Distance Between Points worksheet

1. Distance: 7.87

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=3-1$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=5-2$$=3$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=11-4$$=7$

5. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+3^2+7^2$$=4+9+49$$=62$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{62}$$=7.874008$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{62}$$=\text{√62}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}3\text{|}+\text{|}7\text{|}$$=12$

2. Distance: 9.17

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=3-1$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=6-2$$=4$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=12-4$$=8$

5. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+4^2+8^2$$=4+16+64$$=84$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{84}$$=9.165151$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{84}$$=\text{2√21}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}4\text{|}+\text{|}8\text{|}$$=14$

3. Distance: 4.47

1. The sideways and up-and-down gaps form a right triangle, so the Pythagorean theorem gives the distance

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=6-2$$=4$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=5-3$$=2$

4. Square each gap and add them up, as the Pythagorean theorem says

   $4^2+2^2$$=16+4$$=20$

5. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{20}$$=4.472136$

6. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{20}$$=\text{2√5}$

7. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}4\text{|}+\text{|}2\text{|}$$=6$

4. Distance: 13.93

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=4-1$$=3$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=7-3$$=4$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=15-2$$=13$

5. Square each gap and add them up, as the Pythagorean theorem says

   $3^2+4^2+{13}^2$$=9+16+169$$=194$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{194}$$=13.928388$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{194}$$=\text{√194}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}3\text{|}+\text{|}4\text{|}+\text{|}13\text{|}$$=20$

5. Distance: 15.81

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=5-1$$=4$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=5-2$$=3$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=18-3$$=15$

5. Square each gap and add them up, as the Pythagorean theorem says

   $4^2+3^2+{15}^2$$=16+9+225$$=250$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{250}$$=15.811388$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{250}$$=\text{5√10}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}4\text{|}+\text{|}3\text{|}+\text{|}15\text{|}$$=22$

6. Distance: 7.28

1. The sideways and up-and-down gaps form a right triangle, so the Pythagorean theorem gives the distance

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=4-2$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=8-1$$=7$

4. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+7^2$$=4+49$$=53$

5. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{53}$$=7.28011$

6. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{53}$$=\text{√53}$

7. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}7\text{|}$$=9$

7. Distance: 2.83

1. The sideways and up-and-down gaps form a right triangle, so the Pythagorean theorem gives the distance

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=3-1$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=5-3$$=2$

4. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+2^2$$=4+4$$=8$

5. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{8}$$=2.828427$

6. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{8}$$=\text{2√2}$

7. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}2\text{|}$$=4$

8. Distance: 13.75

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=3-1$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=6-2$$=4$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=15-2$$=13$

5. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+4^2+{13}^2$$=4+16+169$$=189$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{189}$$=13.747727$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{189}$$=\text{3√21}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}4\text{|}+\text{|}13\text{|}$$=19$

9. Distance: 15.65

1. In 3D, the distance formula is the Pythagorean theorem with a third gap added for height

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=3-1$$=2$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=6-2$$=4$

4. Find the gap in z: the second point's z minus the first point's

   $\text{Δ}z$$=18-3$$=15$

5. Square each gap and add them up, as the Pythagorean theorem says

   $2^2+4^2+{15}^2$$=4+16+225$$=245$

6. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{245}$$=15.652476$

7. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{245}$$=\text{7√5}$

8. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}2\text{|}+\text{|}4\text{|}+\text{|}15\text{|}$$=21$

10. Distance: 4.24

1. The sideways and up-and-down gaps form a right triangle, so the Pythagorean theorem gives the distance

   $\text{d = √((x₂ − x₁)² + (y₂ − y₁)²)}$

2. Find the gap in x: the second point's x minus the first point's

   $\text{Δ}x$$=4-1$$=3$

3. Find the gap in y: the second point's y minus the first point's

   $\text{Δ}y$$=5-2$$=3$

4. Square each gap and add them up, as the Pythagorean theorem says

   $3^2+3^2$$=9+9$$=18$

5. Take the square root to undo the squaring. That's the straight-line distance

   $d$$=\sqrt{18}$$=4.242641$

6. For the exact form, pull the largest perfect square (like 4 or 9) out from under the root

   $\sqrt{18}$$=\text{3√2}$

7. For comparison, the grid (Manhattan) distance adds the gaps, ignoring minus signs, as if you could only move along the axes

   $\text{|}3\text{|}+\text{|}3\text{|}$$=6$