Practice · All radii in one circle are equal · Sensim Math

Variant 1 easy

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $20\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $4\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

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1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 20 cm and the radius to a corner is 4 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 20 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 4 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

20 \div 4 = 5

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 4 cm.

4 \times 2 = 8

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 5 cm) plus the diagonal DB (8 cm). Add them for the perimeter.

5 + 5 + 8 = 18

Perimeter is just the total of the side lengths once each side is known.

Answer: 18 cm

4 · Reviewdoes it hold up?

The diagonal (8 cm) matches the picture where the diameter stretches across the square. Two sides of 5 plus a 8 give 18 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 2 easy

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $24\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $5\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 24 cm and the radius to a corner is 5 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 24 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 5 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

24 \div 4 = 6

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 5 cm.

5 \times 2 = 10

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 6 cm) plus the diagonal DB (10 cm). Add them for the perimeter.

6 + 6 + 10 = 22

Perimeter is just the total of the side lengths once each side is known.

Answer: 22 cm

4 · Reviewdoes it hold up?

The diagonal (10 cm) matches the picture where the diameter stretches across the square. Two sides of 6 plus a 10 give 22 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 3 easy

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $36\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $6\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 36 cm and the radius to a corner is 6 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 36 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 6 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

36 \div 4 = 9

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 6 cm.

6 \times 2 = 12

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 9 cm) plus the diagonal DB (12 cm). Add them for the perimeter.

9 + 9 + 12 = 30

Perimeter is just the total of the side lengths once each side is known.

Answer: 30 cm

4 · Reviewdoes it hold up?

The diagonal (12 cm) matches the picture where the diameter stretches across the square. Two sides of 9 plus a 12 give 30 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 4 medium

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $40\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $7\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 40 cm and the radius to a corner is 7 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 40 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 7 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

40 \div 4 = 10

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 7 cm.

7 \times 2 = 14

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 10 cm) plus the diagonal DB (14 cm). Add them for the perimeter.

10 + 10 + 14 = 34

Perimeter is just the total of the side lengths once each side is known.

Answer: 34 cm

4 · Reviewdoes it hold up?

The diagonal (14 cm) matches the picture where the diameter stretches across the square. Two sides of 10 plus a 14 give 34 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 5 medium

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $40\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $8\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 40 cm and the radius to a corner is 8 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 40 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 8 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

40 \div 4 = 10

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 8 cm.

8 \times 2 = 16

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 10 cm) plus the diagonal DB (16 cm). Add them for the perimeter.

10 + 10 + 16 = 36

Perimeter is just the total of the side lengths once each side is known.

Answer: 36 cm

4 · Reviewdoes it hold up?

The diagonal (16 cm) matches the picture where the diameter stretches across the square. Two sides of 10 plus a 16 give 36 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 6 medium

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $48\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $9\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 48 cm and the radius to a corner is 9 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 48 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 9 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

48 \div 4 = 12

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 9 cm.

9 \times 2 = 18

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 12 cm) plus the diagonal DB (18 cm). Add them for the perimeter.

12 + 12 + 18 = 42

Perimeter is just the total of the side lengths once each side is known.

Answer: 42 cm

4 · Reviewdoes it hold up?

The diagonal (18 cm) matches the picture where the diameter stretches across the square. Two sides of 12 plus a 18 give 42 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 7 medium

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $60\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $10\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 60 cm and the radius to a corner is 10 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 60 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 10 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

60 \div 4 = 15

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 10 cm.

10 \times 2 = 20

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 15 cm) plus the diagonal DB (20 cm). Add them for the perimeter.

15 + 15 + 20 = 50

Perimeter is just the total of the side lengths once each side is known.

Answer: 50 cm

4 · Reviewdoes it hold up?

The diagonal (20 cm) matches the picture where the diameter stretches across the square. Two sides of 15 plus a 20 give 50 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 8 hard

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $80\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $14\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 80 cm and the radius to a corner is 14 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 80 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 14 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

80 \div 4 = 20

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 14 cm.

14 \times 2 = 28

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 20 cm) plus the diagonal DB (28 cm). Add them for the perimeter.

20 + 20 + 28 = 68

Perimeter is just the total of the side lengths once each side is known.

Answer: 68 cm

4 · Reviewdoes it hold up?

The diagonal (28 cm) matches the picture where the diameter stretches across the square. Two sides of 20 plus a 28 give 68 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 9 hard

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $100\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $18\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 100 cm and the radius to a corner is 18 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 100 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 18 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

100 \div 4 = 25

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 18 cm.

18 \times 2 = 36

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 25 cm) plus the diagonal DB (36 cm). Add them for the perimeter.

25 + 25 + 36 = 86

Perimeter is just the total of the side lengths once each side is known.

Answer: 86 cm

4 · Reviewdoes it hold up?

The diagonal (36 cm) matches the picture where the diameter stretches across the square. Two sides of 25 plus a 36 give 86 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

Variant 10 hard

The figure on the right shows a square ABCD drawn inside a circle, with all four vertices on the circle. The perimeter of square ABCD is $120\text{ cm}$ . The diagonal DB passes through the center, so it is a diameter, and the radius from the center to a vertex is $21\text{ cm}$ . What is the perimeter of triangle DBC, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square ABCD is inscribed in a circle (all corners on the circle). Its perimeter is 120 cm and the radius to a corner is 21 cm. The diagonal DB is a diameter. Find the perimeter of triangle DBC, whose sides are two square sides and that diagonal.

Givens

Square ABCD is inscribed in the circle, all four vertices on the circle.
Perimeter of the square is 120 cm.
Diagonal DB passes through the center and is a diameter of the circle.
The radius from the center to a vertex is 21 cm.

Unknowns

The perimeter of triangle DBC in cm.

Constraints

All four sides of a square are equal.
All radii in one circle are equal, so the diameter is twice the radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #7 Identify Subproblems

The picture carries the key facts: the diagonal is a diameter (so its length comes from the radius), and the triangle's other two sides are sides of the square. Find each side as a small subproblem, then add the three sides.

3 · Execute3 carry out the plan

1Find one side of the square

#7 Identify Subproblems 3.G.A.1

A square has four equal sides, so each side is the perimeter divided by 4.

120 \div 4 = 30

Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.

2Find the diagonal (a diameter)

#1 Draw a Diagram 3.G.A.1

The diagonal DB goes through the center, so it is a diameter. A diameter is twice the radius, and the radius is 21 cm.

21 \times 2 = 42

Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.

3Add the three sides of the triangle

#7 Identify Subproblems 3.MD.D.8

Triangle DBC has two sides that are sides of the square (each 30 cm) plus the diagonal DB (42 cm). Add them for the perimeter.

30 + 30 + 42 = 102

Perimeter is just the total of the side lengths once each side is known.

Answer: 102 cm

4 · Reviewdoes it hold up?

The diagonal (42 cm) matches the picture where the diameter stretches across the square. Two sides of 30 plus a 42 give 102 cm, a sensible perimeter.

Another way: Guess and check (tool 6): the square side must make four equal sides total the perimeter, and the diameter must be two radii in a row; adding confirms the answer.

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Using that a square has four equal sides and that a line through the center is a diameter equal to twice the radius.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Adding the three side lengths to get the triangle's perimeter.

💡Takeaway. This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!

All radii in one circle are equal · 10 practice problems

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