Area — triangle, parallelogram, trapezium, composite shapes

Why Area Matters

Area tells us:

Physical space: paint needed, land size, carpet required
Optimization: maximize garden space within a fence
Foundation for calculus: integration is "area under curves"

The key insight: all area formulas derive from the rectangle. Every shape either is a rectangle in disguise, or can be cut and rearranged into one.

Triangle Area

Derivation from First Principles

WHY does this work?

Start with a paralelogram: Draw any triangle. Copy it, flip it 180°, and attach it along one side.
You get a paralelogram: Same base $b$ , same height $h$ (the perpendicular doesn't change when flipping).
Paralelogram area: We'll prove this is $bh$ in the next section, but intuitively: it's a "squashed rectangle" with the same base and height as the original rectangle it came from.
Triangle is half: Since we made1 parallelogram from 2 identical triangles: $A_{\triangle} = \frac{1}{2} A_{\text{parallelogram}} = \frac{1}{2}bh$

WHAT is the height? The perpendicular distance from the base to the opposite vertex. Not a slant side! If your triangle sits on its longest side, the height might be shorter than the other two sides.

HOW to use it:

Pick any side as base
Find the perpendicular height to that base (may need Pythagoras)
Multiply and halve

Figure — Area — triangle, parallelogram, trapezium, composite shapes

Worked example Triangle Area — Three Calculations

Given triangle with sides forming a right angle: base = 6cm, perpendicular height = 4 cm, hypotenuse = 7.2 cm (approximate). Using the right-angle sides: $A = \frac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$

Why this step? The two perpendicular sides serve as base and height directly.

Using hypotenuse as base: If we use the 7.2 cm side as base, we need the perpendicular height from the right-angle vertex to the hypotenuse. Using the formula $h = \frac{2A}{b}$ (rearranged from area formula): $h = \frac{2 \times 12}{7.2} = 3.33 \text{ cm}$ Why this step? Area is constant regardless of base choice, so we solve for the unknown height.

Verify: $A = \frac{1}{2} \times 7.2 \times 3.33 \approx 12 \text{ cm}^2$ ✓

Parallelogram Area

Derivation from Rectangle

WHY is it base × height, not base × slant side?

Start with a paralelogram: It has opposite sides parallel and equal.
Cut rearrange: Drop a perpendicular from one top corner to the base, creating a right triangle on the left side. Cut this triangle off.
Slide the triangle: Move it to the right side of the shape. The slant edge of the parallelogram becomes the hypotenuse of the triangle, which perfectly fits.
You get a rectangle: Base $b$ , height $h$ (the perpendicular distance between parallel sides).
Rectangle area is $bh$ : Therefore, paralelogram area is $bh$ .

WHAT'S the trick? The perpendicular height $h$ is not the slant side length. The slant side is longer, but doesn't count for area—only the vertical "squeeze" matters.

HOW to avoid mistakes:

Always find the perpendicular distance between parallel sides
If given slant side $s$ and angle $\theta$ , computeh = s \sin\theta$

Trapezium Area

Derivation from Parallelogram

WHY average the parallel sides?

Duplicate and flip: Copy the trapezium, rotate it 180°, and attach it to the original along one of the non-parallel sides.
You get a parallelogram: The two parallel sides $a$ and $b$ now form the two bases of the parallelogram, stacked. Total base length = $a + b$ . Height is still $h$ .
Parallelogram area: $(a+b) \times h$
Trapezium is half: $A_{\text{trapezium}} = \frac{1}{2}(a+b)h$

Intuition: $\frac{a+b}{2}$ is the average length of the parallel sides—imagine the trapezium as a rectangle with that average width.

HOW to measure:

Identify the two parallel sides (often top and bottom)
Measure perpendicular distance between them (not the slant sides)
Sum the parallel sides, multiply by height, divide by 2

Composite Shapes

The Method

WHAT to do:

Identify components: Outline the simple shapes that make up the composite
Measure each piece: Find necessary dimensions (bases, heights, radii)
Calculate individual areas: Use the formulas for each simple shape
Combine: Add areas if shapes are joined; subtract if one is cut out of another

HOW to handle tricky cases:

Overlaps: Subtract the overlap area to avoid double-counting
Cutouts: Total area = outer shape − inner shape(s)
Irregular pieces: Extend lines to create simpler shapes, then subtract extra parts

Strategy Summary

Shape	Formula	Key Point
Triangle	$\frac{1}{2}bh$	Height is perpendicular to base
Parallelogram	$bh$	Not base × slant side!
Trapezium	$\frac{1}{2}(a+b)h$	Average of parallel sides × height
Composite	Sum or difference	Break into known shapes

Recall Feynman Explanation (Explain to a 12-Year-Old)

Imagine you're laying square tiles on the floor, each tile is 1 meter × 1 meter. Area is just counting how many tiles fit inside a shape.

Triangle: If you take two identical triangles and stick them together like puzzle pieces, you get a parallelogram. So one triangle is half of that parallelogram. That's why we multiply base × height and then divide by 2. Parallelogram: It's like a rectangle that's been pushed sideways. If you cut off the triangle from one side and move it to the other side, you get a perfect rectangle. So it has the same area as a rectangle with the same base and height.

Trapezium: It's like a rectangle, but one side is shorter. If you make two of them and flip one upside down, you can fit them together into a parallelogram. That paralelogram has a base equal to both parallel sides added together. So one trapezium is half of that.

Composite shapes: When shapes are stuck together (like an L-shape or a house with a roof), just find the area of each simple piece separately, then add them up. If there's a hole (like a picture frame), find the big area and subtract the hole.

The secret: everything comes back to rectangles, which are just "how many tiles across" times "how many tiles up."

Connections

1.2.1-Basic-shapes — definitions of these shapes
1.2.11-Perimeter — perimeter vs area distinction
1.3.5-Pythagorean-theorem — finding heights in right triangles
2.1.4-Trigonometric-ratios — using sine to find perpendicular heights
3.4.2-Integrationas-area — calculus generalizes area to curves
1.2.13-Circle-area-and-circumference — area of circular regions

#flashcards/maths

What is the formula for triangle area? :: $A = \frac{1}{2}bh$ where $b$ is base and $h$ is perpendicular height

Why is triangle area half of base × height?

A triangle is half of a parallelogram with the same base and height (duplicate and flip the triangle to see this)

What is the formula for parallelogram area?

A = bh

where

b

is base and

h

is perpendicular height between parallel sides

Why isn't parallelogram area base × slant side?

The slant side is longer but at angle; only the perpendicular height counts for how much vertical space the shape occupies

What is the formula for trapezium area?

A = \frac{1}{2}(a+b)h

where

a

and

b

are parallel sides and

h

is perpendicular height

What does

(a+b)/2

represent in trapezium area?

The average length of the two parallel sides—the trapezium acts like a rectangle with this average width

How do you find area of a composite shape?

Break it into simple shapes (triangles, rectangles, etc.), find each area separately, then add them together (or subtract if there are cutouts)

If a paralelogram has base 12 cm, slant side 8 cm, and angle 30°, what is its area? :: Height = $8\sin(30°) = 4$ cm, so area = $12 \times 4 = 48$ cm²

A trapezium has parallel sides 5 m and 9 m, with height 4 m. What is its area?

A = \frac{1}{2}(5+9) \times 4 = \frac{1}{2}(14)(4) = 28

m²

What is the most common mistake when calculating paralelogram area?

Using the slant side length instead of the perpendicular height

What must you remember to do when a shape has a cutout?

Subtract the area of the cutout from the outer shape's area

What does "height" always mean in area formulas?

The perpendicular distance from the base to the opposite side or vertex—never a slant measurement

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Area matlab kitna jagah ek shape cover karta hai — jaise floor pe kitne tiles fit honge. Sabse important baat: height hamesha perpendicular honi chahiye base se. Agar tumne slant side use kari, galat answer ayega!

Triangle ka area base × height ka adha hota hai kyunki agar tum ek aur triangle flip karke laga do, ek parallelogram ban jata hai. Parallelogram ka area seedha base × height hai — rectangle jaise, bas thoda teda. Trapezium ke liye dono parallel sides ka average lo, aur height se multiply karo, phir half karo. Agar koi complex shape hai (jaise L-shaped room), toh usko chhote pieces mein tod do — rectangle aur triangle — phir sabka area add kar do. Agar bech mein koi hole hai (jaise photo frame), toh bade shape ka area se chote shape ka area minusaro. Yeh basics sab geometry aur physics mein kaam ayenge — land measurement se lekar calculus tak!

Area — triangle, parallelogram, trapezium, composite shapes

Why Area Matters

Triangle Area

Derivation from First Principles

Parallelogram Area

Derivation from Rectangle

Trapezium Area

Derivation from Parallelogram

Composite Shapes

The Method

Strategy Summary

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Basic Geometry

Connections