Area of Triangles & Quadrilaterals

Every area formula in this unit comes from a rectangle that has been cut up or rearranged.

Draw an 8 by 6 rectangle and cut it corner to corner.

You get two triangles. Slide one on top of the other and they match exactly.

one cut, corner to corner 8 by 6 two of these the two halves are identical — just turned around so a triangle is half of the rectangle around it

So each triangle is exactly half the rectangle — 24, not something that needs a new idea.

The rectangle had area 48, and the cut divided it into two equal pieces.

That is where 12 bh comes from. Every formula here is a rectangle in disguise, which is why you can rebuild any of them from a picture.

A parallelogram rearranged into a rectangle of equal area parallelogram: b × h same rectangle, same area h cut the triangle off one end, slide it to the other — nothing is lost

Triangles are half a rectangle

Area = 12 bh. The height must be perpendicular to the base, not along a slanted side.

For a right triangle the two legs are the base and height, which is why those are easiest.

For a slanted triangle the height is measured straight down, sometimes landing outside the shape.

Three triangles, each with base 6 and height 4: each is half of a 6 by 4 rectangle, area 12 right trianglethe legs are b and h slanted triangleh is not the side 5 obtuse triangleh lands outside 64 ½ × 6 × 4 = 12 4 655 ½ × 6 × 4 = 12 not ½ × 6 × 5 4 6 ½ × 6 × 4 = 12 base 6 and height 4 every time — each has half the area of a 6 × 4 rectangle

Slant is not height

A triangle with sides 5, 5 and base 6 has height 4, not 5. Using a slanted side gives an area that is too big every time.

Parallelograms

Area = bh, with no half. Cut the triangle off one end and slide it to the other, and you have a rectangle.

Nothing was added or removed, so the area never changed.

Again the height is perpendicular, not the slanted side.

A parallelogram with base 7, slanted side 5 and height 4: cutting the triangle off one end and sliding it to the other makes a 7 by 4 rectangle cut the triangle off one end, slide it to the other h = 4 5 b = 7 parallelogram 7 × 4 = 28 74 rectangle nothing added, nothing removed: A = bh = 28 — the slanted 5 would give 35, too big

Why there is no half here

A parallelogram is the rectangle, rearranged. A triangle is only half of one, which is where the 12 comes from.

Trapezoids

Area = (b1+b2)2 × h — the average of the two parallel sides, times the height.

Put two identical trapezoids together, one upside down, and they form a parallelogram of base b1+b2.

Half of that is one trapezoid, which is exactly the formula.

A trapezoid with parallel sides b1 and b2 and a copy turned upside down fit together into a parallelogram of base b1 plus b2 two copies, one turned upside down, make a parallelogram h trapezoid the same trapezoid, upside down b₁b₂ b₂b₁ b₁ + b₂ parallelogram: (b₁ + b₂) × h one trapezoid, half of that: ½(b₁ + b₂) × h

Average the parallel sides

If the bases were equal it would be a parallelogram, and the average would just be that base. The formula agrees.

Composite shapes

Cut an awkward shape into rectangles and triangles, find each area, and add them up.

Or find the area of a large rectangle and subtract the pieces that are missing.

Either route works; pick whichever needs fewer measurements.

One shape, area 40, found two ways: adding a rectangle, a rectangle and a triangle, or subtracting a missing triangle from an 8 by 6 rectangle add the piecesA + B + C subtract what is missingbig rectangle − corner AB C 824 4 8×2 + 4×4 + ½×4×4 = 16 + 16 + 8 = 40 −8 86 44 8 × 6 − ½ × 4 × 4 = 48 − 8 = 40 the same 40 either way — use the route that needs fewer measurements

Mark the pieces on the drawing

Draw the cutting lines and label each piece before calculating. Most errors here are pieces counted twice or forgotten.

Units of area

Area is always in square units: square centimetres, square feet, square metres.

That is because you are counting how many unit squares fit inside.

Perimeter, by contrast, is a length, so it is measured in ordinary units.

Squared, not doubled

An answer of "24 cm" for an area is wrong; it is 24 cm². The little two is part of the answer.

Worked examples

A triangle

Find the area of a triangle with base 10 cm and height 6 cm.

  1. Area = 12 bh. Half the surrounding rectangle.
  2. = 12 × 10 × 6. Substitute.
  3. = 30 cm². The rectangle would have been 60 cm².

30 cm²

A trapezoid

Find the area of a trapezoid with parallel sides 8 and 14 and height 5.

  1. Average the parallel sides: 8+142 = 11. The formula is an average times a height.
  2. Multiply by the height: 11 × 5. As if it were a rectangle 11 wide.
  3. = 55 square units. Between 8 × 5 and 14 × 5, as it should be.

55

A composite shape

An L-shape is a 10 by 8 rectangle with a 4 by 3 corner removed. Find its area.

  1. Whole rectangle: 10 × 8 = 80. Start with the shape it came from.
  2. Missing corner: 4 × 3 = 12. The piece taken away.
  3. 80 - 12 = 68 square units. Subtracting is easier here than cutting into two pieces.

68

Words to know

Base
Any side of a triangle, chosen as the one to measure from.
Height
The perpendicular distance from the base to the opposite corner.
Parallelogram
A quadrilateral with both pairs of opposite sides parallel.
Trapezoid
A quadrilateral with exactly one pair of parallel sides.
Composite figure
A shape made by joining simpler ones.

Mistakes to watch for

Using a slanted side as the height. A triangle with base 6 and equal sides 5 has height 4; using 5 gives 15 instead of the correct 12. The height must meet the base at a right angle.

Halving a parallelogram. Its area is bh with no half — it is a whole rectangle rearranged, not half of one.

Adding the parallel sides of a trapezoid without averaging. The formula uses b1+b22; forgetting the division doubles the answer.

Leaving off the square units. Area is counted in unit squares, so the answer is 30 cm², not 30 cm.

Double-counting in a composite shape. Draw the cutting lines and label each piece; overlapping rectangles quietly add the same region twice.

Egyptian surveyors were computing field areas this way four thousand years ago, after the Nile floods erased the boundaries each year.

A triangle’s area depends only on its base and its height, not on how lopsided it is — sliding the top corner sideways changes nothing.

That fact is why a stack of paper stays the same volume however much you push the top sideways.

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