Sector area: how much of the disk you keep

Explain it like I am five

Picture a pizza. The whole round pie is a disk, and a slice cut from the center is a sector. Sector area, A, measures how much surface the slice covers, in square units such as c $m^{2}$ (square centimeters).

Example: the pizza's radius is r = 6 cm and the slice's angle at the center is 150°. The formula A = $1 over 2$ $r^{2}$ θ needs θ (theta) in radians, so convert first: 150° × $π over 180°$ = $5 π over 6$ . Square the radius, multiplying it by itself: $r^{2}$ = 6 × 6 = 36. Then A = $1 over 2$ × 36 × $5 π over 6$ = 18 × $5 π over 6$ = 15π c $m^{2}$ , about 47.12 c $m^{2}$ .

Why the formula is true: the slice is $θ over 2 π$ of a full turn, so it holds that same fraction of the whole pie, whose area is π $r^{2}$ . $θ over 2 π$ × π $r^{2}$ = $1 over 2$ $r^{2}$ θ, because the π on top cancels the π on the bottom and a 2 stays below. Check: 150° is $150 over 360$ = $5 over 12$ of the pie, and the pie is π × 36 = 36π c $m^{2}$ . $5 over 12$ × 36π = 15π c $m^{2}$ .

In plain words

Imagine cutting a slice from a round pizza. The curved crust is an arc, while the whole piece of pizza between the two cuts is a sector. The whole filled circle is called a disk. Arc length measures how much crust you have; sector area (A) measures how much surface you have. A quarter-turn slice keeps one quarter of the crust and one quarter of the pizza. Radians express that same fraction using a full turn of 2π. Use the nonnegative opening of the physical slice, because area cannot be negative. Keep track of the units: crust uses centimeters, while pizza surface uses square centimeters.

The arc is the curved boundary; the sector is the filled quarter of the disk.

Reminder

Squared quantities. $7 squared$ = 7 × 7 = 49, and (7 cm $) squared$ = 49 c $m^{2}$ .
Fraction multiplication. $1 over 2$ × 49 × $2 π over 5$ = $98 π over 10$ = $49 π over 5$ , because the numerator and denominator factors multiply before reduction.
Arc length. s = rθ uses radians. Substituting that into A = $1 over 2$ $r^{2}$ θ gives A = $1 over 2$ r × s.
Semicircular. Semicircular means half a circle. Its central opening is 180° = π radians.

Why it works. A full disk has area π, and a full turn has radian measure 2π. A sector of θ radians occupies of the disk, so A = × π = θ. Canceling π leaves the factor one half. The square matters: doubling a radius doubles both length directions across the slice, multiplying area by 4. Degrees also describe the correct slice, but their fraction is , so the radian formula requires conversion first.

RuleA = θ, with θ in radians; for degrees use A = π.
For one physical slice, r > 0 and 0 ≤ θ ≤ 2π in radians; sector area is a nonnegative size, also called its magnitude.

The same idea, five ways

Say it

Say sector area is one half times radius squared times the central angle in radians.

Write it

A sector is the filled circular slice bounded by two radii and their arc; its area is the same fraction of the disk as its opening is of one complete turn.

In math

A = $1 over 2$ $r^{2}$ θ, with θ in radians
A = $θ over 360°$ π $r^{2}$ , with θ in degrees
A = $θ over 2 π$ π $r^{2}$ , with θ in radians
A = $1 over 2$ rs, using s = rθ
for a physical sector: r > 0 and 0 ≤ θ ≤ 2π in radians

Arc length measures the pizza crust along the edge; sector area measures the pizza surface between the cuts.

See it

The curved arc is a length, while the shaded sector covers square centimeters.

The same idea, other ways

As a fraction of a pizza

The sector is the whole pizza slice between two cuts, not its curved crust. If the slice takes one fifth of the full turn, it takes one fifth of the whole disk's area. Multiply that fraction by π $r^{2}$ to find the covered surface.

One fifth of the angle keeps one fifth of the filled disk.

Why the radius is squared

Widen a slice to twice its radius. It doubles in both length directions, like doubling a rectangle's width and height. The area becomes four times as large. That is why the area formula uses r × r, while the curved crust formula uses only one factor of r.

Doubling both dimensions produces four copies of the original area.

.1Arc versus sector

The crust and the pizza surface belong to the same slice, but they measure different things. Arc length is a distance along a curve. Sector area is the amount of filled surface between the two radii. The same opening supplies the same fraction of the whole circumference and the whole disk.

Arc length (s): s = rθ, measured in length units.
Sector area (A): A = $1 over 2$ $r^{2}$ θ, measured in square units.
Both formulas require radians and the physical opening's nonnegative size.
Combining the formulas gives A = $1 over 2$ rs because s = rθ.
Scope: linear and angular speed are omitted from this section.
From the half-turn anchor, rebuild conversion factors $π radians over 180°$ and $180° over π radians$ , then check that a quarter turn is 90° = $π over 2$ .

The same 60° slice has an arc length in centimeters and an area in square centimeters.

Reminder

Squaring a radius. For r = 5 cm, $r^{2}$ = 5 × 5 = 25 c $m^{2}$ ; doubling gives 10 cm and does not square the radius.

Worked exampleMeasure both the crust and the surface

You are finding both the curved boundary length and the filled slice area. For radius 5 cm and angle $π over 3$ radians, find the arc length and sector area exactly.

The curved edge measures length, and the filled slice measures area.

s = 5 × $π over 3$ = $5 π over 3$ cm.Arc length multiplies the radius by the already-radian angle.
$r^{2}$ = 5 × 5 = 25 c $m^{2}$ .Sector area uses a square of the radius.
A = $1 over 2$ × 25 × $π over 3$ = $25 π over 6$ c $m^{2}$ .Multiply the top factors and the bottom factors: 2 × 3 = 6.

Answer

Arc length: $5 π over 3$ cm.
Sector area: $25 π over 6$ c $m^{2}$ .

Check = 60°, which is one sixth of a full turn. The circumference is 10π cm, and = cm. The disk area is 25π c, and one sixth is c.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: A sector area can be reported in centimeters because its boundary is an arc.

Area measures filled surface in two directions. Arc length measures only distance along the curve.

✓ Instead: Report arc length in centimeters and sector area in square centimeters.

Tips and tricks

Picture crust for arc length and pizza surface for sector area. Check the units beside each answer.

.2Area from a fraction of the circle

If a slice uses one eighth of the turn, it keeps one eighth of the disk. You can find that fraction using degrees or radians. The degree fraction is the opening divided by 360°; the radian fraction is the opening divided by 2π. Both fractions describe the same piece of pizza.

Full disk area is π $r^{2}$ .
For degrees, A = $θ over 360°$ π $r^{2}$ .
For radians, A = $θ over 2 π$ π $r^{2}$ = $1 over 2$ $r^{2}$ θ.
Do not use 360 as the full-turn divisor for a radian measurement, or 2π as the divisor for a degree measurement.

A 45° slice is one eighth of the disk, so its area is one eighth of 16π.

Reminder

Fraction of a disk. 45° is $45 over 360$ = $1 over 8$ of a turn. With radius 4, one eighth of the 16π disk area is 2π square units.

Worked exampleA 45° slice with radius 4

You are finding the amount of filled surface in the specified circular slice. Find the area of a sector with radius 4 units and central angle 45°, exactly and to two decimal places.

The curved edge measures length, and the filled slice measures area.

45° × $π radians over 180°$ = $π over 4$ radians.45 is one quarter of 180, and the radian formula requires this conversion.
$r^{2}$ = 4 × 4 = 16 square units.Squaring the radius means multiplying it by itself.
A = $1 over 2$ × 16 × $π over 4$ = $16 π over 8$ = 2π square units.The bottom factors multiply to 8, and 16 ÷ 8 = 2.
A ≈ 6.28 square units.2π ≈ 6.2831853, which rounds to 6.28 at two decimal places.

Answer

Exact area: 2π square units.
Rounded area: approximately 6.28 square units.

Check 45° is = of a disk. Its full area is 16π square units, and × 16π = 2π square units.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: A radian angle's fraction of a turn is .

360 counts a full turn in degrees. A full turn in radians is 2π, so the numerator and denominator must use matching angle units.

✓ Instead: Use for radians, or for a degree measurement.

Tips and tricks

Find the slice's fraction of a full turn, then multiply by the whole disk area π $r^{2}$ .

Strategy: step by step

Identify the sector, the filled slice between two radii and their arc, because area measures the inside rather than the rim.
Use the physical slice's nonnegative opening. Convert a degree measurement to radians before using A = $1 over 2$ $r^{2}$ θ. A 300° sector is the large slice; replacing it by 60° changes the chosen region. Magnitude means the nonnegative size of the stated opening.
Square the radius first: $r^{2}$ means r × r. Multiply that result by the radian angle and by $1 over 2$ .
Keep π exact when possible, then round only the final requested decimal. Label the result in square units.
Check with the slice's fraction of 360° times π $r^{2}$ . A slice no larger than the full disk cannot have more than the disk's area.

Strategy

Find the filled slice's area

Is the opening in degrees?

YesConvert to radians, or use its fraction of 360° times π.

NoUse a known radian opening directly.

↓

Is the opening missing but the arc length known?

YesCompute θ = to find the opening, or use the equivalent A = rs.

NoSquare the radius and apply the chosen sector-area formula.

↓

Does the question request a decimal answer?

YesKeep extra digits until the final rounding and state its precision.

NoKeep the exact fraction or π expression with square units.

Identify the filled sector and its nonnegative opening.
If the opening is in degrees, convert it to radians for A = $1 over 2$ $r^{2}$ θ.
If only an arc length is given, divide s by r to find the radian opening first.
Square the radius by multiplying it by itself; this supplies the area scale.
Multiply $1 over 2$ $r^{2}$ θ, keep exact factors such as π, and round only when requested.
Label square units and check using the opening's fraction of a whole disk.

Worked exampleA quarter-disk sector

You are finding the amount of filled surface in the specified circular slice. Find the exact area of a sector with radius 7 cm and central angle 90°.

The curved edge measures length, and the filled slice measures area.

90° × $π radians over 180°$ = $π over 2$ radians.The sector formula uses radians, and 90 is half of 180.
$r^{2}$ = 7 × 7 = 49 c $m^{2}$ .Area uses the square of the radius, so the length unit is squared too.
A = $1 over 2$ × 49 × $π over 2$ = $49 π over 4$ c $m^{2}$ .Multiply the numerators 1 × 49 × π and denominators 2 × 2.

Answer

A = c, exactly.

Check The angle is = of the disk. Its full area is π × = 49π c; one quarter is c.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Rung 1: half a disk

You are finding the filled area of one half of the circle. Find the exact area of a semicircular sector, meaning a half-circle slice, with radius 3 cm and angle π radians.

The curved edge measures length, and the filled slice measures area.

$r^{2}$ = 3 × 3 = 9 c $m^{2}$ .The radius must be squared for an area calculation.
A = $1 over 2$ × 9 × π = $9 π over 2$ c $m^{2}$ .The angle is already in radians, so use the sector formula directly.

Answer

A = c.

Check π radians = 180°, half a full turn. The full disk has area 9π c, so half has area c.

Rung 2Rung 2: one radian of surface

You are finding the amount of filled surface in the specified circular slice. Find the sector area for r = 5 cm and θ = 1 radian.

The curved edge measures length, and the filled slice measures area.

$r^{2}$ = 5 × 5 = 25 c $m^{2}$ .Squaring the radius provides the area scale.
A = $1 over 2$ × 25 × 1 = $25 over 2$ = 12.5 c $m^{2}$ .The angle is in radians; taking half of 25 gives 12.5 exactly.

Answer

A = 12.5 c, exactly.

Check One radian uses of a full turn. That fraction of the 25π c disk is × 25π = c.

Rung 3Rung 3: convert a degree opening

You are finding the amount of filled surface in the specified circular slice. Find the sector area for r = 6 cm and θ = 110°, exactly and to two decimal places.

The curved edge measures length, and the filled slice measures area.

110° × $π radians over 180°$ = $110 π over 180$ = $11 π over 18$ radians.The radian formula requires conversion; divide 110 and 180 by 10.
$r^{2}$ = 6 × 6 = 36 c $m^{2}$ .Area uses the squared radius.
A = $1 over 2$ × 36 × $11 π over 18$ = $396 π over 36$ = 11π c $m^{2}$ .Multiply the numerator and denominator factors, then use 396 ÷ 36 = 11.
A ≈ 34.56 c $m^{2}$ .11π ≈ 34.5575192, which rounds to 34.56 at two decimal places.

Answer

Exact area: 11π c $m^{2}$ .
Rounded area: approximately 34.56 c $m^{2}$ .

Check 110° uses = of the disk. The full disk area is 36π c; × 36π = 11π c.

Rung 4Rung 4: an arc determines its sector

You are finding the turn from its arc and radius, then the filled slice area. An arc of 10 cm lies on a circle of radius 5 cm. Find its central angle and the area of its sector.

The curved edge measures length, and the filled slice measures area.

θ = $10 cm over 5 cm$ = 2 radians.The arc contains two radius-lengths, so divide s by r to find the radian opening.
$r^{2}$ = 5 × 5 = 25 c $m^{2}$ .The sector formula uses the squared radius.
A = $1 over 2$ × 25 × 2 = 25 c $m^{2}$ .The recovered angle is in radians, and the factors $1 over 2$ and 2 multiply to 1.

Answer

Central angle: 2 radians.
Sector area: 25 c $m^{2}$ .

Check The circumference is 10π cm, so the 10 cm arc uses = of the rim. The same fraction of the 25π c disk is × 25π = 25 c. Also A = rs gives × 5 × 10 = 25 c.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: With r = 7 cm, use 2r = 14 in place of in the sector formula.

Squaring multiplies a number by itself. Doubling adds two copies. Area scales in two length directions, so the formula needs the square.

✓ Instead: = 7 × 7 = 49 c.

✗ Not this: A sector area can be reported in centimeters.

Centimeters measure length along a path. Area counts covered squares.

✓ Instead: Write square centimeters, c, for an area answer.

✗ Not this: Use θ with θ written as 72 degrees.

The one-half formula comes from the full-turn radian measure 2π. Substituting a degree count changes its meaning.

✓ Instead: Convert 72° to radians first, or use the degree fraction times π.

Tips and tricks

Draw and shade the filled slice so you distinguish its surface from the curved edge.
Write $r^{2}$ = r × r before multiplying other factors.
Check the result as a fraction of the full disk area π $r^{2}$ . For one sector, the result must fit inside that full area.

Trap. Using r instead of , or reporting area in centimeters. Write the radius multiplication explicitly, such as = 7 × 7 = 49, and give square units. The angle must also be in radians for A = θ.

Keep in mind

Square the radius first: with r = 6, $r^{2}$ = 6 × 6 = 36, not 6 × 2 = 12.
Area comes in square units: 15π c $m^{2}$ , not 15π cm.
The formula needs radians; with degrees use A = $θ over 360°$ π $r^{2}$ , as in $150° over 360°$ × 36π = 15π.
Doubling the radius multiplies the area by 4, not 2, because r is squared: r = 12 gives $r^{2}$ = 144, which is 4 × 36.

Memory hookCrust versus cheese: crust is s = rθ, cheese is A = θ. Cheese covers area, so the radius gets squared.

Flash cards: say the answer out loud, then flip

What is a sector?

The slice of a disk between two radii and their arc, like a slice of pizza.

What is a disk?

The whole filled circle: the full pizza, not only its crust.

State the sector area formula and its unit rule.

A = θ, with θ in radians

r = 4 m and θ = 3 radians. Find A.

24 , from × 16 × 3

r = 12 cm and θ = 35°. Find A exactly.

35° = , so A = × 144 × = 14π c, about 43.98 c

A student writes A = × 6 × for r = 6. What went wrong?

The radius was not squared: use = 36, which gives 15π.