E = mc² Calculator

Mass (m)

Energy (E)

Speed of light (c)

Include motion (relativistic energy)

E = m c^2 gives the REST energy. Switch this on to add a velocity and see the total energy, kinetic energy, momentum, and Lorentz factor for a moving body.

Enter a mass to get its rest energy, or enter an energy to get the equivalent mass. The speed of light is already filled in. Switch the energy unit to read the answer in megatons of TNT, kilowatt-hours, food calories, or electron-volts.

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This E=mc² calculator allows you to convert mass into equivalent energy or vice versa using Einstein's most famous equation. Enter a mass and find out the amount of rest energy it contains, or enter an amount of energy and calculate the corresponding mass.

The speed of light is preset, and results are shown in joules, megatons (TNT equivalent), kilowatt hours, calories, and electron volts.

What does E = mc2 mean?

E = mc2 is an equation formulated by Albert Einstein in 1905 as part of his special theory of relativity that describes the equivalence between mass and energy. It states that mass and energy are two different forms of the same phenomenon. The energy contained within a stationary object is equal to its mass multiplied by the speed of light squared, which is referred to as rest energy.

This equation links the mass of everyday objects to the huge energies involved in nuclear physics. As the speed of light squared is a very large number, even a small amount of mass corresponds to an enormous amount of energy. This simple concept forms the basis for nuclear power, stellar energy and even reactions between matter and antimatter used in medical imaging.

The formula of E = mc2:

The rest energy of a given mass is equal to that mass multiplied by the square of the speed of light.

E = m c^2

Here, E is energy, m is mass and c is the speed of light in a vacuum. The speed of light is a defined constant.

c = 299{,}792{,}458\ \tfrac{\text{m}}{\text{s}} \quad\Rightarrow\quad c^2 \approx 8.988 \times 10^{16}\ \tfrac{\text{m}^2}{\text{s}^2}

If the mass unit is kilograms and the speed of light unit is meters per second, then the calculated energy will be in joules. This calculator does conversions so even if you mix grams, megatons (TNT equivalent), or electron volts, consistent results are achieved.

To use this calculator:

Enter the known values and leave the unknown fields blank. To calculate the energy corresponding to a given mass, enter the mass and leave the field for energy blank. To calculate the mass corresponding to a given energy, enter the energy and leave the field for mass blank.

You do not need to select a mode here as the calculator will detect what is to be calculated from the empty fields. The speed of light is entered by default as a fixed value, so you can usually leave it unchanged. If necessary, you can delete the value for the speed of light and allow the calculator to calculate the speed of light from known values of energy or mass.

Calculating mass or light speed.

E=mc2 is an equation that contains three variables and can therefore be manipulated to solve for any one of them. A calculator internally uses the following form:

m = \frac{E}{c^2} \qquad c = \sqrt{\frac{E}{m}}

Thus the converted mass can be calculated if the released energy and the speed of light are known. Physicists use this to determine the mass deficits from the energy released in nuclear reactions.

Transformations of E = mc2

This table shows the same equation in three different ways, each form corresponding to a possibly unknown quantity.

To find	Formula	In words
Energy	E = m c squared	Mass times the speed of light squared
Mass	m = E / c squared	Energy divided by the speed of light squared
Speed of light	c = square root of (E / m)	Square root of energy over mass

Example of a calculation:

How much energy is contained in one kilograms of matter? Set the mass to 1 kg and the speed of light to 299,792,458 meters per second.

E = (1)\,(299{,}792{,}458)^2 \approx 8.987551 \times 10^{16}\ \text{J}

This is equivalent to about 89 petajoules, or about 21 megatons of TNT. This means that from just 1 kilograms of matter you could get the energy of about 1,440 Hiroshima-type atomic bombs. Of course a matter-antimatter annihilation would have to take place in order for all this energy to be released. Even if a nuclear reactor were able to convert only a tiny fraction (say one thousandth) of the mass into energy, it would still be millions of times more than burning the same amount of coal.

Why light speed is squared.

Squaring c leads to a very large value for energy but that is not because mass is moving at the speed of light. The basis for this is the geometric relationship between space and time in relativity theory. From dimensional analysis, energy equals mass times the square of velocity, and since the speed of light is the natural rate that connects space and time it appears in the formula.

When you square such large numbers, the result is a number with about 9 followed by 16 zeros. So even the mass of one paper clip contains enough energy for a huge bomb. This also explains why the change in mass that happens during normal chemical reactions is so small it can't be measured - only a tiny fraction of the energy is released.

Mass loss and nuclear energy:

In nuclear reactions the mass of the products is slightly less than that of the reactants. This lost mass, called mass defect, is carried away as energy and its value exactly matches what is predicted by the equation E = mc2. In fission, heavy nuclei like uranium (e.g., uranium-235) convert about one-thousandth of their mass into energy. In fusion, light nuclei such as hydrogen convert about seven thousandths of their mass into energy, powering the Sun.

The Sun converts about 4 million tons of matter into energy every second, which is the source of its light and heat. However, since this only represents a very small fraction of the total mass of the Sun, it will continue to shine for billions more years.

Energy of moving masses (relativistic consideration)

The equation E = mc2 describes the rest energy, that is, the energy contained in a mass at rest. When the motion option is selected, the calculator adds the energy of the moving mass. As speed increases, a factor called the Lorentz factor increases and the total energy is the product of the Lorentz factor and the rest energy.

\gamma = \frac{1}{\sqrt{1 - (v/c)^2}} \qquad E_{\text{total}} = \gamma\, m c^2

Kinetic energy is the difference between total and rest energy. As speed approaches the speed of light, both the Lorentz factor and the energy approach infinity. This explains why objects with mass can never reach the speed of light. The curves for energy and velocity in the calculator show this steep rise corresponding to the entered mass.

Applications of E = mc2

The equivalence of mass and energy is a fundamental concept in modern physics and technology. This relationship determines the power output of nuclear reactors and the explosive force of atomic weapons. It also explains how stars shine, as hydrogen fuses to helium releasing a small amount of mass in the form of light.

In particle accelerators one can predict the amount of energy needed to create new particles and also the amount of energy released when a particle meets its antiparticle. Positron emission tomography (PET scan) is based on exactly this principle. Electrons and positrons mutually destroy each other, turning their mass into gamma rays that can be detected by a scanner.

Can you turn energy into mass?

Yes, this equation is true both ways around. In pair production, a high-energy gamma ray passing near an atomic nucleus can disappear and create an electron and positron. Pure energy has thus been converted into matter. This phenomenon occurs frequently in particle physics experiments, and is also the reverse of the annihilation process in PET scans.

The law of conservation of energy always holds true. Mass is just one of several forms that energy can take on, so when there's a conversion between the two, no net energy is gained or lost.

Common mistakes:

There are some common misconceptions. One is that any object can easily convert its mass into energy. In chemical reactions only a negligible amount of mass is converted, while in nuclear reactions and subatomic processes a measurable fraction of the energy is released. Another frequent mistake is forgetting to square the speed of light. The energy is not proportional to c but to c2.

The third point is the confusion of units. To get Joules directly, mass must be given in kilograms and speed of light in meters per second. This calculator avoids this error by converting all selected units to consistent base units before performing calculations.

For accurate results, please follow these tips:

Watch out for powers of ten. Here the energy term is often very large while mass is usually small. Leave the unit conversions to the calculator and do not perform any manual conversions. If it helps you understand the numbers better, then switch to the appropriate units of energy when converting to megatons (TNT equivalent) or electron volts. Leave the default value for the speed of light unless you want to change it on purpose.

Frequently asked questions

What does E=mc2 mean?

E = mc² is the equivalence relationship between mass and energy proposed by Einstein in his 1905 paper on special relativity. E stands for energy, m for mass, c for the speed of light, which is 299,792,458 meters per second. It shows that mass and energy can be converted into each other. Since the square of the speed of light is a very large number, a small amount of mass corresponds to an enormous amount of energy.

How much energy is contained in a kilogram of mass?

One kilogram of matter contains about the number of joules calculated by multiplying 8.988 with ten to the power of 16, which is approximately 90 petajoules of energy. This corresponds to about 21.5 megatons of TNT and is the amount of energy from about 1,400 Hiroshima-type atomic bombs. To release all the energy, a matter-antimatter annihilation would be required, while in nuclear fission only a very small fraction of the mass is converted into energy, namely only about one thousandth.

How do you calculate mass or speed of light when given energy?

Enter the known values and leave the unknown fields blank. To calculate mass, the calculator uses the formula m = E / c^2. To calculate speed of light, the calculator uses the formula c = √(E/m). You do not need to select a mode; which column you leave blank indicates what you want to solve for.

Why is the speed of light squared in the equation E = mc^2?

The squaring of the speed of light comes from the relationship between space and time in special relativity, not that mass moves at the speed of light. From a dimensional point of view energy is equal to mass times the square of velocity, and since the speed of light is the natural rate that connects space and time it appears in the equation. It's the squaring of this large number that makes even a small amount of mass correspond to an enormous amount of energy.

Does E = mc2 apply to nuclear reactions?

This is the case for nuclear energy. In fission, a heavy atom such as uranium-235 splits and about one thousandth of its mass is converted into energy. In fusion, light atoms like hydrogen combine and about seven thousandths of their mass is converted to energy. The mass difference, that is, the difference between the masses of reactants and products, equals the released energy divided by c squared (the speed of light).

Can energy be converted back to mass?

Yes. In pair production, a high-energy gamma ray near an atomic nucleus creates an electron and positron, converting energy into matter. This is the reverse process of annihilation. Particles and their antiparticles convert mass back to energy, and this effect is used in PET imaging.

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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: Mass-energy equivalence
Definition, history, the rest-energy and relativistic forms, and the role of the mass defect in nuclear reactions.
NIST CODATA: speed of light in vacuum
The exact defined value of the speed of light, c = 299,792,458 m/s.
Khan Academy: mass-energy equivalence
Plain-language lessons on special relativity, rest energy, and the Lorentz factor.