The two quadrilaterals with only one thing each going for them — and each one has a formula that uses it.
Find the area
A trapezoid has bases 14 and 22 with height 6.
The median is the average of the bases, (14 + 22) ÷ 2 = 18, and the area is median × height = 18 × 6 = 108. D skips the halving. The median shortcut is just ½(b₁ + b₂)h with the halving done first.
Those two are the bases. The other two are the legs, and they are not parallel.
| Part | Which one |
|---|---|
| bases | the parallel pair — 14 and 22 here |
| legs | the slanted pair |
| height | the perpendicular gap between the bases, never a leg |
| median | joins the midpoints of the legs |
Unless the trapezoid is right-angled, the leg is longer than the height. Questions hand you a leg on purpose and hope you use it in the area formula.
m = b₁ + b₂2
the midsegment rule
m = 14 + 222
drop the bases in
m = 18
36 ÷ 2
Halfway up, the trapezoid has had exactly half its widening done. The bottom is 8 wider than the top, so the middle is 4 wider than the top and 4 narrower than the bottom — 18 either way.
It runs the other way too: given the median and one base, double the median and subtract. m = 18 and b₁ = 14 gives b₂ = 36 − 14 = 22.
A = b₁ + b₂2 · h
A = 18 × 6
the median is already the average
A = 108
square units
Which means a trapezoid is a rectangle in disguise: the same area as one that is 18 wide and 6 tall. The formula is median × height, and every trapezoid area question is that rectangle.
Equal legs, and everything below follows.
| Property | What it gives you |
|---|---|
| legs equal | the definition |
| base angles equal | the two at the bottom match, the two at the top match |
| same-leg angles | a top and bottom pair adds to 180° |
| diagonals equal | not necessarily perpendicular, and they do not bisect |
bottom angle = 72°
given
other bottom angle = 72°
base angles match
each top angle = 180 − 72 = 108°
co-interior, along a leg
Adjacent, not opposite — opposite pairs would make it a parallelogram.
| In every kite | Detail |
|---|---|
| diagonals meet at 90° | always |
| the symmetry diagonal bisects the other one | only that way round — it joins the two angles where the equal sides meet |
| one pair of opposite angles equal | the two where the unequal sides meet |
| the symmetry diagonal is a mirror line | it splits the kite into two congruent triangles |
A = d₁ · d₂2
half the product of the diagonals
A = 10 × 162
order does not matter
A = 80
square units
The long diagonal cuts the kite into two triangles, each with base 16 and height 5. Two of ½ × 16 × 5 = 40 gives 80 — which is exactly d₁d₂ ÷ 2.
An isosceles trapezoid has bases 6 and 16 and legs 13. Find its area.
overhang = (16 − 6) ÷ 2 = 5
the extra bottom splits evenly
h = √13² − 5² = √144 = 12
leg is the hypotenuse
A = (6 + 16) ÷ 2 × 12
median × height
A = 11 × 12 = 132
square units
The overhang is half the difference of the bases — not the whole difference. And 11 × 13 = 143 is the leg used as the height, the trap built into these questions.
Backwards: if a trapezoid’s area and height are given, area ÷ height is the median, and double the median is the sum of the bases.
A right trapezoid has bases 9 and 17 and a slanted leg of 10. Find its area.
overhang = 17 − 9 = 8
all on the slanted side
h = √10² − 8² = √36 = 6
the slanted leg is the hypotenuse
A = (9 + 17) ÷ 2 × 6 = 13 × 6 = 78
median × height
Different from step 6: one leg is already the height, so the whole difference of the bases sits on the other side — do not halve it.
A trapezoid has bases 7 and 12 and height 4. Find the median and the area.
m = (7 + 12) ÷ 2 = 9.5
an odd sum gives a half
A = 9.5 × 4 = 38
or 19 × 4 ÷ 2
Different: the median need not be whole. Multiply the sum by the height first, then halve, to keep the arithmetic in whole numbers.
Kite ABCD has AB = AD and CB = CD, with ∠A = 70° and ∠C = 50°. Find ∠B.
∠B = ∠D
the corners between unequal sides
70 + 50 + 2∠B = 360
quadrilateral total
2∠B = 240, ∠B = 120°
and ∠D = 120°
∠A and ∠C sit where the equal sides meet, and they are usually different. The equal pair is the other one, ∠B and ∠D, on the line of symmetry’s two sides. Opposite angles are not equal in pairs as in a parallelogram.
A kite has sides 13 and 20, and its shorter diagonal is 24. Find its area.
half the short diagonal = 12
the symmetry diagonal bisects it at 90°
√13² − 12² = √25 = 5
piece on the short-side end
√20² − 12² = √256 = 16
piece on the long-side end
long diagonal = 5 + 16 = 21
add the two pieces
A = 24 × 21 ÷ 2 = 252
d₁d₂ ÷ 2
The formulas are rarely asked straight. Expect a missing base from the median or the area, an isosceles trapezoid with only its legs given, or a kite given by its sides. Each time, find the missing piece first — usually with one right triangle — and then use median × height or d₁d₂ ÷ 2. Ten-second check: a trapezoid’s area must lie between short base × height and long base × height.
Find the other diagonal
A kite has area 84 and one diagonal 12.
A kite's area is ½ × d₁ × d₂, so 84 = ½(12)d₂ = 6d₂ and d₂ = 14. A divides by 12 without first undoing the ½. Check: ½ × 12 × 14 = 84 ✓
