Practice · Cut a formula-less shape at the centre into sectors and triangles · Sensim Math

Variant 1 easy

In the figure at the right, a circle with a radius of $42\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 42 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 42 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 120, 30, 120, 90.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 120, 30, 120, 90 degrees.

120 + 30 + 120 + 90 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

1846.32,\quad 461.58,\quad 1846.32,\quad 882

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

1846.32 + 461.58 + 1846.32 + 882 = 5036.22

5036.22 cm2 shaded.

Answer: 5036.22 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 5036.22 cm2 is less than the whole circle's 5538.96 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 5538.96 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 2 easy

In the figure at the right, a circle with a radius of $12\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 12 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 12 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 60, 90, 120.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 60, 90, 120 degrees.

90 + 60 + 90 + 120 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

72,\quad 75.36,\quad 72,\quad 150.72

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

72 + 75.36 + 72 + 150.72 = 370.08

370.08 cm2 shaded.

Answer: 370.08 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 370.08 cm2 is less than the whole circle's 452.16 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 452.16 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 3 easy

In the figure at the right, a circle with a radius of $48\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 48 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 48 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 30, 120, 90, 120.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 30, 120, 90, 120 degrees.

30 + 120 + 90 + 120 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

602.88,\quad 2411.52,\quad 1152,\quad 2411.52

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

602.88 + 2411.52 + 1152 + 2411.52 = 6577.92

6577.92 cm2 shaded.

Answer: 6577.92 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 6577.92 cm2 is less than the whole circle's 7234.56 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 7234.56 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 4 easy

In the figure at the right, a circle with a radius of $6\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 6 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 6 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 90, 60, 120.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 90, 60, 120 degrees.

90 + 90 + 60 + 120 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

18,\quad 18,\quad 18.84,\quad 37.68

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

18 + 18 + 18.84 + 37.68 = 92.52

92.52 cm2 shaded.

Answer: 92.52 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 92.52 cm2 is less than the whole circle's 113.04 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 113.04 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 5 medium

In the figure at the right, a circle with a radius of $54\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 54 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 54 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 60, 150, 60, 90.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 60, 150, 60, 90 degrees.

60 + 150 + 60 + 90 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

1526.04,\quad 3815.1,\quad 1526.04,\quad 1458

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

1526.04 + 3815.1 + 1526.04 + 1458 = 8325.18

8325.18 cm2 shaded.

Answer: 8325.18 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 8325.18 cm2 is less than the whole circle's 9156.24 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 9156.24 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 6 medium

In the figure at the right, a circle with a radius of $54\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 54 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 54 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 30, 90, 150.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 30, 90, 150 degrees.

90 + 30 + 90 + 150 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

1458,\quad 763.02,\quad 1458,\quad 3815.1

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

1458 + 763.02 + 1458 + 3815.1 = 7494.12

7494.12 cm2 shaded.

Answer: 7494.12 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 7494.12 cm2 is less than the whole circle's 9156.24 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 9156.24 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 7 medium

In the figure at the right, a circle with a radius of $12\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 12 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 12 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 30, 90, 180, 60.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 30, 90, 180, 60 degrees.

30 + 90 + 180 + 60 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

37.68,\quad 72,\quad 226.08,\quad 75.36

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

37.68 + 72 + 226.08 + 75.36 = 411.12

411.12 cm2 shaded.

Answer: 411.12 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 411.12 cm2 is less than the whole circle's 452.16 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 452.16 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 8 medium

In the figure at the right, a circle with a radius of $48\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 48 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 48 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 60, 180, 30.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 60, 180, 30 degrees.

90 + 60 + 180 + 30 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

1152,\quad 1205.76,\quad 3617.28,\quad 602.88

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

1152 + 1205.76 + 3617.28 + 602.88 = 6577.92

6577.92 cm2 shaded.

Answer: 6577.92 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 6577.92 cm2 is less than the whole circle's 7234.56 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 7234.56 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 9 hard

In the figure at the right, a circle with a radius of $24\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 24 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 24 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 180, 30, 60.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 180, 30, 60 degrees.

90 + 180 + 30 + 60 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

288,\quad 904.32,\quad 150.72,\quad 301.44

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

288 + 904.32 + 150.72 + 301.44 = 1644.48

1644.48 cm2 shaded.

Answer: 1644.48 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 1644.48 cm2 is less than the whole circle's 1808.64 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 1808.64 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 10 hard

In the figure at the right, a circle with a radius of $24\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 24 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 24 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 90, 30, 60, 180.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 90, 30, 60, 180 degrees.

90 + 30 + 60 + 180 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

288,\quad 150.72,\quad 301.44,\quad 904.32

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

288 + 150.72 + 301.44 + 904.32 = 1644.48

1644.48 cm2 shaded.

Answer: 1644.48 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 1644.48 cm2 is less than the whole circle's 1808.64 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 1808.64 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 11 hard

In the figure at the right, a circle with a radius of $48\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 48 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 48 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 180, 60, 30, 90.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 180, 60, 30, 90 degrees.

180 + 60 + 30 + 90 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

3617.28,\quad 1205.76,\quad 602.88,\quad 1152

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

3617.28 + 1205.76 + 602.88 + 1152 = 6577.92

6577.92 cm2 shaded.

Answer: 6577.92 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 6577.92 cm2 is less than the whole circle's 7234.56 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 7234.56 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Variant 12 hard

In the figure at the right, a circle with a radius of $24\ \text{cm}$ has its circumference divided into $12$ equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in $\text{cm}^2$ ?

Your answer

Show solution

1 · Understandwhat's really being asked

A circle of radius 24 cm has its rim cut into 12 equal parts, and a region bounded by chords and arcs is shaded. We want its area.

Givens

The radius is 24 cm.
The rim is divided into 12 equal parts, so each step is 30 degrees.
The shaded region's four wedges have central angles 30, 30, 90, 210.

Unknowns

The area of the shaded region.

Constraints

The region is bounded partly by chords and partly by arcs.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems#8 Analyze the Units

The whole shape has no formula. Cutting it at the centre gives four wedges that do -- but not all of the same kind: a right-angled wedge with a chord is a triangle, not a sector.

3 · Execute4 carry out the plan

1Find what one step is worth

#8 Analyze the Units 6.RP.A.3

12 equal parts of a full turn.

360 \div 12 = 30

Each step is 30 degrees.

2Cut the region at the centre

#1 Draw a Diagram 7.G.B.5

Four wedges, with angles 30, 30, 90, 210 degrees.

30 + 30 + 90 + 210 = 360

Four pieces to measure.

3Measure each by its kind

#7 Identify Subproblems 7.G.B.4

A wedge cut off by a chord at a right angle is a triangle on two radii; the others are sectors of the circle.

150.72,\quad 150.72,\quad 288,\quad 1055.04

Not all of them are sectors.

4Add the four

#7 Identify Subproblems 7.G.B.4

They do not overlap, so the areas add.

150.72 + 150.72 + 288 + 1055.04 = 1644.48

1644.48 cm2 shaded.

Answer: 1644.48 cm²

4 · Reviewdoes it hold up?

The four angles add to 360 degrees, a full turn, and the shaded area 1644.48 cm2 is less than the whole circle's 1808.64 cm2 -- less by exactly the bits the chords cut off.

Another way: Treating every wedge as a sector would give 1808.64 cm2, the whole circle: the chords are what make the difference, and they only appear if each wedge is read for what it is.

Standardsmin grade 7

7.G.B.4 Know the formulas for area and circumference of a circle — Measuring the sectors as shares of the circle.
7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles — Using the central angles the twelve points create.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning each angle into a share of the whole turn.

💡Takeaway. Cut an odd shape at the centre. Then look at each piece and ask what it actually is -- some wedges are triangles.

Cut a formula-less shape at the centre into sectors and triangles · 12 practice problems

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