 degrees with builders maintaining a surprisingly consistent slope for all four sides. The tool also shows the vertex angle of each side face, which is the angle at the top point of the side. This gives an idea of how sharp or blunt the side is.

A full example:

Consider a regular square pyramid with base side length of 6 cm and height of 4 cm. The distance from the center of the square base to one of its sides is half the side length, or 3 cm. First we calculate the slant height using the Pythagorean theorem.

s = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \ \text{cm}

Since the base area is four sides, the lateral surface area is equal to the perimeter multiplied by half of the slant height.

\text{LSA} = \tfrac{1}{2}\times 24 \times 5 = 60 \ \text{cm}^2

The base is a square so the area is equal to the side length squared. If you add this to the lateral surface area then you get the total surface area.

A = 6^2 + 60 = 36 + 60 = 96 \ \text{cm}^2

An example of a pyramid with a pentagon as its base:

Consider a pyramid with a pentagon base where the base side is 10 centimeters and the height is 15 centimeters. The distance from the center of a regular pentagon to its sides is about seven centimeters, so the slant height will be approximately 16.5 centimeters.

\text{LSA} = \tfrac{1}{2}\times 50 \times 16.5 \approx 412.58 \ \text{cm}^2

A \approx 172.05 + 412.58 \approx 584.63 \ \text{cm}^2

Pyramid formulas summary:

This table summarizes the individual relationships used for pyramids with regular bases. The base has n sides each of length a, the height is h and the slant height is s.

Quantity	Symbol	Formula
Base apothem	m	m = a divided by (2 tan(pi over n))
Base perimeter	P	P = n a
Base area	B	B = one half P m
Slant height	s	s = square root of (m squared + h squared)
Lateral surface area	LSA	LSA = one half P s
Total surface area	A	A = one half P s + B
Volume	V	V = one third B h

The surface of some pyramids.

These regular tetrahedrons all have a base area of 5 centimeters. It turns out that with the same base area, the surface area of a pyramid becomes larger as it is taller. The values are rounded to two decimal places.

Base edge	Height	Slant height	Total surface area
5 cm	5 cm	5.59 cm	80.90 sq cm
5 cm	10 cm	10.31 cm	128.08 sq cm
5 cm	12 cm	12.26 cm	147.58 sq cm
5 cm	20 cm	20.16 cm	226.56 sq cm

Applications of the surface area of a pyramid:

If you look closely, you can find pyramids all around us. The most famous examples are the stone pyramids of Egypt and Central America, but also the glass pyramid of the Louvre, pointed roofs of temples and towers as well as tents stretched out are pyramid bodies.

When the lateral surface area is known, a manufacturer can calculate how much material (such as wooden planks, canvas or foil) will be required to cover the sloping sides. This serves as the basis for making tents, pointed roofs or decorative gift boxes. For closed pyramids, the total surface area is important because it indicates how much material is needed to coat them, while the volume indicates how much capacity there is inside.

Tips for accurate results:

Measure the base along a full side of the polygon and check before entering if the length you get is the vertical height or diagonal height. These are different lines. Use the same unit for all measurements.

Remember that surface area increases proportionally to the square of the dimensions, so small errors in length can lead to large errors in area. Also make sure that the base is actually a regular polygon with equal side lengths. These formulas assume that the vertices are exactly above the center of the base.

Frequently asked questions

What is the difference between the lateral area and total surface area of a cone?

The lateral surface area is the area of the triangular sides only, calculated by multiplying the perimeter of the base by half the slant height (apothem). The total surface area includes the area of the polygonal base in addition. If the base is open (such as a tent), use the lateral surface area; if the cone is closed, use the total surface area.

What value do you use to calculate the surface area of a cone: height or apothem?

Use the apothem. The lateral area is calculated by multiplying the perimeter of the base by half the height (the apothem), where the apothem runs along the middle of the triangular sides. The vertical height is only used to calculate volume. If the height is known but the apothem is not, this tool will first calculate the apothem from the sum of the square of the distance from the center of a side to the center and the square of the height.

How do you calculate the height of a pyramid's side?

Since the slant height, distance to center of base and vertical height form a right triangle, the slant height is equal to the square root of the sum of the squares of the distance to center of base and the vertical height. For a regular square pyramid, the distance to center of base is half the length of one side of the base. So for a regular square pyramid with a base of 6 by 4, the slant height would be equal to the square root of the sum of 3 squared and 4 squared, which is 5. If you enter any two values into the tool, it will calculate the remaining data.

What is the distance to the center of the base side? Why is it required for the tool?

The distance from the center to a side is the distance from the center of the polygonal base to any point on the midpoint of one of its sides. The distance from the center to a side determines the area of the base, and in combination with the height also the lateral surface area. For square bases it corresponds to half the length of the side, while for pentagons or hexagons trigonometric functions are used. This is why the general formula contains the term tan(pi divided by n).

Can this calculation tool also calculate pyramids with triangular, pentagonal or hexagonal bases?

Yes. By selecting the shape of the base above, the tool can be used for triangles, squares, pentagons, hexagons and octagons. The calculation process which starts with two arbitrary input values is applied to all these shapes. The tool is designed for regular pyramids, that is pyramids where the base is a regular polygon with equal length sides and the apex is directly above the center of the base.

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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: Pyramid (geometry)
Definitions of the base, apex, lateral faces, slant height, and surface area of a pyramid.
MathWorld: Pyramid
Formal treatment of the regular right pyramid, its surface area, and its volume.
Math is Fun: Pyramids
Plain-language explanation of a pyramid's parts, surface area, and volume.