Key area formulas for the most common shapes — all built from the same base-times-height idea.
The Mnemonic
One Idea, Four Formulas
Rectangle: Area = length × width (a special case of base × height where the sides are perpendicular). Parallelogram: Area = base × height, where height is measured PERPENDICULAR to the base, not along the slanted side. Triangle: Area = ½ × base × height. Trapezoid: Area = ½ × (base₁ + base₂) × height, averaging the two parallel sides before multiplying by height.
Every one of these formulas is really the same core idea — base times height — just adjusted for the shape's structure: cut in half for triangles, averaged for trapezoids with two different-length bases.
Each formula reduces to the same underlying idea — base × height, adjusted for the shape
💡 Memory Trick
"Key area formulas for the most common shapes." Rectangle and parallelogram: full base × height. Triangle: HALF of base × height (a triangle is literally half of a parallelogram sharing the same base and height). Trapezoid: average the two bases, then multiply by height — treating it like a rectangle with a single 'in-between' base length.
Why It Works
Why a Triangle Is Exactly Half
Take any triangle and duplicate it, rotating the copy 180° — the two triangles fit together perfectly to form a parallelogram with the same base and height as the original triangle. Since that parallelogram's area is base × height, and it's made of exactly two identical triangles, each triangle must be exactly half: ½ × base × height.
The trapezoid formula follows a similar doubling trick: take a trapezoid and duplicate it, rotating the copy 180° and attaching it to the original. The result is a parallelogram whose base equals the SUM of the trapezoid's two bases (b₁ + b₂), with the same height. That parallelogram's area is (b₁+b₂) × h, and since it's two trapezoids, each trapezoid is half of that: ½(b₁+b₂)h.
Using It In A Proof
Measuring Height Correctly
The single most common error across all of these formulas is measuring height incorrectly — height must always be perpendicular to the base, never along a slanted side.
1
Identify the true base and height
For a parallelogram or triangle drawn at a slant, the height is the perpendicular (straight up-and-down) distance to the base — not the length of the slanted side itself.
2
For trapezoids, identify both parallel bases
A trapezoid has exactly one pair of parallel sides — these are b₁ and b₂. The height is the perpendicular distance between those two parallel sides.
3
Apply the matching formula
Match the shape to its formula and substitute the correctly identified base(s) and height.
Full Worked Example
Finding the Area of a Trapezoid
Given: A trapezoid has parallel sides of 8 and 14, with a height of 6. Find: its area.
1
Identify the two bases and height
b₁ = 8, b₂ = 14, h = 6.
2
Average the two bases
(b₁ + b₂) = 8 + 14 = 22, so the average base is 22/2 = 11 — though the formula handles this in one step below.
3
Apply the formula
Area = ½(8 + 14)(6) = ½(22)(6) = ½(132).
4
Calculate
Area = 66 square units.
Sanity check: this trapezoid's area should sit between a rectangle using the shorter base (8×6=48) and one using the longer base (14×6=84) — 66 sits right in that range, as expected for an 'in-between' shape.
🎯 Quick Worked Example
A parallelogram has a base of 12 and a height of 5. A triangle shares that same base and height. Compare their areas.
1
Find the parallelogram's area. Area = base × height = 12 × 5 = 60.
2
Find the triangle's area. Area = ½ × base × height = ½ × 12 × 5 = 30.
3
Compare. The triangle's area (30) is exactly half the parallelogram's area (60) — confirming the relationship directly, since they share the same base and height.
📌 Exam Application
Trapezoid and parallelogram problems frequently draw the shape at a slant specifically to tempt you into using the slanted side length as the height — always look for (or calculate) the perpendicular distance between the base and its opposite side/vertex, since using the slanted side directly will always overstate the true height.
⚠️ Most Common Quadrilateral and Triangle Areas Mistakes
Trap 1 — Using the slanted side as height: Height must be measured perpendicular to the base — the slanted side of a parallelogram or trapezoid is almost always longer than the true perpendicular height, and using it directly produces an inflated area.
Trap 2 — Forgetting to average the trapezoid's bases: Using only one base (like a rectangle formula) instead of averaging both parallel sides ignores the trapezoid's actual shape — always add both bases and divide by 2 (or equivalently, multiply the sum by ½ within the full formula).
✓ Quick Self-Test
1) State the area formula for a parallelogram. 2) State the area formula for a triangle, and explain why it's half of the parallelogram formula. 3) State the area formula for a trapezoid. 4) A triangle has a base of 10 and height of 7 — find its area. 5) A trapezoid has bases of 6 and 10 with a height of 4 — find its area.