The angles of a triangle are in the ratio 3:4:5. Find the smallest angle in radian measure.

Exercise: Angle Measurement · Q11

3:4:5

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Concept understanding — Angle Conversion

Angle Conversion: Why We Need It, and How It Works

Imagine you're measuring the length of a table. You could use centimetres, inches, or feet — all are valid, but the number changes depending on which unit you pick. The same idea applies to angles. An angle is a measure of rotation or opening between two lines, and we have different units to describe that same physical turn.

The two most important units you'll meet are degrees and radians. Degrees are what you likely already know: a full circle is $360^\circ$ . Radians are less familiar but far more natural in mathematics — they're based on the geometry of the circle itself.

The Intuition: What Is a Radian?

Take a circle of radius $r$ . Now walk along the circumference a distance equal to $r$ . The angle you've swept out at the centre is 1 radian.

Note

A radian is the angle that subtends an arc length equal to the radius.

If the arc length $s = r$ , then the angle $\theta = 1$ radian.

So radians directly connect the angle to the arc length:

\theta \text{ (in radians)} = \frac{s}{r}

This is why radians are the "natural" unit — they come from the circle's own geometry, not an arbitrary number like 360.

The Key Relationship

A full circle is $360^\circ$ . In radians, a full circle is the circumference divided by the radius:

\frac{2\pi r}{r} = 2\pi \text{ radians}

So the fundamental conversion is:

360^\circ = 2\pi \text{ radians}

From this, you get the two conversion factors you'll use constantly:

1^\circ = \frac{\pi}{180} \text{ radians}

1 \text{ radian} = \frac{180^\circ}{\pi}

How to Convert: Two Simple Steps

Degrees to radians: Multiply by $\frac{\pi}{180}$ .

Example: Convert $60^\circ$ to radians.

60^\circ \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3} \text{ radians}

Radians to degrees: Multiply by $\frac{180^\circ}{\pi}$ .

Example: Convert $\frac{5\pi}{6}$ radians to degrees.

\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = \frac{5 \times 180^\circ}{6} = 150^\circ

Tip

Memorise these common conversions — they appear constantly:

$0^\circ = 0$ , $30^\circ = \frac{\pi}{6}$ , $45^\circ = \frac{\pi}{4}$ , $60^\circ = \frac{\pi}{3}$ , $90^\circ = \frac{\pi}{2}$ , $180^\circ = \pi$ , $270^\circ = \frac{3\pi}{2}$ , $360^\circ = 2\pi$

Why Bother With Radians?

Degrees are fine for everyday use (a right angle is $90^\circ$ , easy). But in calculus, physics, and advanced trigonometry, radians are essential. Here's why:

The derivative of $\sin x$ is $\cos x$ only if $x$ is in radians. In degrees, you'd get an ugly constant factor.
Arc length and area formulas become simple: $s = r\theta$ , $A = \frac{1}{2}r^2\theta$ — these only work with radians.
Many natural phenomena (like oscillations, waves, circular motion) are described cleanly using radians. …

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Let the angles be $3k,4k,5k$ . Since the angle sum of a triangle is $180^\circ$ ,

$3k+4k+5k=180^\circ \Rightarrow 12k=180^\circ \Rightarrow k=15^\circ$ . The smallest angle is …

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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 19 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$40°20'$
(a) $\dfrac{12\pi}{540}$
(b) $\dfrac{121\pi}{540}$
(c) $\dfrac{\pi}{540}$
(d) None of these
›Reveal solutionSolution

Convert minutes to a decimal degree, then multiply by $\pi/180$ to get radians.

First express $40°20'$ as a single degree value. Since $1° = 60'$ , $20' = \dfrac{20}{60}° = \dfrac{1}{3}°$ . So

$40°20' = \left(40 + \dfrac{1}{3}\right)° = \dfrac{121}{3}°$ …

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CBSE 2026Set ANNUAL1 markMCQ
$120°$
(a) $\frac{4\pi}{3}$
(b) $\frac{\pi}{3}$
(c) $\frac{2\pi}{3}$
(d) $\frac{\pi}{6}$
›Reveal solutionSolution

$120° = \frac{2\pi}{3}$ radians, option (c).

To convert an angle from degrees to radians, multiply by $\frac{\pi}{180}$ :

…

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CBSE 2025Set ANNUAL1 markMCQ
$\dfrac{\pi}{2}$
(a) $180°$
(b) $90°$
(c) $120°$
(d) $360°$
›Reveal solutionSolution

$\dfrac{\pi}{2}$ radian $= 90°$ .

The conversion between radians and degrees uses $\pi$ radians $= 180°$ .

…

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CBSE 2025Set ANNUAL1 markMCQ
$\left(\dfrac{3}{4}\right)^c$
(a) $42°50''$
(b) $42°57'16''$
(c) $49°61'$
(d) $41°57'18''$
›Reveal solutionSolution

Using the standard approximation $1$ radian $\approx 57°16'22''$ (from $\pi\approx\frac{22}{7}$ ), three-quarters of a radian works out to $42°57'16''$ .

$\text{Degree measure} = \text{Radian measure} \times \frac{180°}{\pi}$

Using $\pi \approx \dfrac{22}{7}$ : $\dfrac{180}{\pi} \approx \dfrac{180\times7}{22} = \dfrac{1260}{22} = 57.2727...°$ , i.e. $1$ radian $\approx 57°16'22''$ .

$\left(\frac34\right)^c = \frac34 \times 57.2727...° = 42.9545...°$

Converting the decimal part to minutes/seconds:

$0.9545° \times 60 = 57.27' \implies 42°57.27'$ …

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CBSE 2024Set ANNUAL1 markMCQ
$1:3:5$
(a) $\dfrac{\pi}{9}$
(b) $\dfrac{\pi}{3}$
(c) $\dfrac{5\pi}{9}$
(d) None of these
›Reveal solutionSolution

Use the angle-sum property of a triangle to find each angle from the given ratio, then convert the smallest to radians.

Let the angles be $x, 3x, 5x$ (in the ratio $1:3:5$ ). The angles of a triangle sum to $180^\circ$ :

$x + 3x + 5x = 180^\circ \implies 9x = 180^\circ \implies x = 20^\circ$

So the angles are $20^\circ, 60^\circ, 100^\circ$ . The smallest is $20^\circ$ .

…

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CBSE 2024Set ANNUAL1 mark
$\dfrac{5\pi}{3}$
›Reveal solutionSolution

The angle $\dfrac{5\pi}{3}$ (given in radian form) equals $300^\circ$ in degree measure.

…

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CBSE 2024Set ANNUAL1 markMCQ
$\dfrac{5\pi}{3}$
(a) $210^\circ$
(b) $510^\circ$
(c) $300^\circ$
(d) None of these.
›Reveal solutionSolution

$\dfrac{5\pi}{3}$ radians equals $300^\circ$ .

We use the relation $180^\circ = \pi$ radians, so $1$ radian $=\dfrac{180^\circ}{\pi}$ . …

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CBSE 2024Set ANNUAL1 markMCQ
$\pi$
(a) 60°
(b) 180°
(c) 90°
(d) 45°
›Reveal solutionSolution

By definition of radian measure, $\pi$ radian equals $180°$ .

Step 1. A full circle is $360°$ , which corresponds to $2\pi$ radians.

…

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CBSE 2023Set ANNUAL1 markMCQ
$=$
(a) $\dfrac{\pi}{180°}$
(b) $\dfrac{180°}{\pi}$
(c) $180°$
(d) $360°$
›Reveal solutionSolution

Since $\pi \text{ rad} = 180°$ , one radian equals $180°/\pi \approx 57.3°$ .

The relation between radian and degree measure comes from the fact that a full circle subtends an angle of $2\pi$ radians, which is also $360°$ . So:

$\pi \text{ radians} = 180°$

…

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CBSE 2023Set ANNUAL1 markMCQ
$240°$
(a) $\dfrac{26\pi}{9}$
(b) $\dfrac{4\pi}{3}$
(c) $\dfrac{5\pi}{36}$
(d) $\dfrac{7\pi}{6}$
›Reveal solutionSolution

$240° = \dfrac{4\pi}{3}$ radians.

To convert an angle from degrees to radians, multiply by $\dfrac{\pi}{180}$ :

$240° \times \frac{\pi}{180} = \frac{240\pi}{180} = \frac{4\pi}{3}$

…

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CBSE 2023Set ANNUAL1 markMCQ
$Q. Degree measure of 7π/6 is:$
(a) 30°
(b) 11°/3
(c) 210°
(d) None of these
›Reveal solutionSolution

Multiply the radian measure by $\dfrac{180^\circ}{\pi}$ to convert to degrees.

Since $\pi$ radians $= 180^\circ$ : …

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CBSE 2023Set ANNUAL1 mark
$\dfrac{11}{16}$
›Reveal solutionSolution

$\dfrac{11}{16}$ radians $\approx39^\circ22'30''$ .

Using $\text{degree} = \text{radian}\times\dfrac{180}{\pi}$ and $\pi=\dfrac{22}{7}$ , so $\dfrac{180}{\pi}=\dfrac{180\times7}{22}=\dfrac{1260}{22}=\dfrac{630}{11}$ :

$\dfrac{11}{16}\times\dfrac{630}{11} = \dfrac{630}{16} = 39.375^\circ.$

…

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Q.The angles of a triangle are in the ratio 3:4:53:4:53:4:5. Find the smallest angle in radian measure.

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Q.The angles of a triangle are in the ratio $3:4:5$ . Find the smallest angle in radian measure.

Unlock everything free for 14 days