Calculate the solid angle subtended by the periphery of an area of 1 cm$^2$ at a point situated symmetrically at a distance of 5 cm from the area.

Calculate the solid angle subtended by the periphery of an area of 1 cm² at a point situated symmetrically at

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Concept understanding — Significant Figures Calculation

Significant Figures: The Art of Honest Measurement

Imagine you're measuring the length of a table with a ruler that has marks every millimeter. You see the table edge falls somewhere between 152.3 cm and 152.4 cm. You estimate it as 152.35 cm. But here's the truth: you're certain about 152.3, pretty sure about the 0.05, and guessing about anything beyond that. Significant figures are simply a way to communicate how much of that number you actually know.

The Core Idea

Every measurement has uncertainty. Significant figures (or "sig figs") are the digits in a number that carry meaningful information about its precision. They include all the digits you're sure of, plus one more that you estimate.

Note

A digit is "significant" if removing it would change the precision of the measurement. Zeros can be tricky — they might just be placeholders.

The Rules (Memorize These)

1. Non-zero digits are always significant

$123.45$ has 5 sig figs. Simple.

2. Zeros between non-zero digits are significant

$1002$ has 4 sig figs. The zeros are "sandwiched" — they're part of the measurement.

3. Leading zeros are never significant

$0.00123$ has 3 sig figs. Those zeros just tell you where the decimal point is.

4. Trailing zeros are significant only if there's a decimal point

$1200$ has 2 sig figs (no decimal — zeros are placeholders)
$1200.$ has 4 sig figs (decimal tells us those zeros were measured)
$1200.0$ has 5 sig figs

5. Exact numbers have infinite sig figs

If you count 5 apples, that's exactly 5 — no uncertainty. Conversion factors like $1 \text{ m} = 100 \text{ cm}$ are exact by definition.

Tip

When in doubt, write the number in scientific notation. $1.20 \times 10^3$ clearly has 3 sig figs, while $1.2 \times 10^3$ has 2.

Why This Matters: Calculations

When you multiply or add measurements, the uncertainty propagates. You can't claim more precision than your least precise measurement.

Multiplication and Division

The result should have the same number of sig figs as the measurement with the fewest sig figs.

$3.14 \times 2.5 = 7.85$ but you report $7.9$ (2 sig figs, because 2.5 has only 2)

Addition and Subtraction

The result should have the same decimal places as the measurement with the fewest decimal places.

$12.11 + 18.0 = 30.11$ but you report $30.1$ (one decimal place, because 18.0 has one) …

Why this formula?

Significant Figures: Why the Rules Work

Let’s start with the core idea: significant figures (sig figs) are a way to honestly report how precise a measurement is. The rules for addition/subtraction and multiplication/division aren’t arbitrary — they come directly from how uncertainty propagates through calculations.

1. The Fundamental Idea: Uncertainty is the Key

Every measurement has an uncertainty (error). When we say a length is 12.3 cm, we mean:

The true value lies somewhere between 12.25 cm and 12.35 cm (assuming ±0.05 cm uncertainty).
The last digit (3) is uncertain; the digits before it (1 and 2) are certain.

Why this matters: When we combine measurements, the uncertainty in the result depends on the uncertainties of the inputs. Sig fig rules are a shortcut for this uncertainty propagation.

2. Rule for Addition and Subtraction

Statement: The result should have the same number of decimal places as the measurement with the fewest decimal places.

Example:

$12.3 + 4.56 = 16.86$ → round to 16.9 (one decimal place, like 12.3)

Why this holds

Consider two measurements:

$A = 12.3 \pm 0.05$ (uncertainty in the tenths place)
$B = 4.56 \pm 0.005$ (uncertainty in the hundredths place)

When we add:

Certain digits: 12.3 has certainty up to the tenths place. 4.56 has certainty up to the hundredths place.
The weaker link: The tenths place of A is uncertain. So in the sum, the hundredths place (from B) is meaningless — because we don’t even know the tenths place of A exactly.

Mathematically, the absolute uncertainty in the sum is:

\Delta(A+B) = \sqrt{(\Delta A)^2 + (\Delta B)^2} \approx \sqrt{0.05^2 + 0.005^2} \approx 0.0502

This uncertainty is ~0.05, which affects the tenths place. So reporting the hundredths place is false precision.

Key takeaway: The result’s last significant digit is in the same decimal place as the least precise measurement’s last digit.

3. Rule for Multiplication and Division

Statement: The result should have the same number of significant figures as the measurement with the fewest significant figures.

Example:

$12.3 \times 4.56 = 56.088$ → round to 56.1 (three sig figs, like both inputs)

Why this holds

Let’s use relative uncertainty (percentage error):

$A = 12.3 \pm 0.05$ → relative uncertainty = $\frac{0.05}{12.3} \approx 0.00407$ (0.407%)
$B = 4.56 \pm 0.005$ → relative uncertainty = $\frac{0.005}{4.56} \approx 0.00110$ (0.110%)

For multiplication, relative uncertainties add (approximately):

\frac{\Delta(A \times B)}{A \times B} \approx \sqrt{\left(\frac{\Delta A}{A}\right)^2 + \left(\frac{\Delta B}{B}\right)^2}

Plugging in:

\approx \sqrt{0.00407^2 + 0.00110^2} \approx 0.00422 \text{ (0.422\%)}

Now, the absolute uncertainty in the product:

\Delta(A \times B) \approx 0.00422 \times (12.3 \times 4.56) \approx 0.00422 \times 56.088 \approx 0.237

This uncertainty (~0.2) affects the tenths place of the result. So the result 56.088 has uncertainty in the first decimal — meaning only three digits (5, 6, and the uncertain 1) are meaningful. That’s three sig figs, matching the input with fewer sig figs (both have three here).

Key takeaway: The number of sig figs in the result is limited by the least precise measurement’s number of sig figs, because relative uncertainty is dominated by the measurement with the largest relative error.

4. Why These Rules Are Different …

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The solid angle measures how large an object appears in three dimensions. For a small area $A$ viewed symmetrically from a distance $r$ , the solid angle $\Omega$ is given by $\Omega = A/r^2$ . Substituting the given values, the solid angle is $\boxed{0.04 \text{ sr}}$ .

The concept of a solid angle is a three-dimensional analogue to the familiar two-dimensional angle. Just as a 2D angle measures the "spread" of lines originating from a point in a plane, a solid angle measures the "spread" of lines originating from a point in 3D space, forming a cone and subtending an area on a sphere centered at that point.

Imagine shining a flashlight onto a wall. The area illuminated on the wall depends on how far you are from it and how wide the beam is. The solid angle quantifies this "apparent size" of the illuminated area as seen from the flashlight's bulb.

The unit of solid angle is the steradian (sr). One steradian is the solid angle subtended at the center of a sphere by a portion of the surface whose area is equal to the square of the sphere's radius.

For a small planar area $dA$ subtending a solid angle $d\Omega$ at a point P at a distance $r$ , the general formula is:

d\Omega = \frac{dA \cos\theta}{r^2}

The problem states that the point is "situated symmetrically" from the area. This is a crucial piece of information. It means the point lies directly above the center of the area, along the normal to the area. In this specific configuration, the angle $\theta$ between the area's normal and the line connecting the area to the point is $0^\circ$ . Since $\cos(0^\circ) = 1$ , the formula simplifies significantly for this symmetrical case:

\Omega = \frac{A}{r^2}

This simplified formula applies when the area is perpendicular to the line of sight from the point.

Now, let's calculate the solid angle step-by-step.

Identify the given parameters.
- Area, $A = 1 \text{ cm}^2$
- Distance, $r = 5 \text{ cm}$
Recall the appropriate formula for solid angle.

Since the point is situated symmetrically, the simplified formula $\Omega = \frac{A}{r^2}$ is applicable.
Substitute the given values into the formula. …

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Here are the common mistakes students make when solving this solid angle problem, along with how to avoid each.

1. Using the Wrong Formula for Solid Angle

The Mistake:

Students often try to use the formula for a point on the axis of a circular disc:

\Omega = 2\pi \left(1 - \cos\theta\right)

without checking if the area is actually a circle or if the point is on its axis. Here, the area is given as $1 \text{ cm}^2$ , but its shape is not specified. The problem says the point is symmetrically situated, which means the point is on the normal through the center of the area, but the area could be a square, a rectangle, or any shape.

How to Avoid:

Always check the geometry first. If the shape is not a circle, do not use the disc formula.
For a small area $dA$ at a distance $r$ from the point, the exact formula is:

d\Omega = \frac{dA \cos\theta}{r^2}

where $\theta$ is the angle between the normal to the area and the line joining the point to the area.

Here, the point is symmetrically situated at 5 cm from the area, so the normal to the area points directly toward the point. Hence $\theta = 0^\circ$ and $\cos\theta = 1$ .
The correct approach: treat the area as a collection of infinitesimal elements, but since the distance (5 cm) is much larger than the size of the area ( $\sqrt{1} = 1$ cm), we can approximate:

\Omega \approx \frac{A}{r^2} = \frac{1 \text{ cm}^2}{(5 \text{ cm})^2} = \frac{1}{25} \text{ sr}

Key takeaway: Use $\Omega = A/r^2$ only when the area is small and perpendicular to the line of sight, and the point is far compared to the size of the area.

2. Forgetting the $\cos\theta$ Factor

The Mistake:

Even when using the correct formula $\Omega = A/r^2$ , students forget that this is valid only when the area is perpendicular to the line joining the point to the area. If the area is tilted, the projected area $A\cos\theta$ must be used.

How to Avoid:

Always draw a diagram. Mark the normal to the area and the line from the point to the center of the area.
If the point is symmetrically situated, the normal points directly at the point, so $\theta = 0$ and $\cos\theta = 1$ . No correction needed.
If the problem had said "point is 5 cm away from the area but not on the normal," you would need to compute $\cos\theta$ and use:

\Omega = \frac{A \cos\theta}{r^2}

Memory aid: The solid angle is the projected area divided by the square of the distance.

3. Confusing Solid Angle with Plane Angle

The Mistake:

Some students try to find a linear angle (like the half-angle $\theta$ of a cone) and then plug it into $\Omega = 2\pi(1-\cos\theta)$ . This is unnecessary and often leads to errors if the area is not circular.

How to Avoid:

Solid angle is a 3D concept (measured in steradians), not a 2D angle.
If the area is small and flat, the simplest path is $\Omega = A_\perp / r^2$ .
Only use the cone formula if the problem explicitly describes a circular disc and the point is on its axis.

Quick check: If your answer comes out in degrees or radians, you’ve used the wrong formula.

4. Incorrect Unit Handling

The Mistake: …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 20 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. How many significant figures does 0.057 have?$
(a) 2
(b) 4
(c) 3
(d) 0
›Reveal solutionSolution
Leading zeros used only to fix the decimal point are never significant. 0.057 has exactly 2 significant figures: 5 and 7.

Rule for counting significant figures:

All non-zero digits are significant.

Zeros between two non-zero digits are significant.

Leading zeros (to the left of the first non-zero digit) are NOT significant -- they only locate the decimal point.

Trailing zeros after a decimal point ARE significant. …
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CBSE 2026Set ANNUAL1 markMCQ
$Q. How many significant figures are there in 0.0052?$
(a) 4
(b) 1
(c) 3
(d) 2
›Reveal solutionSolution

0.0052 has exactly 2 significant figures (5 and 2); leading zeros are not counted.

Significant figures are the meaningful digits in a measured or calculated quantity - they indicate the precision of a measurement. The rules for counting them: (1) all non-zero digits are significant; (2) zeros between non-zero digits are significant; (3) leading zeros (to the left of the first non-zero digit) are NEVER significant - they only serve to locate the decimal point; (4) trailing zeros after a decimal point ARE significant.

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. If π = 3.14, then the value of π^2 is:$
(a) 9.860
(b) 9.9
(c) 9.86
(d) 9.8596
›Reveal solutionSolution

By the rules of significant figures, a result cannot have more significant figures than the least precise value used to calculate it; since pi = 3.14 has 3 significant figures, pi^2 must also be reported to 3 significant figures, i.e., 9.86.

When multiplying or dividing measured quantities, the result should be rounded off to the same number of significant figures as the quantity with the fewest significant figures used in the calculation. This is because a calculation cannot manufacture precision beyond what the original measurement actually had.

Here, pi is given as 3.14, which has 3 significant figures.

pi^2 = 3.14 x 3.14 = 9.8596 (this is the raw arithmetic result, but it has 5 significant figures, more precision than 3.14 actually supports)

Rounding 9.8596 to 3 significant figures: …

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CBSE 2026Set ANNUAL1 mark
$Q. Fill in the blank: The number of significant figures in 2.005 is ______.$
›Reveal solutionSolution

2.005 has 4 significant figures because zeros sandwiched between non-zero digits always count as significant.

Rules for significant figures: (1) all non-zero digits are significant; (2) zeros between two non-zero digits (captive/sandwiched zeros) are significant; (3) leading zeros are not significant; (4) trailing zeros after a decimal point are significant.

…

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CBSE 2026Set ANN1 mark
$Q. Find the number of significant figures in the measurement, 0.04597 g.$
›Reveal solutionSolution

Leading zeros are not significant; the digits 4, 5, 9, 7 count, giving 4 significant figures.

Rule: zeros to the left of the first non-zero digit (leading zeros) are NOT significant; they only locate the decimal point. All non-zero digits are significant. …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Number of significant figures in 2.005 are:$
(a) 1
(b) 2
(c) 3
(d) 4
›Reveal solutionSolution
Every digit in 2.005 is significant because zeros sandwiched between non-zero digits always count.

Rules for counting significant figures:

All non-zero digits are significant.

Zeros between two non-zero digits are significant ("captive zeros").

Leading zeros (before the first non-zero digit) are NOT significant.

Trailing zeros after a decimal point ARE significant.

In $2.005$ : the digits are 2, 0, 0, 5.

'2' is a non-zero digit — significant. …
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CBSE 2024Set ANNUAL1 markMCQ
$Q. Number of significant figures in 2.005 is:$
(a) Two
(b) Three
(c) Four
(d) Infinite
›Reveal solutionSolution
All four digits in 2.005 are significant because the zeros sit between non-zero digits.

Rules for counting significant figures:

All non-zero digits are significant: the digits 2 and 5 count. …
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CBSE 2024Set ANNUAL1 mark
$Q. Number of significant figures in 0.00300 is ____.$
›Reveal solutionSolution

0.00300 has 3 significant figures: the leading zeros used only to fix the decimal point are not counted, but trailing zeros after a decimal point ARE significant.

Rules for significant figures: leading zeros (0.00...) before the first non-zero digit are never significant -- they only locate the decimal point. Zeros appearing after the decimal point AND after a non-zero digit ARE significant, since they indicate …

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CBSE 2023Set ANNUAL1 markMCQ
$Q. Number of significant figures in 2.005:$
(a) 2
(b) 3
(c) 4
(d) Infinite
›Reveal solutionSolution
Captive zeros (zeros sandwiched between non-zero digits) always count as significant, so 2.005 has 4 significant figures.

Rules used:

All non-zero digits (2, 5) are significant.

Zeros between two non-zero digits (captive zeros) are significant. …
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CBSE 2023Set ANNUAL1 markMCQ
$Q. Round off the number 19.95 into three significant figures.$
(a) 20.1
(b) 19.9
(c) 19.5
(d) 20.0
›Reveal solutionSolution

Rounding 19.95 to three significant figures using the 'round half to even' convention gives 20.0.

19.95 has four significant figures (1, 9, 9, 5). To round to three significant figures, we must decide the fate of the third significant digit (the second 9), based on the digit being dropped (a 5, with nothing after it).

Rule for rounding off when the digit to be dropped is exactly 5: if the preceding digit is even, it is left unchanged; if the preceding digit is odd, it is increased by 1 (this is the 'round half to even' convention used for significant figures, and it avoids a systematic upward bias).

…

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CBSE 2023Set ANNUAL1 mark
$Q. 5.74 g of a substance occupies 1.2 cm^3. Express its density in proper significant figures.$
›Reveal solutionSolution

Density = 5.74/1.2 = 4.7833... g/cm^3, rounded to 2 significant figures = 4.8 g/cm^3.

Rule for significant figures in division/multiplication: the result must be reported with the same number of significant figures as the input quantity that has the FEWEST significant figures. This keeps the calculated result from claiming more precision than the actual measurements support.

Step 1: Count significant figures in each given quantity.

Mass = 5.74 g has 3 significant figures.

Volume = 1.2 cm^3 has 2 significant figures.

Step 2: Compute the raw quotient.

Density = mass/volume = 5.74 / 1.2 = 4.7833... g/cm^3.

…

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CBSE 2023Set ANNUAL1 markMCQ
$Q. If radius of circle is 2.12 cm, then express its area must be:$
(a) 14 cm^2
(b) 14.1 cm^2
(c) 14.11 cm^2
(d) 14.1124 cm^2
›Reveal solutionSolution

Rounded to the correct significant figures, the area is 14.1 cm^2.

Area A = πr^2 = 3.14159 × (2.12)^2 = 3.14159 × 4.4944 = 14.1197 cm^2.

…

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Q.Calculate the solid angle subtended by the periphery of an area of 1 cm $^2$ at a point situated symmetrically at a distance of 5 cm from the area.

Concept understanding — Significant Figures Calculation

Significant Figures: The Art of Honest Measurement

The Core Idea

The Rules (Memorize These)

1. Non-zero digits are always significant

2. Zeros between non-zero digits are significant

3. Leading zeros are never significant

4. Trailing zeros are significant only if there's a decimal point

5. Exact numbers have infinite sig figs

Why This Matters: Calculations

Multiplication and Division

Addition and Subtraction

Why this formula?

Significant Figures: Why the Rules Work

1. The Fundamental Idea: Uncertainty is the Key

2. Rule for Addition and Subtraction

Why this holds

3. Rule for Multiplication and Division

Why this holds

4. Why These Rules Are Different …

Concept: Solid Angle and the Area-Squared Law

Method: Direct Projection Method

Steps:

1. Using the Wrong Formula for Solid Angle

2. Forgetting the $\cos\theta$ Factor

3. Confusing Solid Angle with Plane Angle

4. Incorrect Unit Handling

Q.Calculate the solid angle subtended by the periphery of an area of 1 cm2^22 at a point situated symmetrically at a distance of 5 cm from the area.

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Q.Calculate the solid angle subtended by the periphery of an area of 1 cm $^2$ at a point situated symmetrically at a distance of 5 cm from the area.

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