Colorful triangles, parallelograms, and trapezoids tiling a grid plane.
Geometry · 6.G

Lesson 5.1 — Area of Triangles & Quadrilaterals

Every triangle is just half of a rectangle hiding inside it — once you see that, the area formula is easy.

🎙️ Narration script

Welcome! Today we're figuring out the area of triangles and quadrilaterals. And here's the big secret: every triangle is really just half of a rectangle hiding inside it.

So let's start with what area actually means. Area is how many little unit squares fit inside a flat shape. A rectangle holds base times height squares. Simple.

Now a parallelogram, that leaning, slanted rectangle, holds the exact same amount. Why? Because you can slide a triangle off one end, move it to the other, and it snaps right into a rectangle. So its area is also base times height.

And a triangle? A triangle is exactly half of that rectangle. So its area is one half times base times height.

Here's the one thing to watch out for. The height has to be the straight-up distance to the base, the part that makes a right angle with it. Not the slanted side. The slanted side is longer, and it'll give you the wrong answer.

Let's do an example. A little triangle flag with a base of six inches and a height of four inches. We do one half times six times four. Six times four is twenty-four, and half of that is twelve. So twelve square inches of cloth.

Compare that to a parallelogram garden bed, base five meters, height three meters. No halving here, just five times three, which is fifteen square meters.

So, quick recap. Area counts unit squares. A parallelogram is base times height. A triangle is half of that, one half base times height. Always use the perpendicular height, and always write your answer in square units. Nice work!

1 Core idea

Area is how many unit squares fit inside a flat shape. A rectangle holds base × height squares. A parallelogram holds the same amount, because you can slide a triangle off one end to the other and it becomes a rectangle: A = b × h. A triangle is exactly half of that rectangle, so A = ½ × b × h. The height must be the straight-up distance to the base — not a slanted side.

🧩 Think of it like… folding a rectangular flag corner-to-corner. The crease splits it into two identical triangles, so each one covers exactly half the cloth — that's the ½ in the formula.
Where it breaks: the diagonal crease is longer than the flag's height, so if you measure along that slanted edge instead of the straight-up height, you'll overcount the area.

2 Key terms

Area
The number of square units that cover a shape (measured in units², like cm² or in²).
Base (b)
One side you choose to measure from — usually the bottom.
Height (h)
The perpendicular (straight-up) distance from the base to the opposite point or side. It makes a right angle with the base.
Parallelogram
A 4-sided shape with two pairs of parallel sides (a slanted "leaning" rectangle).
Trapezoid
A 4-sided shape with exactly one pair of parallel sides.

3 Real-life examples

  • Triangle flag: base 6 in, height 4 in → ½ × 6 × 4 = 12 in² of cloth.
  • Garden plot: a parallelogram bed, base 5 m and height 3 m → 5 × 3 = 15 m².
  • Pizza slice: a thin triangular slice with base 4 in and height 9 in → ½ × 4 × 9 = 18 in².
  • Tile floor: count how many 1-foot squares cover the floor — that count is the area.
🤔 Pause & think: Why does multiplying a rectangle's area by ½ give a triangle's area?
Reveal the thinking
Take any triangle and make a second copy of it. The two copies always snap together into a parallelogram (or rectangle) with the very same base and height. That rectangle holds b × h squares, and your triangle is exactly one of its two equal halves — so its area is half of b × h.

4 Common doubts

Is the height the same as the slanted side?

No. The height goes straight up from the base and makes a right angle (⟂) with it. A slanted side is longer.

Why do we multiply by ½ for a triangle?

Two copies of the same triangle snap together into a rectangle (or parallelogram). One triangle is half of it.

Does it matter which side I call the base?

No — as long as you use the height that goes with that base, the area comes out the same.

Why "square" units?

Area counts squares, so the unit is squared: cm × cm = cm². Length alone is just cm.

5 Step-by-step (area of a triangle)

  1. Find the base — pick a side and read its length (b = 6).
  2. Find the matching height — the perpendicular distance to that base (h = 4).
  3. Multiply base × height (6 × 4 = 24).
  4. Take half — divide by 2 (24 ÷ 2 = 12).
  5. Label the units — write the answer in square units (12 units²).

📊 See it · a triangle is half its rectangle

base b = 6 h = 4 A = ½ b h = ½×6×4 = 12

The dashed blue rectangle is 6 × 4 = 24 squares. The orange triangle fills exactly half → 12 square units.

Compare with a parallelogram (no ½ — it fills the whole rectangle):

A = b × h → parallelogram formula
A = 6 × 4 → base 6, height 4
= 24 square units → twice the triangle
✅ Check yourself
  1. A triangle has base 10 cm and height 6 cm. What is its area?
    answer ½ × 10 × 6 = 30 cm².
  2. A parallelogram has that same base 10 and height 6. Why is its area 60 cm², not 30?
    answer A parallelogram fills the whole b × h rectangle (10 × 6 = 60), while the triangle is only half of it — so the parallelogram is exactly twice the triangle.
⚡ Quick recap. Area counts unit squares. A parallelogram is A = b h; a triangle is half of that, A = ½ b h. Always use the perpendicular height (the straight-up distance to the base), and write your answer in square units.

Grounded in CA CCSS-M, Grade 6 · 6.G (area of triangles, quadrilaterals, and polygons), California Department of Education. Hero image generated with Gemini Nano Banana Pro.