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← 3-2 · All radii in one circle are equal · Radius and Diameter Relationships

All radii in one circle are equal · 10 practice problems

3.MD.D.83.G.A.1

Generated variants — 10

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

Variant 1 answer: 22 cm

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 cm24\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 cm5\text{ cm}. What is the perimeter of triangle DBC, in cm?

5 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
24÷4=624 \div 4 = 6
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
5×2=105 \times 2 = 10
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=226 + 6 + 10 = 22
Perimeter is just the total of the side lengths once each side is known.
Answer: 22 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 2 answer: 50 cm

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 cm60\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 cm10\text{ cm}. What is the perimeter of triangle DBC, in cm?

10 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
60÷4=1560 \div 4 = 15
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
10×2=2010 \times 2 = 20
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=5015 + 15 + 20 = 50
Perimeter is just the total of the side lengths once each side is known.
Answer: 50 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 3 answer: 30 cm

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 cm36\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 cm6\text{ cm}. What is the perimeter of triangle DBC, in cm?

6 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
36÷4=936 \div 4 = 9
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
6×2=126 \times 2 = 12
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=309 + 9 + 12 = 30
Perimeter is just the total of the side lengths once each side is known.
Answer: 30 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 4 answer: 34 cm

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 cm40\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 cm7\text{ cm}. What is the perimeter of triangle DBC, in cm?

7 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
40÷4=1040 \div 4 = 10
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
7×2=147 \times 2 = 14
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=3410 + 10 + 14 = 34
Perimeter is just the total of the side lengths once each side is known.
Answer: 34 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 5 answer: 86 cm

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 cm100\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 cm18\text{ cm}. What is the perimeter of triangle DBC, in cm?

18 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
100÷4=25100 \div 4 = 25
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
18×2=3618 \times 2 = 36
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=8625 + 25 + 36 = 86
Perimeter is just the total of the side lengths once each side is known.
Answer: 86 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 6 answer: 102 cm

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 cm120\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 cm21\text{ cm}. What is the perimeter of triangle DBC, in cm?

21 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
120÷4=30120 \div 4 = 30
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
21×2=4221 \times 2 = 42
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=10230 + 30 + 42 = 102
Perimeter is just the total of the side lengths once each side is known.
Answer: 102 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 7 answer: 68 cm

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 cm80\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 cm14\text{ cm}. What is the perimeter of triangle DBC, in cm?

14 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
80÷4=2080 \div 4 = 20
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
14×2=2814 \times 2 = 28
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=6820 + 20 + 28 = 68
Perimeter is just the total of the side lengths once each side is known.
Answer: 68 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 8 answer: 36 cm

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 cm40\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 cm8\text{ cm}. What is the perimeter of triangle DBC, in cm?

8 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
40÷4=1040 \div 4 = 10
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
8×2=168 \times 2 = 16
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=3610 + 10 + 16 = 36
Perimeter is just the total of the side lengths once each side is known.
Answer: 36 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 9 answer: 18 cm

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 cm20\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 cm4\text{ cm}. What is the perimeter of triangle DBC, in cm?

4 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
20÷4=520 \div 4 = 5
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
4×2=84 \times 2 = 8
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=185 + 5 + 8 = 18
Perimeter is just the total of the side lengths once each side is known.
Answer: 18 cm

Review

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.

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.

Standards · min 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.
💡 This only needs Grade 3 shape sense: equal square sides, and a diameter is just two radii in a row!
Variant 10 answer: 42 cm

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 cm48\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 cm9\text{ cm}. What is the perimeter of triangle DBC, in cm?

9 cm A B C D
Show solution

Understand

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.

Plan

#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.

Execute

#7 Identify Subproblems 3.G.A.1
A square has four equal sides, so each side is the perimeter divided by 4.
48÷4=1248 \div 4 = 12
Knowing a square has four equal sides is a basic shape-attribute idea taught in grade 3.
#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.
9×2=189 \times 2 = 18
Reading the figure shows the diagonal is a straight line through the center, which makes it a diameter.
#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=4212 + 12 + 18 = 42
Perimeter is just the total of the side lengths once each side is known.
Answer: 42 cm

Review

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.

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.

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