← Sectors of one radius add by angle, and the total can pass a full circle · Circumference and Area of a Circle

Sectors of one radius add by angle, and the total can pass a full circle · 12 practice problems

7.G.B.48.G.A.5

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

In the figure at the right, a part of a circle with radius $5\ \text{cm}$ is drawn at each vertex of a quadrilateral and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 5 cm sits at each of a quadrilateral's 4 vertices. We want their total area.

Givens

The polygon has 4 sides.
Every sector has a radius of 5 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A quadrilateral splits into 2 triangles, each worth 180 degrees.

(4 - 2) \times 180 = 360

360 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

360 \div 360 = 1

1 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 5 cm.

5 \times 5 \times 3.14 = 78.5

78.5 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

78.5 \times 1 = 78.5

78.5 cm2 shaded.

Answer: 78.5 cm²

4 · Reviewdoes it hold up?

78.5 cm2 is less than one full circle of 78.5 cm2, which is right: the 4 angles add to 360 degrees, short of a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 2 easy

In the figure at the right, a part of a circle with radius $4\ \text{cm}$ is drawn at each vertex of a hexagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 4 cm sits at each of a hexagon's 6 vertices. We want their total area.

Givens

The polygon has 6 sides.
Every sector has a radius of 4 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A hexagon splits into 4 triangles, each worth 180 degrees.

(6 - 2) \times 180 = 720

720 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

720 \div 360 = 2

2 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 4 cm.

4 \times 4 \times 3.14 = 50.24

50.24 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

50.24 \times 2 = 100.48

100.48 cm2 shaded.

Answer: 100.48 cm²

4 · Reviewdoes it hold up?

100.48 cm2 is more than one full circle of 50.24 cm2, which is right: the 6 angles add to 720 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 3 easy

In the figure at the right, a part of a circle with radius $6\ \text{cm}$ is drawn at each vertex of a pentagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 6 cm sits at each of a pentagon's 5 vertices. We want their total area.

Givens

The polygon has 5 sides.
Every sector has a radius of 6 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A pentagon splits into 3 triangles, each worth 180 degrees.

(5 - 2) \times 180 = 540

540 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

540 \div 360 = 1.5

1.5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 6 cm.

6 \times 6 \times 3.14 = 113.04

113.04 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

113.04 \times 1.5 = 169.56

169.56 cm2 shaded.

Answer: 169.56 cm²

4 · Reviewdoes it hold up?

169.56 cm2 is more than one full circle of 113.04 cm2, which is right: the 5 angles add to 540 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 4 easy

In the figure at the right, a part of a circle with radius $6\ \text{cm}$ is drawn at each vertex of a heptagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 6 cm sits at each of a heptagon's 7 vertices. We want their total area.

Givens

The polygon has 7 sides.
Every sector has a radius of 6 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A heptagon splits into 5 triangles, each worth 180 degrees.

(7 - 2) \times 180 = 900

900 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

900 \div 360 = 2.5

2.5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 6 cm.

6 \times 6 \times 3.14 = 113.04

113.04 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

113.04 \times 2.5 = 282.6

282.6 cm2 shaded.

Answer: 282.6 cm²

4 · Reviewdoes it hold up?

282.6 cm2 is more than one full circle of 113.04 cm2, which is right: the 7 angles add to 900 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 5 medium

In the figure at the right, a part of a circle with radius $3\ \text{cm}$ is drawn at each vertex of a octagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 3 cm sits at each of a octagon's 8 vertices. We want their total area.

Givens

The polygon has 8 sides.
Every sector has a radius of 3 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A octagon splits into 6 triangles, each worth 180 degrees.

(8 - 2) \times 180 = 1080

1080 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

1080 \div 360 = 3

3 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 3 cm.

3 \times 3 \times 3.14 = 28.26

28.26 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

28.26 \times 3 = 84.78

84.78 cm2 shaded.

Answer: 84.78 cm²

4 · Reviewdoes it hold up?

84.78 cm2 is more than one full circle of 28.26 cm2, which is right: the 8 angles add to 1080 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 6 medium

In the figure at the right, a part of a circle with radius $8\ \text{cm}$ is drawn at each vertex of a quadrilateral and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 8 cm sits at each of a quadrilateral's 4 vertices. We want their total area.

Givens

The polygon has 4 sides.
Every sector has a radius of 8 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A quadrilateral splits into 2 triangles, each worth 180 degrees.

(4 - 2) \times 180 = 360

360 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

360 \div 360 = 1

1 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 8 cm.

8 \times 8 \times 3.14 = 200.96

200.96 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

200.96 \times 1 = 200.96

200.96 cm2 shaded.

Answer: 200.96 cm²

4 · Reviewdoes it hold up?

200.96 cm2 is less than one full circle of 200.96 cm2, which is right: the 4 angles add to 360 degrees, short of a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 7 medium

In the figure at the right, a part of a circle with radius $4\ \text{cm}$ is drawn at each vertex of a nonagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 4 cm sits at each of a nonagon's 9 vertices. We want their total area.

Givens

The polygon has 9 sides.
Every sector has a radius of 4 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A nonagon splits into 7 triangles, each worth 180 degrees.

(9 - 2) \times 180 = 1260

1260 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

1260 \div 360 = 3.5

3.5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 4 cm.

4 \times 4 \times 3.14 = 50.24

50.24 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

50.24 \times 3.5 = 175.84

175.84 cm2 shaded.

Answer: 175.84 cm²

4 · Reviewdoes it hold up?

175.84 cm2 is more than one full circle of 50.24 cm2, which is right: the 9 angles add to 1260 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 8 hard

In the figure at the right, a part of a circle with radius $10\ \text{cm}$ is drawn at each vertex of a hexagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 10 cm sits at each of a hexagon's 6 vertices. We want their total area.

Givens

The polygon has 6 sides.
Every sector has a radius of 10 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A hexagon splits into 4 triangles, each worth 180 degrees.

(6 - 2) \times 180 = 720

720 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

720 \div 360 = 2

2 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 10 cm.

10 \times 10 \times 3.14 = 314

314 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

314 \times 2 = 628

628 cm2 shaded.

Answer: 628 cm²

4 · Reviewdoes it hold up?

628 cm2 is more than one full circle of 314 cm2, which is right: the 6 angles add to 720 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 9 hard

In the figure at the right, a part of a circle with radius $5\ \text{cm}$ is drawn at each vertex of a decagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 5 cm sits at each of a decagon's 10 vertices. We want their total area.

Givens

The polygon has 10 sides.
Every sector has a radius of 5 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A decagon splits into 8 triangles, each worth 180 degrees.

(10 - 2) \times 180 = 1440

1440 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

1440 \div 360 = 4

4 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 5 cm.

5 \times 5 \times 3.14 = 78.5

78.5 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

78.5 \times 4 = 314

314 cm2 shaded.

Answer: 314 cm²

4 · Reviewdoes it hold up?

314 cm2 is more than one full circle of 78.5 cm2, which is right: the 10 angles add to 1440 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 10 medium

In the figure at the right, a part of a circle with radius $10\ \text{cm}$ is drawn at each vertex of a pentagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 10 cm sits at each of a pentagon's 5 vertices. We want their total area.

Givens

The polygon has 5 sides.
Every sector has a radius of 10 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A pentagon splits into 3 triangles, each worth 180 degrees.

(5 - 2) \times 180 = 540

540 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

540 \div 360 = 1.5

1.5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 10 cm.

10 \times 10 \times 3.14 = 314

314 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

314 \times 1.5 = 471

471 cm2 shaded.

Answer: 471 cm²

4 · Reviewdoes it hold up?

471 cm2 is more than one full circle of 314 cm2, which is right: the 5 angles add to 540 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 11 hard

In the figure at the right, a part of a circle with radius $20\ \text{cm}$ is drawn at each vertex of a pentagon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 20 cm sits at each of a pentagon's 5 vertices. We want their total area.

Givens

The polygon has 5 sides.
Every sector has a radius of 20 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A pentagon splits into 3 triangles, each worth 180 degrees.

(5 - 2) \times 180 = 540

540 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

540 \div 360 = 1.5

1.5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 20 cm.

20 \times 20 \times 3.14 = 1256

1256 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

1256 \times 1.5 = 1884

1884 cm2 shaded.

Answer: 1884 cm²

4 · Reviewdoes it hold up?

1884 cm2 is more than one full circle of 1256 cm2, which is right: the 5 angles add to 540 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Variant 12 hard

In the figure at the right, a part of a circle with radius $2\ \text{cm}$ is drawn at each vertex of a 12-sided polygon and shaded. What is the total area of the shaded parts, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A sector of radius 2 cm sits at each of a 12-sided polygon's 12 vertices. We want their total area.

Givens

The polygon has 12 sides.
Every sector has a radius of 2 cm.
Each sector's angle is the polygon's angle at that vertex.

Unknowns

The total area of the shaded sectors.

Constraints

The polygon's own angles are never given individually.

2 · Planchoose the strategy

#16 Count the Complement · also uses: #7 Identify Subproblems#5 Look for a Pattern

The sectors share a radius, so only their angles matter -- and nobody needs the angles one at a time, only their sum, which any polygon's angles have without being measured.

3 · Execute4 carry out the plan

1Add the polygon's angles

#5 Look for a Pattern 8.G.A.5

A 12-sided polygon splits into 10 triangles, each worth 180 degrees.

(12 - 2) \times 180 = 1800

1800 degrees in all.

2Compare that with a full turn

#16 Count the Complement 7.G.B.4

A whole circle is 360 degrees, so the sectors together are that many circles.

1800 \div 360 = 5

5 circles' worth.

3Find one whole circle

#7 Identify Subproblems 7.G.B.4

Radius 2 cm.

2 \times 2 \times 3.14 = 12.56

12.56 cm2 for a full circle.

4Take that many circles

#7 Identify Subproblems 7.G.B.4

The sectors can be slid together into that much of a circle.

12.56 \times 5 = 62.8

62.8 cm2 shaded.

Answer: 62.8 cm²

4 · Reviewdoes it hold up?

62.8 cm2 is more than one full circle of 12.56 cm2, which is right: the 12 angles add to 1800 degrees, past a full turn.

Another way: Measuring each angle and taking each sector separately gives the same total, but the polygon never says what the individual angles are -- only their sum is fixed.

Standardsmin grade 8

7.G.B.4 Know the formulas for area and circumference of a circle — Getting a circle's area and taking a share of it.
8.G.A.5 Use informal arguments to establish facts about angle sum and exterior angles — Using the angle sum of a polygon.

💡Takeaway. Sectors with the same radius add up by their angles alone -- and the total can be more than one whole circle.

Where next?

← Back to the problem: Sectors of one radius add by angle, and the total can pass a full circle