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The only way we can measure a planet's mass is through its gravity. This has been the way Earth's mass was measured, too (we can't directly probe what's in Earth's interior, but we can measure the gravity on the surface). Since nobody ever visited other planets and was able to measure gravity on the spot, we usually have to resort to other methods. The most commonly used technique is to observe a body orbiting or passing close to the planet and see how its path is affected by the planet's gravity. For example, if we see a moon orbiting a planet at certain distance from it, the orbital period of the moon at that particular distance will depend on the planet's mass only. Heavier the planet, stronger it attracts the moon and faster the moon moves. It is straightforward for astronomers to calculate the planet's mass after they have observed the motion of one of its moons for a while. Mercury and Venus have no moons, so their exact mass was not known until a few decades ago. Before spaceflight was developed, the only way to measure their gravity was to see how they affect other planets' orbits. Astronomers would measure very small changes in, say, Earth's orbit, that were caused by the attraction of Venus. These changes are small and it was hard to get the exact mass of Venus by this technique. But once the spacecraft were launched to Venus and they flew close to it, scientists could easily measure its mass by tracking how were these probes attracted while passing by Venus. The same technique was used for Mercury when the Mariner X spacecraft flew by it.
The Earth is the easier problem of the two. Remember that from Newton's Law of Gravitation, (1) Fgrav. = GMm/(R^2) where Fgrav is gravitational force, G is the universal gravitational constant, M and m are the masses of the two objects attracting each other, and R is the distance between their centers of mass. Now from Newton's Second Law, (2) F = ma where a is acceleration, F is force, and m is the mass of the accelerated object. So since we know G, all we have to do is drop an object and measure its acceleration a. Then we know F/m, which is the same as Fgrav/m since our object is moving under the influence of gravity alone. R, the radius of the Earth (the center of mass of a sphere, such as the Earth, is just its geometric center, so R is also the distance between the objects' center of mass) has been known reasonably ever since Eratosthenes of Cyrene did his experiments with the sunlight going down the well at Syene but not at Alexandria at the summer solstice. (The Sun's rays are parallel, so if you know the distance between Syene and Alexandria, and also the angle at which the sun's rays at Alexandria fall on the same date, you can figure out the angle between them and hence the radius of the Earth. Let me know if you need further explanation of this and I can go into more detail-- but try drawing a picture with a circle and the parallel rays and see if you can figure out the geometry.) Another way to measure R is to move around from north to south and obtain your latitudes by measuring the elevation of the North Star above the horizon. If you know how far you have traveled in miles across the surface of the earth, you know the relationship between angular and linear distance, and dividing miles by angle (measured in radians) will give you the Earth's radius in miles. Once you know Fgrav./m, G, and R, you can rearrange equation (1): M = (R^2)*Fgrav / G*m where M is the mass of the Earth, and plug in the numbers. If you did not know G beforehand, you would need to determine it experimentally. The simplest way to do this is through the Cavendish experiment, in which a torsion balance is used to measure the attraction between pairs of lead weights. It actually works, too! The Moon is a much trickier problem. The trouble is, that since in both equations (1) and (2) m appears in the same relation to F, it's not possible to use just those two equations to solve for m (the body being accelerated. Try it! The acceleration just doesn't depend on the mass of the accelerated body.). You can estimate it roughly by assuming that the Moon is just as dense as the Earth and then scaling the mass of the Earth down to the volume of the Moon: Mmoon ~ (Vmoon/Vearth)*Mearth but that will give you a mass which is too high, since it turns out that the Moon is less dense than the Earth! Once we sent spacecraft to orbit the Moon, we could measure the force of the Moon's gravity on them and obtain a really accurate measurement of the Moon's mass in exactly the way we measured the Earth's mass. I believe that the real mass of the Moon was known before then because of precise astronomical measurements (the Earth and the Moon really orbit the center of mass of the joint system, which is inside the Earth but not at its center, and how far out it is depends on the mass of the Moon) but that would be beyond the scope of a high-school explanation.