Trapezoid Formulas

Area, perimeter, midsegment and height of a trapezoid, plus relations for the diagonals, the angles at a leg and the bases found from legs and angles. The isosceles trapezoid is covered as a special case.

15 formulas · Grades 8–9

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All 15 formulas on this page
  1. Area of a trapezoid
  2. Midsegment of a trapezoid
  3. Perimeter of a trapezoid
  4. Height of a trapezoid (via area)
  5. Segment between the midpoints of the diagonals
  6. Sum of angles on one leg of a trapezoid
  7. Base of a trapezoid via the midsegment
  8. Area of a trapezoid (via midsegment and height)
  9. Perimeter of an isosceles trapezoid
  10. Sum of the squared diagonals of a trapezoid
  11. Base of a trapezoid via height and base angles
  12. Base of a trapezoid via legs and angles
  13. Leg of a trapezoid via height and angle
  14. Height of a trapezoid via leg and angle
  15. Area of a trapezoid (via diagonals and angle)

Area of a trapezoid

Grade 8
S=a+b2⋅hS = \frac{a + b}{2} \cdot h

The area of a trapezoid equals the product of the half-sum of the bases and the height.

abh
a and b are the bases, h is the height between them.
Legend:
  • a — Base a
  • b — Base b
  • h — Height h

Midsegment of a trapezoid

Grade 8
m=a+b2m = \frac{a + b}{2}

The midsegment of a trapezoid equals the half-sum of its bases.

abm
m is the midsegment, joining the midpoints of the legs.
Legend:
  • a — Base a
  • b — Base b

Perimeter of a trapezoid

Grade 8
P=a+b+c+dP = a + b + c + d

Sum of the lengths of all four sides of a trapezoid.

cdab
The perimeter is the sum of the two bases and the two legs.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c
  • d — Leg d

Height of a trapezoid (via area)

Grade 8
h=2Sa+bh = \frac{2S}{a+b}

Height of a trapezoid expressed through its area and bases.

hab
The height is derived from the trapezoid's area formula.
Legend:
  • S — Area (S)
  • a — Base a
  • b — Base b

Segment between the midpoints of the diagonals

Grade 8
MN=∣a−b∣2MN = \frac{|a-b|}{2}

The segment joining the midpoints of a trapezoid's diagonals equals half the difference of its bases.

ab
The segment between the midpoints of the diagonals is half the difference of the bases.
Legend:
  • a — Base a
  • b — Base b

Sum of angles on one leg of a trapezoid

Grade 8
β=180∘−α\beta = 180^\circ - \alpha

The sum of the angles adjacent to one leg of a trapezoid equals 180°.

αβab
The angles on one leg add up to 180°.
Legend:
  • alpha — Angle α (deg)

Base of a trapezoid via the midsegment

Grade 8
a=2m−ba = 2m - b

A trapezoid's base expressed through the midsegment and the other base.

mab
The midsegment is the half-sum of the bases.
Legend:
  • m — Midsegment m
  • b — Base b

Area of a trapezoid (via midsegment and height)

Grade 8
S=m⋅hS = m \cdot h

Area of a trapezoid as the product of the midsegment and the height.

mh
m is the midsegment, h is the height of the trapezoid.
Legend:
  • m — Midsegment m
  • h — Height h

Perimeter of an isosceles trapezoid

Grade 8
P=a+b+2cP = a + b + 2c

Perimeter of an isosceles trapezoid, whose two legs are equal.

abcc
Both legs c are equal — the trapezoid is isosceles.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c

Sum of the squared diagonals of a trapezoid

Grade 9
d12+d22=2ab+c2+d2d_1^2 + d_2^2 = 2ab + c^2 + d^2

Relates a trapezoid's diagonals to the lengths of its bases and legs.

d₁d₂
d₁ and d₂ are the diagonals of the trapezoid.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c
  • d — Leg d

Base of a trapezoid via height and base angles

Grade 9
a=b+h⋅(ctg⁡α+ctg⁡β)a = b + h \cdot (\operatorname{ctg}\alpha + \operatorname{ctg}\beta)

The larger base of a trapezoid via the smaller base, the height and the angles at the lower base.

αβhab
The height and the base angles complete the larger base.
Legend:
  • b — Base b
  • h — Height h
  • alpha — Angle α (deg)
  • beta — Angle β (deg)

Base of a trapezoid via legs and angles

Grade 9
a=b+ccos⁡α+dcos⁡βa = b + c\cos\alpha + d\cos\beta

The larger base of a trapezoid via the smaller base, the legs and the angles at the lower base.

αβcdab
The legs and the base angles give the larger base.
Legend:
  • b — Base b
  • c — Leg c
  • d — Leg d
  • alpha — Angle α (deg)
  • beta — Angle β (deg)

Leg of a trapezoid via height and angle

Grade 9
c=hsin⁡αc = \frac{h}{\sin \alpha}

The leg of a trapezoid via the height and the adjacent angle at the base.

αchab
The leg is the hypotenuse of a right triangle with leg h.
Legend:
  • h — Height h
  • alpha — Angle α (deg)

Height of a trapezoid via leg and angle

Grade 9
h=c⋅sin⁡αh = c \cdot \sin \alpha

The height of a trapezoid via a leg and the adjacent angle at the base.

αch
α is the angle at the base, c is the leg.
Legend:
  • c — Leg c
  • alpha — Angle α (deg)

Area of a trapezoid (via diagonals and angle)

Grade 9
S=12d1d2sin⁡φS = \frac{1}{2} d_1 d_2 \sin \varphi

Area of a trapezoid from its diagonals and the angle between them.

d₁d₂φ
φ is the angle between the diagonals of the trapezoid.
Legend:
  • d1 — Diagonal d1
  • d2 — Diagonal d2
  • phi — Angle φ (in degrees)

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