Perimeters and Basic Areas

HSG.GPE.B.7 — Lesson 1 of 2

Expressing Geometric Properties with Equations

Learning Objectives

By the end of this lesson you will be able to:

1. Compute the perimeter of a polygon by summing distances between consecutive vertices
2. Compute the area of a rectangle in the coordinate plane using side lengths
3. Compute the area of a triangle using the base-height method and the bounding rectangle method

Hook: How Long Is the Fence? How Big Is the Lot?

A city surveyor records the corners of a triangular lot:

• How much fencing is needed? (perimeter)
• How large is the lot? (area)

Both answered exactly — using only coordinates.

Perimeter = Sum of All Side Lengths

Each side is a segment between consecutive vertices.

Procedure:

1. List all sides (returning to start)
2. Compute for each side
3. Sum the results

The Closing-Side Rule

An -sided polygon has exactly sides — requiring distance computations.

Always return to the starting vertex.

The 3-4-5 Triangle: Distance in Action

Two axis-aligned sides use simple subtraction. The diagonal side requires the full distance formula.

Triangle Perimeter: Step by Step (1 of 2)

Given: , ,

Sides: , ,

Triangle Perimeter: Step by Step (2 of 2)

From previous slide: ,

Exact form preferred — use decimals only if requested.

Quadrilateral Perimeter: Side Lengths

Given: , , ,

Quadrilateral Perimeter: Final Sum

Side lengths: , , ,

Decimal approximation: units

Check-In: Triangle Perimeter

Compute the perimeter of triangle :

1. List all three sides: , ,
2. Apply the distance formula to each
3. Sum the three lengths

Leave your answer in exact form.

Check-In: Answer

Triangle is isosceles: .

From Perimeter to Area

We computed perimeters by summing side lengths. Now we turn to area.

Three methods in this lesson:

• Axis-aligned rectangles — read dimensions directly
• Tilted rectangles — verify with slopes, measure with distance formula
• Triangles — base-height or bounding rectangle subtraction

Area of an Axis-Aligned Rectangle

If sides are parallel to the axes, read dimensions directly:

Example: , , ,

• Width: · Height:
• Area: square units

Area of a Tilted Rectangle

When sides are not parallel to the axes:

1. Verify it is a rectangle (adjacent slope product )
2. Compute side lengths using the distance formula
3. Multiply adjacent sides for the area

Tilted Rectangle: Worked Example

Given: , , ,

• Slopes: , . Product: ✓
• Sides: ;
• Area: sq units

Triangle Area: Base-Height Method

Strategy: Choose a horizontal or vertical side as the base when available. The height is the perpendicular distance from the opposite vertex.

Example: , ,

• Base : horizontal, length
• Height:
• Area sq units

The Bounding Rectangle Method

Enclose the triangle in the smallest axis-aligned box. Subtract the three shaded corner triangles.

Bounding Rectangle: Full Calculation

Given: , , . Box: to , area

Quick Check: Bounding Box Logic

In the bounding rectangle method:

1. What kind of triangles are the corner triangles?
2. How do you find the lengths of their legs?
3. Why do we subtract them from the box?

Quick Check: Answer

1. They are always right triangles (aligned with the axes)
2. Use coordinate differences (simple subtraction)
3. We subtract them to isolate the target triangle's area

Key Takeaways

1. Perimeter = sum of side lengths (distance formula for diagonals)
2. Closing-side rule — always return to vertex 1 ( sides = distances)
3. Rectangle area — width height (direct or via distance formula)
4. Triangle area — base-height or bounding rectangle subtraction
5. Units — perimeter is linear; area is square

Summary of Watch-Outs

Coming Up: Lesson 2

Lesson 2 of 2 — HSG.GPE.B.7

Two powerful shortcuts:

• Coordinate area formula — triangle area directly from vertices
• Shoelace formula — area of any simple polygon
• Real-world applications — land surveying, irregular regions

Both produce exact answers efficiently.