Practice · Center to edge distance is the radius · Sensim Math

Variant 1 answer: 8 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $32\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 32 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 32 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 32 cm, so divide to find one side, which is the radius.

32 \div 4 = 8

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 8 cm.

AB = r = 8

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 8 cm

Review

AB (8 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (8 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 8 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 32 cm to find one side: 32 / 4 = 8.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 2 answer: 10 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $40\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 40 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 40 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 40 cm, so divide to find one side, which is the radius.

40 \div 4 = 10

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 10 cm.

AB = r = 10

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 10 cm

Review

AB (10 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (10 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 10 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 40 cm to find one side: 40 / 4 = 10.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 3 answer: 9 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $36\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 36 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 36 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 36 cm, so divide to find one side, which is the radius.

36 \div 4 = 9

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 9 cm.

AB = r = 9

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 9 cm

Review

AB (9 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (9 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 9 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 36 cm to find one side: 36 / 4 = 9.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 4 answer: 6 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $24\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 24 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 24 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 24 cm, so divide to find one side, which is the radius.

24 \div 4 = 6

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 6 cm.

AB = r = 6

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 6 cm

Review

AB (6 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (6 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 6 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 24 cm to find one side: 24 / 4 = 6.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 5 answer: 2 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $8\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 8 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 8 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 8 cm, so divide to find one side, which is the radius.

8 \div 4 = 2

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 2 cm.

AB = r = 2

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 2 cm

Review

AB (2 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (2 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 2 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 8 cm to find one side: 8 / 4 = 2.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 6 answer: 3 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $12\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 12 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 12 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 12 cm, so divide to find one side, which is the radius.

12 \div 4 = 3

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 3 cm.

AB = r = 3

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 3 cm

Review

AB (3 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (3 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 3 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 12 cm to find one side: 12 / 4 = 3.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 7 answer: 5 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $20\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 20 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 20 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 20 cm, so divide to find one side, which is the radius.

20 \div 4 = 5

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 5 cm.

AB = r = 5

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 5 cm

Review

AB (5 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (5 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 5 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 20 cm to find one side: 20 / 4 = 5.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 8 answer: 4 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $16\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 16 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 16 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 16 cm, so divide to find one side, which is the radius.

16 \div 4 = 4

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 4 cm.

AB = r = 4

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 4 cm

Review

AB (4 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (4 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 4 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 16 cm to find one side: 16 / 4 = 4.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 9 answer: 7 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $28\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 28 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 28 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 28 cm, so divide to find one side, which is the radius.

28 \div 4 = 7

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 7 cm.

AB = r = 7

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 7 cm

Review

AB (7 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (7 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 7 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 28 cm to find one side: 28 / 4 = 7.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Variant 10 answer: 12 cm

In the figure on the right, the two circles are the same size and each one passes through the center of the other. When the perimeter of the shaded quadrilateral is $48\ \text{cm}$ , what is the length of segment $\overline{\text{AB}}$ , in cm?

Show solution

Understand

Two equal circles overlap so that each one passes through the other's center. A rhombus is shaded inside the lens-shaped overlap; its four corners are the two circle centers and the two tips of the lens. Segment AB is the horizontal diagonal of this rhombus, joining the two circle centers. Given the rhombus has perimeter 48 cm, we must find the length of AB.

Givens

The two circles are the same size.
Each circle passes through the other circle's center.
The shaded quadrilateral is a rhombus whose vertices are the two centers and the two lens tips.
Segment AB joins the two circle centers (the horizontal diagonal of the rhombus).
The perimeter of the rhombus is 48 cm.

Unknowns

The length of segment AB, the segment joining the two circle centers.

Constraints

Every side of the rhombus joins a circle's center to a point on that circle (a lens tip), so each side equals one radius.
Because each circle passes through the other's center, the distance between the two centers equals one radius.

Plan

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

Mark every side of the rhombus as a radius to find the radius from the perimeter. Then recognize that AB, joining the two centers, is also exactly one radius because each circle passes through the other's center.

Execute

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

Each side runs from a circle's center to a point that lies on that same circle (a lens tip), so each side equals the radius. All four sides are equal radii, which is why the shape is a rhombus.

\text{side} = r

The distance from a center to any point on its circle is always the radius.

#7 Identify Subproblems 3.MD.D.8

The rhombus has 4 equal sides and perimeter 48 cm, so divide to find one side, which is the radius.

48 \div 4 = 12

Perimeter of a polygon is the sum of its sides; equal sides let us divide, a Grade 3 perimeter idea.

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

Segment AB runs from the center of the left circle to the center of the right circle. Because each circle passes through the other circle's center, the center of one circle lies exactly on the other circle. So the distance between the two centers is one radius. AB is therefore the radius, 12 cm.

AB = r = 12

If a point sits on a circle, it is exactly one radius from that circle's center -- and the far center sits right on the circle.

Answer: 12 cm

Review

AB (12 cm) equals one side of the rhombus, which makes sense: the two centers and a lens tip form an equilateral triangle with all sides equal to the radius, so the center-to-center distance matches a side. Units are centimeters, correct for a length.

Find the radius (12 cm) the same way, then note that the center of the right circle lies on the left circle (since each passes through the other's center), so it is exactly one radius from the left center; thus AB = 12 cm (Tool 7, Identify Subproblems).

Standards · min grade 3

3.G.A.1 Understand that shapes in different categories share attributes — Recognizing every rhombus side and the center-to-center segment AB as equal radii.
3.MD.D.8 Solve real-world problems involving perimeters of polygons — Using the rhombus perimeter 48 cm to find one side: 48 / 4 = 12.

💡 Every side is a radius, so divide the perimeter by 4 -- and since each circle reaches the other's center, the segment between centers is that same radius!

Center to edge distance is the radius · 10 practice problems

Generated variants — 10

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