How much is one cubic centimeter of gold worth?

Esitmating with a ring

The plan is to estimate the value of gold with the price of a gold ring. This works well because gold is the most expensive part of the ring, and strongly influences the price of the cheapest rings, but the purity is measured and reported. I've seen 14 carat gold rings for around $300, which I'm going to use as a baseline.

Here's the plan for calculating it:

graph BT; A[Volume of Gold in Ring] -->|-1| B[Cost/Volume of Gold] I[Cost of Gold in Ring] --> B

Volume of Gold

First we estimate the area of the annulus, and then the height to get the ring volume. Finally we estimate the purity of the ring to get the volume of gold.

graph BT; A[Volume of Gold in Ring] D[Area of Ring] --> F E[Height of Ring] --> F F[Volume of Ring] --> A G[% of Ring that is Gold] --> A

The area of an annulus with inner radius r and outer radius R is \( \pi (R^2 - r^2) \). My ring finger is about 1cm across, so has a radius of about 5mm. A ring is typically around 2mm thick. So the area of the annulus, looking at the ring top down, is \( \pi \left((5+2 \rm{mm})^2 - (5 \rm{mm})^2\right) \approx 3 \times (49 - 25)\rm{mm}^2 \), so approximately 75 mm2.

The height of a ring is roughly twice it's thicknesss; call it 4mm. So the volume of the ring is about 300 mm3.

Now a 14 carat gold is 14/24 gold by weight, the rest being other cheaper harder metals. Let's assume that the metals are similar density to gold, so that by volume of gold is about 14/24, say roughly half the volume of the ring. So the volume of gold in the ring is about 150mm3, which is 0.15 cm3.

Putting this calculation into a graph:

graph BT; A[Volume of Gold in Ring
0.15 cm3] D[Area of Ring
0.75 cm2] --> F J[Inner Area of Ring
0.75cm2] -. -1 .-> D K[Outer Area of Ring
1.5cm2] -.-> D L[Radius of finger
0.5 cm] -->|2| J M[Radius of finger and ring
0.7cm] -->|2| K N[π] --> J O[π] --> K E[Height of Ring
0.4 cm] --> F F[Volume of Ring
0.3 cm3] --> A G[% of Ring that is Gold
50%] --> A

Estimating Value of Gold

We know the 14 carat gold ring costs around $300. I think most of the cost of the cheaper rings is going to be the gold itself; the material filler is quite cheap, but there's some payment for craftsmanship and margins. Let's assume 80% of the total cost is gold. Then the cost of gold is 80% of $300, which is about $250.

graph BT; C[Cost of Ring
$300] --> I[Cost of Gold
$250] H[% of Ring Cost is Gold
80%] --> I

Putting it together

Combining our two estimates gives $250 for 0.15 cm3 of gold, so about $2000 per cubic centimeter.

graph BT; A[Volume of Gold in Ring
0.15 cm3] -->|-1| B[Cost/Volume of Gold
$2000 cm-3] I[Cost of Gold in Ring
$250] --> B

Checking

Whenever we do these kinds of estimates it's always good to check it where possible to see whether we ended up in the right place. Using that a 14 carat ring is $300 is a great way of using pedestrian knowledge to calculate the value of gold. But I can look up buying gold bars directly.

I can see a gold bar online of 255mm by 85mm by 35mm for just over a million dollars. The volume of this bar is about 25cm by 8cm by 4cm which is about 800 cubic centimeters. So my estimated cost would be $2000/cubic centimeter by 800 cubic centimeters which is $1.6 million.

So it looks like I overestimated the value by about 60%, it's closer to $1300 per cubic centimeter. But this is a pretty good outcome for such crude assumptions.