Perimeter and Area of Plane Figures

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

• Calculate the perimeter of triangles, quadrilaterals, and composite figures
• Calculate the area of rectangles, triangles, parallelograms, and trapezoids
• Solve real-life problems involving perimeter and area

Wire or Tiles — Which Question?

Your family wants to fence a rectangular garden 5 m × 3 m and tile the floor inside.

• How many metres of wire do you need?
• How many 1 m² tiles do you need?

Same garden — different answers with different units.

Perimeter — Walking Around the Garden

Trace every side of the 5 m × 3 m garden:

Equal opposite sides shortcut:

Perimeter is a length — unit is metres, never metres².

Perimeter: Add Every Side Length

For any figure:

For a rectangle:

Unit rule: Perimeter is a distance → cm, m (never cm² or m²).

Perimeter of a Three-Sided Figure

Triangle with sides 6 cm, 8 cm, 10 cm — no shortcut:

Sum all three sides.

Unit check: 24 cm — write cm, not cm².

Check-In: Perimeter of Two Figures

Find the perimeter of each figure. State the unit every time.

Figure A: A rectangle with length 7 cm and width 4 cm.

Figure B: A triangle with sides 5 cm, 12 cm, and 13 cm.

Write your working, then write the unit. Is it cm or cm²?

From Wire to Tiles — A Different Question

You have the wire: 16 m of fencing. Now a new question: how many 1 m² tiles cover the courtyard floor?

Walking the edge → distance.
Covering the inside → different operation, different unit.

Central distinction: perimeter (distance around) vs area (space inside).

Area: Covering the Inside with Unit Squares

• 4 cm × 3 cm rectangle → 4 columns × 3 rows = 12 unit squares →
• Area is the space enclosed by a figure — measured in square units (cm², m²).

Area Formula from Tiling; Height Recall

Rectangle area:

Area is length × length → cm² — always a squared unit.

Perpendicular-height recall:
Height = perpendicular distance from base to opposite vertex — not the slant side.

Triangle = Half Its Bounding Rectangle

Draw the bounding rectangle around a right triangle.
The triangle fills exactly half of it.

The ½ is not optional — the other half is always there.

Skipping the ½ gives the area of the full rectangle, not the triangle.

Worked: Rectangle and Triangle Areas

Rectangle: 6 cm × 4 cm

Triangle: base 6 cm, height 4 cm

Triangle = half the rectangle. Both answers in cm².

Check-In: Rectangle and Triangle Areas

Find each area. State the unit every time.

Figure A: A rectangle with length 5 m and width 3 m.

Figure B: A triangle with base 8 cm and perpendicular height 6 cm.

Which formula do you use for each? Write the unit — is it cm or cm²?

From Tiling to Paving Stones — Enter Parallelograms

The family's courtyard has paving stones shaped like parallelograms.

A parallelogram is not a rectangle — you cannot tile it with a neat grid. Or can you?

Any parallelogram can be rearranged into a rectangle with the same base and the same perpendicular height.

Cut-and-Shift: Parallelogram Becomes a Rectangle

1. Drop a perpendicular to the base → true height
2. Cut off the left triangle
3. Shift it to the right → a rectangle with base and height

Probe: Which Measurement Is the Height?

• Given: base = 8 cm, slant side = 6 cm, dashed perpendicular height = 5 cm
• Which value goes into ? The shorter, dashed perpendicular — not the slant side.

Worked Example: Parallelogram Area Calculation

Parallelogram: base 8 cm, height 5 cm, slant 6 cm

Wrong — using slant side:

40 cm² is correct. Slant side is irrelevant to area.

Trapezoid: Average of Two Parallel Sides

A trapezoid has two parallel sides (shorter) and (longer), and perpendicular height .

Think of it as the average of a rectangle with base and one with base :

The is the average parallel side.

Worked Example: Trapezoid Area Calculation

Trapezoid: cm, cm, height cm

32 cm² — a covering. Height is the perpendicular 4 cm.

Check-In: Parallelogram and Trapezoid Areas

Figure A: Parallelogram — base 10 cm, perpendicular height 6 cm, slant side 7 cm.

Figure B: Trapezoid — parallel sides 5 cm and 9 cm, perpendicular height 4 cm.

Find each area. Which measurement does Figure A need — the 7 cm or the 6 cm?

Composite Figures: Decompose to Solve

The family's garden is L-shaped — no single formula covers it.

Every formula you just learned covers part of it.

Strategy: Break the L into shapes you know. Find each area. Combine.

Composite Area: Two Methods, Same Answer

Method 1 — Split and add:

Method 2 — Whole minus notch:

Composite Perimeter: Outer Boundary Only

The internal cut line is a tool — put no fence along it.

Six outer edges of the same L-shape:

Internal cut excluded. Two hidden edges computed from given dimensions.

Worked: Real Cost in UGX

L-shaped garden: 68 m². Turf costs 5,000 UGX/m².

Steps:

1. Find composite area → 68 m²
2. Multiply by rate → 340,000 UGX

m² × UGX/m² → UGX. Units cancel correctly.

Your Turn: Solve Without Scaffolding

L-shaped plot: 12 m × 9 m overall; notch is 5 m × 4 m.

Find:

1. Area of the plot (m²)
2. Perimeter (m) — outer boundary only
3. Cost to pave at 4,500 UGX/m²

No hints. Decompose, check units, show all working.

Summary: Perimeter and Area Both Solved

What you can now do:

• Perimeter: sum outer sides → m
• Area (rectangles, triangles, parallelograms, trapezoids, composites) → m²
• Real-life: area × rate → cost in UGX

Essential check: Distance around, or space inside?

Next lesson (measurement-02): Circles — circumference and area.