Area of Triangles and Parallelograms

Review: Area of a Rectangle

Area = length × width

Example: Rectangle 8 cm by 5 cm

• Area = 8 × 5 = 40 cm²

We'll use this to understand triangles and parallelograms!

Area of a Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel and equal.

Formula: Area = base × height

A = bh

Where:

• base (b): length of the bottom side
• height (h): perpendicular distance from base to top

Important: Height is NOT the slanted side! It must be perpendicular (90°) to the base.

Example 1: Parallelogram

Base = 10 cm, Height = 6 cm

Formula: A = bh

Calculate: A = 10 × 6 = 60 cm²

Answer: 60 cm²

Why the Formula Works

If you "cut off" a triangle from one side and move it to the other, a parallelogram becomes a rectangle!

The area stays the same: base × height

Example 2: Parallelogram

Base = 12 m, Height = 7 m

A = 12 × 7 = 84 m²

Note: If the problem gives you the slanted side (not height), you cannot use it directly in the formula!

Area of a Triangle

A triangle is half of a parallelogram!

Formula: Area = (1/2) × base × height

A = (1/2)bh or A = bh/2

Where:

• base (b): length of any side (usually the bottom)
• height (h): perpendicular distance from base to opposite vertex

Example 1: Triangle

Base = 8 cm, Height = 5 cm

Formula: A = (1/2)bh

Calculate: A = (1/2) × 8 × 5 = (1/2) × 40 = 20 cm²

Answer: 20 cm²

Example 2: Triangle

Base = 10 m, Height = 6 m

A = (1/2) × 10 × 6 = (1/2) × 60 = 30 m²

Tip: You can multiply first, then divide by 2, or divide one number by 2 first:

• (1/2) × 10 × 6 = 5 × 6 = 30 ✓

Example 3: Right Triangle

Right triangle with legs 6 ft and 8 ft

The legs ARE the base and height!

A = (1/2) × 6 × 8 = 24 ft²

Finding Missing Dimensions

If you know the area and one dimension, you can find the other!

Example: Find Height of Triangle

Area = 36 cm², Base = 9 cm, Height = ?

Formula: A = (1/2)bh

Substitute: 36 = (1/2) × 9 × h

Solve:

36 = 4.5h
36 ÷ 4.5 = h
h = 8 cm

Example: Find Base of Parallelogram

Area = 56 m², Height = 7 m, Base = ?

Formula: A = bh

Substitute: 56 = b × 7

Solve:

56 ÷ 7 = b
b = 8 m

Choosing the Right Base and Height

Any side can be the base! Just make sure:

• Height is perpendicular to the base you choose
• Base and height must correspond to each other

Example: Triangle with Different Orientations

The same triangle can be measured with different base-height pairs, giving the same area!

Important: The height is always the perpendicular distance, not a slanted side.

Units for Area

Area is always in square units:

• cm² (square centimeters)
• m² (square meters)
• ft² (square feet)
• in² (square inches)

Why? Because we multiply length × length

Composite Shapes

Break complex shapes into triangles and parallelograms!

Example: Pentagon = rectangle + triangle

• Find each area separately
• Add them together

Real-World Applications

Flag design: Triangle flags

• Base 3 ft, height 4 ft → Area = 6 ft²

Roof section: Triangular roof end

• Base 20 ft, height 12 ft → Area = 120 ft²

Garden bed: Parallelogram plot

• Base 15 m, height 8 m → Area = 120 m²

Road sign: Triangular yield sign

• Calculate area for materials needed

Practice