← Perimeter splits into radius arcs and straight parts · Radius and Diameter Relationships

Perimeter splits into radius arcs and straight parts · 10 practice problems

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

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Generated variants — 10

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

Variant 1 easy

The figure on the right is made by drawing, at each vertex of a square with perimeter $24\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

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1 · Understandwhat's really being asked

A square has perimeter 24 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 2 cm. We must find the radius of the circle.

Givens

The square has perimeter 24 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 2 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

24 \div 4 = 6

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (6 cm) is made of one radius at the left corner, the 2 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

6 - 2 = 4

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

4 \div 2 = 2

Two equal radii share the leftover length equally, so halve it.

Answer: 2 cm

4 · Reviewdoes it hold up?

A radius of 2 cm is smaller than half the 6 cm side, which it must be so the two corner arcs do not overlap the 2 cm straight middle. Units are centimeters, correct for a radius. Check: 2 + 2 + 2 = 6 cm matches the side.

Another way: Set the side as radius + 2 + radius = 6 and solve 2 x radius = 4, giving radius = 2 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (24 / 4 = 6) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 4 cm to get one radius: 4 / 2 = 2.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 2 easy

The figure on the right is made by drawing, at each vertex of a square with perimeter $28\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 28 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 3 cm. We must find the radius of the circle.

Givens

The square has perimeter 28 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 3 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

28 \div 4 = 7

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (7 cm) is made of one radius at the left corner, the 3 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

7 - 3 = 4

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

4 \div 2 = 2

Two equal radii share the leftover length equally, so halve it.

Answer: 2 cm

4 · Reviewdoes it hold up?

A radius of 2 cm is smaller than half the 7 cm side, which it must be so the two corner arcs do not overlap the 3 cm straight middle. Units are centimeters, correct for a radius. Check: 2 + 3 + 2 = 7 cm matches the side.

Another way: Set the side as radius + 3 + radius = 7 and solve 2 x radius = 4, giving radius = 2 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (28 / 4 = 7) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 4 cm to get one radius: 4 / 2 = 2.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 3 medium

The figure on the right is made by drawing, at each vertex of a square with perimeter $32\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 32 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 2 cm. We must find the radius of the circle.

Givens

The square has perimeter 32 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 2 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

32 \div 4 = 8

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (8 cm) is made of one radius at the left corner, the 2 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

8 - 2 = 6

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

6 \div 2 = 3

Two equal radii share the leftover length equally, so halve it.

Answer: 3 cm

4 · Reviewdoes it hold up?

A radius of 3 cm is smaller than half the 8 cm side, which it must be so the two corner arcs do not overlap the 2 cm straight middle. Units are centimeters, correct for a radius. Check: 3 + 2 + 3 = 8 cm matches the side.

Another way: Set the side as radius + 2 + radius = 8 and solve 2 x radius = 6, giving radius = 3 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (32 / 4 = 8) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 6 cm to get one radius: 6 / 2 = 3.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 4 easy

The figure on the right is made by drawing, at each vertex of a square with perimeter $32\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

Givens

The square has perimeter 32 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 4 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

32 \div 4 = 8

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (8 cm) is made of one radius at the left corner, the 4 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

8 - 4 = 4

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

4 \div 2 = 2

Two equal radii share the leftover length equally, so halve it.

Answer: 2 cm

4 · Reviewdoes it hold up?

A radius of 2 cm is smaller than half the 8 cm side, which it must be so the two corner arcs do not overlap the 4 cm straight middle. Units are centimeters, correct for a radius. Check: 2 + 4 + 2 = 8 cm matches the side.

Another way: Set the side as radius + 4 + radius = 8 and solve 2 x radius = 4, giving radius = 2 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (32 / 4 = 8) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 4 cm to get one radius: 4 / 2 = 2.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 5 medium

The figure on the right is made by drawing, at each vertex of a square with perimeter $36\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 36 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 5 cm. We must find the radius of the circle.

Givens

The square has perimeter 36 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 5 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

36 \div 4 = 9

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (9 cm) is made of one radius at the left corner, the 5 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

9 - 5 = 4

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

4 \div 2 = 2

Two equal radii share the leftover length equally, so halve it.

Answer: 2 cm

4 · Reviewdoes it hold up?

A radius of 2 cm is smaller than half the 9 cm side, which it must be so the two corner arcs do not overlap the 5 cm straight middle. Units are centimeters, correct for a radius. Check: 2 + 5 + 2 = 9 cm matches the side.

Another way: Set the side as radius + 5 + radius = 9 and solve 2 x radius = 4, giving radius = 2 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (36 / 4 = 9) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 4 cm to get one radius: 4 / 2 = 2.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 6 medium

The figure on the right is made by drawing, at each vertex of a square with perimeter $40\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 40 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 4 cm. We must find the radius of the circle.

Givens

The square has perimeter 40 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 4 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

40 \div 4 = 10

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (10 cm) is made of one radius at the left corner, the 4 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

10 - 4 = 6

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

6 \div 2 = 3

Two equal radii share the leftover length equally, so halve it.

Answer: 3 cm

4 · Reviewdoes it hold up?

A radius of 3 cm is smaller than half the 10 cm side, which it must be so the two corner arcs do not overlap the 4 cm straight middle. Units are centimeters, correct for a radius. Check: 3 + 4 + 3 = 10 cm matches the side.

Another way: Set the side as radius + 4 + radius = 10 and solve 2 x radius = 6, giving radius = 3 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (40 / 4 = 10) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 6 cm to get one radius: 6 / 2 = 3.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 7 medium

The figure on the right is made by drawing, at each vertex of a square with perimeter $40\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

Givens

The square has perimeter 40 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 6 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

40 \div 4 = 10

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (10 cm) is made of one radius at the left corner, the 6 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

10 - 6 = 4

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

4 \div 2 = 2

Two equal radii share the leftover length equally, so halve it.

Answer: 2 cm

4 · Reviewdoes it hold up?

A radius of 2 cm is smaller than half the 10 cm side, which it must be so the two corner arcs do not overlap the 6 cm straight middle. Units are centimeters, correct for a radius. Check: 2 + 6 + 2 = 10 cm matches the side.

Another way: Set the side as radius + 6 + radius = 10 and solve 2 x radius = 4, giving radius = 2 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (40 / 4 = 10) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 4 cm to get one radius: 4 / 2 = 2.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 8 hard

The figure on the right is made by drawing, at each vertex of a square with perimeter $48\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 48 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 4 cm. We must find the radius of the circle.

Givens

The square has perimeter 48 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 4 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

48 \div 4 = 12

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (12 cm) is made of one radius at the left corner, the 4 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

12 - 4 = 8

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

8 \div 2 = 4

Two equal radii share the leftover length equally, so halve it.

Answer: 4 cm

4 · Reviewdoes it hold up?

A radius of 4 cm is smaller than half the 12 cm side, which it must be so the two corner arcs do not overlap the 4 cm straight middle. Units are centimeters, correct for a radius. Check: 4 + 4 + 4 = 12 cm matches the side.

Another way: Set the side as radius + 4 + radius = 12 and solve 2 x radius = 8, giving radius = 4 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (48 / 4 = 12) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 8 cm to get one radius: 8 / 2 = 4.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 9 hard

The figure on the right is made by drawing, at each vertex of a square with perimeter $56\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 56 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 6 cm. We must find the radius of the circle.

Givens

The square has perimeter 56 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 6 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

56 \div 4 = 14

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (14 cm) is made of one radius at the left corner, the 6 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

14 - 6 = 8

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

8 \div 2 = 4

Two equal radii share the leftover length equally, so halve it.

Answer: 4 cm

4 · Reviewdoes it hold up?

A radius of 4 cm is smaller than half the 14 cm side, which it must be so the two corner arcs do not overlap the 6 cm straight middle. Units are centimeters, correct for a radius. Check: 4 + 6 + 4 = 14 cm matches the side.

Another way: Set the side as radius + 6 + radius = 14 and solve 2 x radius = 8, giving radius = 4 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (56 / 4 = 14) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 8 cm to get one radius: 8 / 2 = 4.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Variant 10 hard

The figure on the right is made by drawing, at each vertex of a square with perimeter $64\ \text{cm}$ , a part of a circle of equal size and shading it. What is the radius of the circle, in cm?

Your answer

Show solution

1 · Understandwhat's really being asked

A square has perimeter 64 cm. At each of its four vertices, an equal quarter circle is drawn into the interior and shaded. On the top side, the straight middle part between the two corner arcs is labeled 8 cm. We must find the radius of the circle.

Givens

The square has perimeter 64 cm.
At every vertex a quarter circle of the same size is drawn inward, so each corner uses one radius along each side meeting that vertex.
On the top side, the straight middle portion left between the two corner arcs is 8 cm.

Unknowns

The radius of the circle in centimeters.

Constraints

The quarter circle at a vertex reaches one radius along each side from that vertex.
On any side, the two corner arcs eat up one radius at each end, so side length = radius + straight middle + radius.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #11 Work Backwards

First find the square's side from its perimeter. Then work backwards on the top side: it is one radius, the straight middle, and another radius, so removing the middle and halving gives the radius.

3 · Execute3 carry out the plan

1Find the side of the square

#11 Work Backwards 3.MD.D.8

A square has 4 equal sides, so divide the perimeter by 4.

64 \div 4 = 16

Perimeter of a square is 4 equal sides; dividing recovers one side, a Grade 3 perimeter skill.

2Split the top side into its parts

#1 Draw a Diagram 3.MD.D.8

The top side (16 cm) is made of one radius at the left corner, the 8 cm straight middle, and one radius at the right corner. So the two radii together fill the rest of the side.

16 - 8 = 8

Each corner arc starts one radius from the vertex, so the leftover length is two radii.

3Find one radius

#1 Draw a Diagram 3.OA.C.7

The two radii together are the leftover length, so one radius is half of that.

8 \div 2 = 4

Two equal radii share the leftover length equally, so halve it.

Answer: 4 cm

4 · Reviewdoes it hold up?

A radius of 4 cm is smaller than half the 16 cm side, which it must be so the two corner arcs do not overlap the 8 cm straight middle. Units are centimeters, correct for a radius. Check: 4 + 8 + 4 = 16 cm matches the side.

Another way: Set the side as radius + 8 + radius = 16 and solve 2 x radius = 8, giving radius = 4 cm (Tool 13, Convert to Algebra).

Standardsmin grade 3

3.MD.D.8 Solve real-world problems involving perimeters of polygons — Finding the square's side (64 / 4 = 16) and splitting the side into radius and straight parts.
3.OA.C.7 Fluently multiply and divide within 100 — Halving the leftover 8 cm to get one radius: 8 / 2 = 4.

💡Takeaway. Each side is radius + straight part + radius -- take away the middle and split in two, easy Grade 3 perimeter thinking!

Where next?

← Back to the problem: Perimeter splits into radius arcs and straight parts