Worksheet: Using Distance Formula to Find Perimeter in Grade 3

Master perimeter calculation using distance concepts through comprehensive practice problems: grids, coordinates, and step-by-step solutions with 5 detailed exercises.

Complete Summary: Distance Formula for Perimeter
Key Concepts and Definitions

Distance: The length between two points. For points on a grid that are aligned horizontally or vertically, distance is simply the difference in coordinates.

Perimeter: The total distance around the outside of a shape. Found by adding up all the side lengths.

Grid System: A coordinate system where points are located using horizontal (x) and vertical (y) positions.

Horizontal Distance: Distance between points with the same y-coordinate: |x₂ - x₁|.

Vertical Distance: Distance between points with the same x-coordinate: |y₂ - y₁|.

Basic Distance Formulas

Horizontal Distance: |x₂ - x₁| (same y-coordinate)

Vertical Distance: |y₂ - y₁| (same x-coordinate)

Perimeter: Sum of all side lengths

Rectangle Perimeter: 2 × (Length + Width)

Problem-Solving Steps

1. Identify the coordinates of all vertices

2. Calculate the distance between consecutive points

3. Add all distances to find perimeter

4. Verify your answer makes sense

5. Write the answer with correct units

Tips and Tricks

Memory Aid: Count the spaces between points on the grid to find distances.

Coordinate System: X moves left/right, Y moves up/down.

Visualization: Plot points on grid paper to see the shape clearly.

Checking Work: Count grid squares along each side to verify distances.

Common Mistakes to Avoid

• Forgetting to count the distance as the number of units between points

• Not considering absolute values when calculating distances

• Missing a side when adding up distances

• Confusing horizontal and vertical distances

• Forgetting to include units in the final answer

Horizontal: |x₂ - x₁|
Horizontal Distance
Vertical: |y₂ - y₁|
Vertical Distance
Tip 1: For adjacent points on a grid, just count the spaces between them.
Tip 2: Always subtract the smaller coordinate from the larger one for positive distance.
Tip 3: Draw lines connecting the points to visualize the shape.
Solution: Exercises 1 to 2
1 Rectangle on Grid
Exercise 1
Find the perimeter of a rectangle with vertices at (2,3), (2,7), (6,7), and (6,3) on a coordinate grid.
Definition:

Coordinate distance: For horizontal lines, distance = |x₂ - x₁|. For vertical lines, distance = |y₂ - y₁|.

Solution method:
  1. Identify the coordinates of all vertices
  2. Calculate the distance between consecutive points
  3. Add all distances to find perimeter
  4. Verify the answer makes sense
Vertices
(2,3), (2,7), (6,7), (6,3)
Distances
4, 4, 4, 4
Perimeter
16 units
Step 1: Identify the coordinates

Vertices: A(2,3), B(2,7), C(6,7), D(6,3)

Step 2: Calculate distance AB (vertical)

Points A(2,3) and B(2,7) have same x-coordinate

Distance = |7 - 3| = 4 units

Step 3: Calculate distance BC (horizontal)

Points B(2,7) and C(6,7) have same y-coordinate

Distance = |6 - 2| = 4 units

Step 4: Calculate distance CD (vertical)

Points C(6,7) and D(6,3) have same x-coordinate

Distance = |7 - 3| = 4 units

Step 5: Calculate distance DA (horizontal)

Points D(6,3) and A(2,3) have same y-coordinate

Distance = |6 - 2| = 4 units

Step 6: Calculate total perimeter

Perimeter = 4 + 4 + 4 + 4 = 16 units

Perimeter = 16 units
Final answer:

The perimeter of the rectangle is 16 units

Applied rules:

• Horizontal distance: |x₂ - x₁| when y-coordinates are equal

• Vertical distance: |y₂ - y₁| when x-coordinates are equal

• Perimeter: Sum of all side lengths

2 Square Perimeter
Exercise 2
A square has vertices at (1,1), (1,5), (5,5), and (5,1). Use the distance formula to find the perimeter of the square.
Definition:

Square: A quadrilateral with all sides equal and all angles 90 degrees. Perimeter = 4 × side length.

Vertices
(1,1), (1,5), (5,5), (5,1)
Side Length
4 units
Perimeter
16 units
Step 1: Identify the coordinates

Vertices: A(1,1), B(1,5), C(5,5), D(5,1)

Step 2: Calculate distance AB (vertical)

Points A(1,1) and B(1,5) have same x-coordinate

Distance = |5 - 1| = 4 units

Step 3: Calculate distance BC (horizontal)

Points B(1,5) and C(5,5) have same y-coordinate

Distance = |5 - 1| = 4 units

Step 4: Calculate distance CD (vertical)

Points C(5,5) and D(5,1) have same x-coordinate

Distance = |5 - 1| = 4 units

Step 5: Calculate distance DA (horizontal)

Points D(5,1) and A(1,1) have same y-coordinate

Distance = |5 - 1| = 4 units

Step 6: Calculate total perimeter

All sides are equal (4 units each)

Perimeter = 4 × 4 = 16 units

Perimeter = 16 units
Final answer:

The perimeter of the square is 16 units

Applied rules:

• Distance formula: Use absolute difference for aligned points

• Square property: All sides are equal in length

• Perimeter of square: 4 × side length

Solution: Exercises 3 to 5
3 Irregular Shape
Exercise 3
Find the perimeter of a shape with vertices at (0,0), (0,4), (3,4), (3,2), (5,2), (5,0), and back to (0,0).
Definition:

Irregular polygon: A shape with unequal sides. Perimeter is still the sum of all side lengths.

Solution method:
  1. List all vertices in order
  2. Calculate distance between each pair of consecutive vertices
  3. Add all distances to find total perimeter
  4. Verify the shape closes properly
Step 1: List all vertices in order

A(0,0), B(0,4), C(3,4), D(3,2), E(5,2), F(5,0)

Step 2: Calculate distance AB (vertical)

A(0,0) to B(0,4): |4 - 0| = 4 units

Step 3: Calculate distance BC (horizontal)

B(0,4) to C(3,4): |3 - 0| = 3 units

Step 4: Calculate distance CD (vertical)

C(3,4) to D(3,2): |4 - 2| = 2 units

Step 5: Calculate distance DE (horizontal)

D(3,2) to E(5,2): |5 - 3| = 2 units

Step 6: Calculate distance EF (vertical)

E(5,2) to F(5,0): |2 - 0| = 2 units

Step 7: Calculate distance FA (horizontal)

F(5,0) to A(0,0): |5 - 0| = 5 units

Step 8: Calculate total perimeter

Perimeter = 4 + 3 + 2 + 2 + 2 + 5 = 18 units

Perimeter = 18 units
Final answer:

The perimeter of the irregular shape is 18 units

Applied rules:

• Distance calculation: Use absolute difference for aligned points

• Irregular polygon: Add all side lengths regardless of shape

• Order matters: Follow vertices in sequence to close the shape

4 Path Around Garden
Exercise 4
A rectangular garden has vertices at (2,2), (2,6), (8,6), and (8,2). There's a path 1 unit wide around the garden. Find the perimeter of the outer edge of the path.
Definition:

Border calculation: Add the border width to both sides of each dimension.

Step 1: Find the coordinates of the outer rectangle

Inner rectangle: (2,2), (2,6), (8,6), (8,2)

With 1-unit path around: subtract/add 1 to each coordinate

Outer rectangle: (1,1), (1,7), (9,7), (9,1)

Step 2: Calculate distance between consecutive outer vertices

(1,1) to (1,7): |7 - 1| = 6 units (vertical)

(1,7) to (9,7): |9 - 1| = 8 units (horizontal)

(9,7) to (9,1): |7 - 1| = 6 units (vertical)

(9,1) to (1,1): |9 - 1| = 8 units (horizontal)

Step 3: Calculate total perimeter

Perimeter = 6 + 8 + 6 + 8 = 28 units

Step 4: Alternative calculation using rectangle formula

Length = 9 - 1 = 8 units, Width = 7 - 1 = 6 units

Perimeter = 2 × (8 + 6) = 2 × 14 = 28 units

Perimeter = 28 units
Final answer:

The perimeter of the outer edge of the path is 28 units

Applied rules:

• Border calculation: Add border width to both sides of each dimension

• Distance formula: Use absolute difference for aligned points

• Rectangle perimeter: 2 × (length + width)

5 Multi-Step Problem
Exercise 5
A building has vertices at (3,3), (3,7), (7,7), and (7,3). A parking lot extends 2 units out from each side of the building. What is the perimeter of the outer edge of the parking lot? If fencing costs $3 per unit, how much will it cost to fence the parking lot?
Definition:

Extended perimeter: Add the extension to both sides of each dimension, then calculate the perimeter of the larger shape.

Step 1: Find the coordinates of the outer rectangle (parking lot boundary)

Building vertices: (3,3), (3,7), (7,7), (7,3)

With 2-unit parking lot around: subtract/add 2 to each coordinate

Parking lot vertices: (1,1), (1,9), (9,9), (9,1)

Step 2: Calculate distances between consecutive outer vertices

(1,1) to (1,9): |9 - 1| = 8 units (vertical)

(1,9) to (9,9): |9 - 1| = 8 units (horizontal)

(9,9) to (9,1): |9 - 1| = 8 units (vertical)

(9,1) to (1,1): |9 - 1| = 8 units (horizontal)

Step 3: Calculate the perimeter of the parking lot

Perimeter = 8 + 8 + 8 + 8 = 32 units

Step 4: Calculate the cost of fencing

Cost = Perimeter × Price per unit

Cost = 32 × $3 = $96

Step 5: Verify the calculation

Alternative: Length = 9 - 1 = 8, Width = 9 - 1 = 8

Perimeter = 2 × (8 + 8) = 32 units ✓

Perimeter: 32 units, Cost: $96
Final answer:

The perimeter of the outer edge of the parking lot is 32 units, and it will cost $96 to fence the parking lot

Applied rules:

• Extension calculation: Add extension to both sides of each dimension

• Distance formula: Use absolute difference for aligned points

• Cost calculation: Multiply perimeter by unit price

Advanced Distance Formula Strategies
Horizontal: |x₂ - x₁|, Vertical: |y₂ - y₁|
Coordinate Distance Formulas
Advanced concepts:

Grid systems: Understanding how coordinates correspond to positions on a grid.

Border calculations: Adding extensions to all sides of a shape.

Multi-step problems: Problems requiring multiple calculations.

Advanced problem-solving steps:
  1. Visualize: Plot points on a grid to see the shape
  2. Sequence: List vertices in the order they connect
  3. Calculate: Find distance between each pair of consecutive points
  4. Sum: Add all distances to find perimeter
  5. Verify: Check that your answer makes sense
Tip 1: For rectangles, just find length and width, then use P = 2(L + W).
Tip 2: For borders, add the border width to both ends of each dimension.
Tip 3: Always go around the shape in the same direction to avoid missing sides.
Tip 4: Draw the shape on grid paper to visualize and verify your work.
Coordinate rules: X changes for horizontal movement, Y changes for vertical.
Distance principle: Distance is always positive, so use absolute values.
Essential formulas to memorize:

• Horizontal distance: |x₂ - x₁| when y-coordinates are equal

• Vertical distance: |y₂ - y₁| when x-coordinates are equal

• Rectangle perimeter: 2 × (Length + Width)

• Square perimeter: 4 × Side

• Any polygon perimeter: Sum of all side lengths

Common mistakes to avoid:

• Subtracting coordinates in the wrong order and getting a negative distance

• Forgetting to use absolute values for distance calculations

• Missing a side when calculating the perimeter of irregular shapes

• Not accounting for all sides when calculating extended boundaries

Direction Formula Condition Example
Horizontal |x₂ - x₁| y-coordinates equal (2,3) to (5,3) = |5-2| = 3
Vertical |y₂ - y₁| x-coordinates equal (2,3) to (2,7) = |7-3| = 4
Perimeter Sum of all sides All sides included Rectangle: 2(L+W)

Questions & Answers

Question: Why do we use absolute value when calculating distance between points?

Answer: We use absolute value because distance is always positive. Distance measures how far apart two points are, regardless of direction.

For example, if you move from point (3,2) to point (1,2), the distance is |3 - 1| = 2 units.

If you move from point (1,2) to point (3,2), the distance is |1 - 3| = |-2| = 2 units.

Both movements cover the same distance of 2 units, even though they go in opposite directions. The absolute value ensures we always get a positive distance measurement.

Question: How do I find the perimeter when there's a border around a shape?

Answer: When there's a border around a shape, you need to account for the border on both sides of each dimension:

  1. Identify the original dimensions of the inner shape
  2. Add the border width to both ends of each dimension
  3. Calculate the perimeter of the new, larger shape

For example, if a rectangle is 4 units wide and has a 1-unit border around it:

  • New width = 4 + 1 + 1 = 6 units (add 1 unit for left border and 1 unit for right border)
  • Similarly, add 2 units to the length (1 for top, 1 for bottom)

The key is remembering that the border extends outward on both sides of each dimension.

Question: How can I check if I've calculated the perimeter correctly using the distance formula?

Answer: There are several ways to check your perimeter calculation:

  1. Count grid squares: On a grid, count the actual units along each side
  2. Draw the shape: Sketch the shape to verify you haven't missed any sides
  3. Use alternative formulas: For rectangles, try P = 2(L + W) after finding length and width
  4. Verify closure: Make sure your path returns to the starting point

For example, if you calculated a rectangle's perimeter as 20 units, verify that 2 × (length + width) also equals 20.

Always go back and check that you calculated the distance for every side of the shape.