Triangle & Compound Areas

Mastering the rectangle is only the start. What happens when you cut a rectangle along its diagonal? When you glue several shapes together? This lesson runs on just two ideas: take half, and split it up.

Triangle: half of a rectangle

Any triangle fits inside a rectangle with the same base and height, and it takes up exactly half — here the right angles you met in triangles earn their keep: the height must be perpendicular to the base, never the slanted side. With base $b$ and height $h$ ,

$A = \frac{b \times h}{2}$

A triangle with base 8 cm and height 5 cm: $8 \times 5 \div 2 = 20\ \mathrm{cm^2}$ .

Cut two and fit them together

Cut out two identical triangles and they always assemble into a parallelogram — that is where the "divide by 2" comes from: one triangle is half of that parallelogram.

Parallelogram: a nudged rectangle

Take a paper rectangle, cut off a right triangle along a slanted line, and slide it to the other side — you get a parallelogram. The area did not change: same base, same height, so $A = b \times h$ . It stands in a "double" relationship with the triangle: cut it along a diagonal and it splits into two congruent triangles.

Compound shapes: split them up

An L-shaped rug

An L-shaped rug can be split into a big $8\times 6$ rectangle plus a small $4\times 2$ rectangle:

$8 \times 6 + 4 \times 2 = 48 + 8 = 56\ \mathrm{cm^2}$

The split is not unique: you could also complete an $8\times 8$ square (area 64) and subtract the missing notch $4\times 2 = 8$ , giving $64 - 8 = 56$ . Two routes, one answer — which makes a fine built-in check.

InteractiveAdvanced Area Grid

Width w5Height h4

Area = length × width=5 × 4 = 20unit squares

Place a triangle on the grid: count the squares first, then compute "base times height divided by two" — over and over you will see it take exactly half of the matching rectangle.

Trapezoids and units

The trapezoid (one pair of parallel sides) wears a formula that resembles the triangle's. With parallel sides $a$ and $b$ , and height $h$ :

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

For example $a=6$ , $b=10$ , $h=4$ : $(6+10) \times 4 \div 2 = 32$ — think "average the two parallel sides, then multiply by the height".

Area work also demands unit discipline: lengths in cm give areas in cm². Converting lengths means squaring the factor: $1\ \mathrm{m} = 100\ \mathrm{cm}$ , but $1\ \mathrm{m^2} = 10000\ \mathrm{cm^2}$ . If the units disagree, unify them before computing — the grid thinking from the area lesson all pays off here.

Check yourself

Quick quiz

1. A triangle has base 10 cm and height 6 cm. What is its area?
2. An L-shape splits into 8×6 and 4×2 pieces. What is the total area?
3. A trapezoid has parallel sides 6 and 10 and height 4. What is its area?