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Here's the detailed breakdown with the mathematical formulas (Non-SegWit):

Single-Sig Transaction Size: A single-sig transaction with 1 input and 2 outputs is approximately 226 bytes.

2-of-2 Multisig Transaction Size: A 2-of-2 multisig transaction with similar characteristics is about 334 bytes.
To compare it to the single-sig transaction, we use the formula:
Size of Single-SigSize of 2-of-2 Multisig​\=226 bytes334 bytes​≈1.48.
This shows that the 2-of-2 multisig transaction is 1.48 times larger than the single-sig transaction.

2-of-3 Multisig Transaction Size: A 2-of-3 multisig transaction, on the other hand, is roughly 367 bytes. Comparing this to the single-sig transaction, the formula is:
Size of 2-of-3 Multisig​\=226 bytes367 bytes​≈1.62.
Hence, a 2-of-3 multisig transaction is 1.62 times larger than a single-sig transaction.

In conclusion, the fees for a 2-of-2 and 2-of-3 multisig transaction are approximately 1.48 times and 1.62 times more than those for a single-sig transaction, respectively. These calculations assume constant fee rates and offer a clear understanding of how the increased size of multisig transactions leads to higher transaction fees in the Bitcoin network.

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Here's the detailed breakdown with the mathematical formulas (SegWit):

In Bitcoin, SegWit is an implemented protocol upgrade that changes the way data is stored in Bitcoin transactions. It effectively separates (or 'segregates') the witness (signature) data from the transaction data. This separation has a few key implications:

Reduced Transaction Size: SegWit transactions are typically smaller in size when compared in terms of block weight, a new concept introduced by SegWit that combines the size of a transaction with and without the signature data.

Fee Calculation: For SegWit transactions, fees are often calculated based on the weight units (WU) rather than the size in bytes. This typically results in lower fees for SegWit transactions compared to non-SegWit transactions of similar economic value.

Backward Compatibility: SegWit is backward compatible. Non-SegWit nodes see SegWit transactions as regular transactions but without signatures, while SegWit-enabled nodes see and validate the full transactions with signatures.

Calculating the size of SegWit transactions, including single-sig, 2-of-2 multisig, and 2-of-3 multisig, requires understanding the concept of "virtual size" or "vsize." In SegWit transactions, the size of the transaction is calculated differently due to the separation of witness data. SegWit transactions are measured in weight units (WU), and 4 WU are equivalent to 1 vbyte (virtual byte).
Let's calculate the approximate vsize for each type of transaction assuming they use SegWit:
1. Single-Sig SegWit Transaction (P2WPKH)
Inputs: Each P2WPKH input is approximately 68 vbytes (down from 148 bytes non-SegWit).
Outputs: Each output is around 31 vbytes for P2WPKH (similar to non-SegWit).
Overhead: The base transaction overhead is about 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+68vbytes(1input) +31vbytes(output1)+31vbytes(output2) = 140.5 vbytes

2. 2-of-2 Multisig SegWit Transaction (P2WSH)
Inputs: Each P2WSH input for a 2-of-2 multisig is approximately 140 vbytes (down from 334 bytes non-SegWit).
Outputs: Each output remains around 31 vbytes.
Overhead: About 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+140vbytes(1input) +31vbytes(output1) +31vbytes(output2) = 212.5 vbytes

3. 2-of-3 Multisig SegWit Transaction (P2WSH)
Inputs: Each P2WSH input for a 2-of-3 multisig is approximately 155 vbytes (down from 367 bytes non-SegWit).
Outputs: Each output remains around 31 vbytes.
Overhead: About 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+155vbytes(1input) +31vbytes(output1)+31vbytes(output2) = 227.5 vbytes

Summary
Single-Sig SegWit Transaction: ~140.5 vbytes
2-of-2 Multisig SegWit Transaction: ~212.5 vbytes
2-of-3 Multisig SegWit Transaction: ~227.5 vbytes
These are approximate values, as the actual size can vary slightly based on the exact length of signatures and scripts. In general, SegWit transactions are more efficient in terms of space usage, leading to potentially lower fees.

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Calculating the size of Taproot transactions, including single-sig, 2-of-2 multisig, and 2-of-3 multisig, involves understanding the efficiency improvements brought by the Taproot upgrade. Taproot transactions use Schnorr signatures and can make complex multisig transactions appear as simple single-sig transactions on the blockchain, significantly optimizing their size.

Let's calculate the approximate sizes for each type of transaction assuming they use Taproot (P2TR - Pay to Taproot):

1. Single-Sig Taproot Transaction
Inputs: Each P2TR input with a single signature is around 57.25 vbytes.
Outputs: Each output is about 43 vbytes for P2TR.
Overhead: The base transaction overhead is about 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+57.25vbytes(1input) +43vbytes(output1)+43vbytes(output2) = 153.75 vbytes

2. 2-of-2 Multisig Taproot Transaction
Taproot allows 2-of-2 multisig transactions to appear like a single-sig transaction when spending conditions are met:

Inputs: The size remains around 57.25 vbytes.
Outputs: Each output remains around 43 vbytes.
Overhead: About 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+57.25vbytes(1input) +43vbytes(output1)+43vbytes(output2) = 153.75 vbytes

3. 2-of-3 Multisig Taproot Transaction
Similarly, a 2-of-3 multisig transaction can also be optimized:
Inputs: The size for the input remains approximately 57.25 vbytes.
Outputs: Each output remains around 43 vbytes.
Overhead: About 10.5 vbytes.
For a transaction with 1 input and 2 outputs:
10.5vbytes(base)+57.25vbytes(1input) +43vbytes(output1)+43vbytes(output2) = 153.75 vbytes

Summary
Single-Sig Taproot Transaction: ~153.75 vbytes
2-of-2 Multisig Taproot Transaction: ~153.75 vbytes
2-of-3 Multisig Taproot Transaction: ~153.75 vbytes
Taproot optimizes the transaction size significantly by enabling even complex multisig transactions to be indistinguishable from single-sig transactions on-chain. This leads to more efficient use of block space and potentially lower transaction fees. These sizes are approximations and can vary slightly based on the transaction's specific details.

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In the context of Bitcoin transactions and their fees, understanding the difference between bytes and vbytes (virtual bytes) is crucial, especially with the introduction of Segregated Witness (SegWit) and Taproot.

Bytes
Pre-SegWit Transactions: Before SegWit, transactions were measured in bytes. This is the total size of a transaction when serialized, including all data such as inputs, outputs, and scripts.
Fee Calculation: Fees were typically calculated based on the size of the transaction in bytes. The fee rate would be satoshis per byte (sat/B).
Vbytes
SegWit Transactions: With SegWit, the concept of vbytes was introduced. A vbyte is a unit of measurement that takes into account the discount on witness data provided by SegWit. This makes SegWit transactions more efficient by reducing the size of the transaction data that counts towards the block size limit.
Weight Units: SegWit introduced the concept of weight units (WU). The maximum block weight is 4 million WU. Non-witness data counts as 4 WU per byte, and witness data (like signatures) counts as 1 WU per byte. Therefore, 4 WU is equivalent to 1 vbyte.
Fee Calculation: Post-SegWit, fees are often calculated in satoshis per vbyte (sat/vB). This can make SegWit transactions cheaper in terms of fees, as the part of the transaction that contains the signatures is discounted.

Conversion and Calculation
- For Non-SegWit Transactions: 1 byte = 1 vbyte.
- For SegWit Transactions: The size in vbytes is typically smaller than the size in bytes due to the discount on witness data. The exact conversion depends on the amount of witness data in the transaction.

How to Measure for Fees
Check Transaction Type: Determine if your transaction is a traditional (non-SegWit) transaction, a SegWit transaction, or a Taproot transaction.
Calculate Size:
- For non-SegWit transactions, measure the total size in bytes.
- For SegWit and Taproot transactions, calculate the size in vbytes, considering the discounted weight of the witness data.
Apply Fee Rate: Use the appropriate fee rate (sat/B for bytes, sat/vB for vbytes) based on current network conditions and urgency of the transaction.

Example - If you have a SegWit transaction that is 200 vbytes and the current fee rate is 50 sat/vB, the fee would be 200×50\=10,000200 \\times 50 = 10,000200×50\=10,000 satoshis.
- For a non-SegWit transaction of the same size (200 bytes) at a rate of 50 sat/B, the fee would also be 10,000 satoshis.
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*Rounding and Averages: This calculator might round numbers or use slightly different averages for transaction components, leading to minor discrepancies in the final calculated size.

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