Practice · Perimeter of shape joining circle centers · Sensim Math

Variant 1 easy

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $18\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 18 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 18 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 18 inches, so divide by 3.

18 \div 3 = 6

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 6 in

4 · Reviewdoes it hold up?

A diameter of 6 inches gives a triangle perimeter of 3 x 6 = 18 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 18 and solve for the unknown d: d = 18 / 3 = 6 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 18 / 3 = 6.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 2 easy

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $24\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 24 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 24 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 24 inches, so divide by 3.

24 \div 3 = 8

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 8 in

4 · Reviewdoes it hold up?

A diameter of 8 inches gives a triangle perimeter of 3 x 8 = 24 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 24 and solve for the unknown d: d = 24 / 3 = 8 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 24 / 3 = 8.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 3 easy

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $27\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 27 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 27 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 27 inches, so divide by 3.

27 \div 3 = 9

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 9 in

4 · Reviewdoes it hold up?

A diameter of 9 inches gives a triangle perimeter of 3 x 9 = 27 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 27 and solve for the unknown d: d = 27 / 3 = 9 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 27 / 3 = 9.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 4 medium

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $30\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 30 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 30 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 30 inches, so divide by 3.

30 \div 3 = 10

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 10 in

4 · Reviewdoes it hold up?

A diameter of 10 inches gives a triangle perimeter of 3 x 10 = 30 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 30 and solve for the unknown d: d = 30 / 3 = 10 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 30 / 3 = 10.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 5 medium

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $33\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 33 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 33 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 33 inches, so divide by 3.

33 \div 3 = 11

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 11 in

4 · Reviewdoes it hold up?

A diameter of 11 inches gives a triangle perimeter of 3 x 11 = 33 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 33 and solve for the unknown d: d = 33 / 3 = 11 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 33 / 3 = 11.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 6 medium

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $36\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 36 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 36 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 36 inches, so divide by 3.

36 \div 3 = 12

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 12 in

4 · Reviewdoes it hold up?

A diameter of 12 inches gives a triangle perimeter of 3 x 12 = 36 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 36 and solve for the unknown d: d = 36 / 3 = 12 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 36 / 3 = 12.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 7 medium

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $45\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 45 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 45 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 45 inches, so divide by 3.

45 \div 3 = 15

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 15 in

4 · Reviewdoes it hold up?

A diameter of 15 inches gives a triangle perimeter of 3 x 15 = 45 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 45 and solve for the unknown d: d = 45 / 3 = 15 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 45 / 3 = 15.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 8 hard

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $48\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 48 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 48 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 48 inches, so divide by 3.

48 \div 3 = 16

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 16 in

4 · Reviewdoes it hold up?

A diameter of 16 inches gives a triangle perimeter of 3 x 16 = 48 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 48 and solve for the unknown d: d = 48 / 3 = 16 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 48 / 3 = 16.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 9 hard

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $60\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 60 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 60 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 60 inches, so divide by 3.

60 \div 3 = 20

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 20 in

4 · Reviewdoes it hold up?

A diameter of 20 inches gives a triangle perimeter of 3 x 20 = 60 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 60 and solve for the unknown d: d = 60 / 3 = 20 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 60 / 3 = 20.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Variant 10 hard

The figure on the right is made of equal circles drawn so that neighboring circles touch. The center of each circle is joined to the centers of its neighbors to form a triangle. If the perimeter of the triangle is $90\text{ in}$ , what is the diameter of one circle, in inches?

Your answer

Show solution

1 · Understandwhat's really being asked

Three equal circles are placed so each touches the others, with one on top and two below. Joining the three centers makes an equilateral triangle, and the triangle's perimeter is 90 inches. We must find the diameter of one circle.

Givens

There are three circles, all the same size.
Neighboring circles touch (are tangent).
The three centers are joined to form an equilateral triangle.
The triangle's perimeter is 90 inches.

Unknowns

The diameter of one circle in inches.

Constraints

When two equal circles touch on the outside, the distance between their centers equals one radius plus one radius, which is the diameter.
So each side of the triangle equals one diameter; the triangle has three equal sides.

2 · Planchoose the strategy

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

Draw two touching circles and mark the center-to-center distance as a diameter. Then each triangle side is one diameter, so divide the perimeter by 3 to get the diameter.

3 · Execute3 carry out the plan

1One triangle side equals one diameter

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

Where two equal circles touch, the touching point is one radius from each center. So the distance between the two centers is radius + radius = diameter. Each side of the triangle joins two touching centers, so each side is one diameter.

\text{side} = r + r = d

Two radii laid end to end across the touch point make exactly one diameter.

2The triangle is three diameters around

#7 Identify Subproblems 3.MD.D.8

The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.

\text{perimeter} = d + d + d = 3 \times d

Perimeter adds up the three equal sides, each one diameter long.

3Divide to find the diameter

#7 Identify Subproblems 3.OA.C.7

Three diameters make 90 inches, so divide by 3.

90 \div 3 = 30

Sharing the perimeter equally among 3 equal sides is division within 100.

Answer: 30 in

4 · Reviewdoes it hold up?

A diameter of 30 inches gives a triangle perimeter of 3 x 30 = 90 inches, matching the given. Units are inches, correct for a length.

Another way: Use 3 x d = 90 and solve for the unknown d: d = 90 / 3 = 30 in (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Seeing each triangle side as a diameter (radius + radius) between two touching circles.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Expressing the triangle perimeter as three equal sides.
3.OA.C.7 Fluently multiply and divide within 100 — Computing 90 / 3 = 30.

💡Takeaway. Each side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!

Perimeter of shape joining circle centers · 10 practice problems

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