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Geometry

Areas of Polygons

Lesson

Each polygon has its own area formula. Memorize these five and you’re set for most geometry problems.

Area formulas

  • Rectangle: A=ℓwA = \ell w.
  • Square: A=s2A = s^2.
  • Triangle: A=12bhA = \tfrac{1}{2} b h.
  • Parallelogram: A=bhA = b h.
  • Trapezoid: A=12(b1+b2)hA = \tfrac{1}{2}(b_1 + b_2) h.
  • Rhombus / Kite (diagonals): A=12d1d2A = \tfrac{1}{2} d_1 d_2.
  • Regular polygon: A=12PaA = \tfrac{1}{2} P a (P = perimeter, a = apothem).

Worked example

Trapezoid bases 5 and 9, height 4:

A=12(5+9)(4)=28A = \tfrac{1}{2}(5 + 9)(4) = 28

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Triangle b=10,h=6\text{Triangle } b=10, h=6

Problem 2

Parallelogram b=8,h=5\text{Parallelogram } b=8, h=5

Problem 3

Trapezoid bases 5, 9, h=4\text{Trapezoid bases 5, 9, } h = 4

Problem 4

Square side 7\text{Square side 7}

Practice

Standard problems matching the lesson.

Problem 5

Triangle b=12,h=5\text{Triangle } b=12, h=5

Problem 6

Triangle b=8,h=15\text{Triangle } b=8, h=15

Problem 7

Parallelogram b=10,h=4\text{Parallelogram } b=10, h=4

Problem 8

Parallelogram b=6,h=9\text{Parallelogram } b=6, h=9

Problem 9

Trapezoid bases 6, 10, h=4\text{Trapezoid bases 6, 10, } h=4

Problem 10

Trapezoid bases 7, 13, h=8\text{Trapezoid bases 7, 13, } h=8

Problem 11

Rhombus diagonals 6 and 10\text{Rhombus diagonals 6 and 10}

Problem 12

Kite diagonals 8 and 12\text{Kite diagonals 8 and 12}

Problem 13

Regular polygon P=40,a=5\text{Regular polygon } P=40, a=5

Problem 14

Regular polygon P=60,a=4\text{Regular polygon } P=60, a=4

Problem 15

Right triangle legs 5 and 12\text{Right triangle legs 5 and 12}

Problem 16

Trapezoid bases 10, 14, h=6\text{Trapezoid bases 10, 14, } h=6

Problem 17

Rhombus diagonals 18 and 24\text{Rhombus diagonals 18 and 24}

Problem 18

Square s=12\text{Square s=12}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Triangle area 30, base 10. Height?\text{Triangle area 30, base 10. Height?}

Problem 20

Parallelogram area 48, base 8. Height?\text{Parallelogram area 48, base 8. Height?}

Problem 21

Trap area 50, bases 4 and 6. Height?\text{Trap area 50, bases 4 and 6. Height?}

Problem 22

Regular polygon area 200, perim 40. Apothem?\text{Regular polygon area 200, perim 40. Apothem?}

Problem 23

Triangle area 78, base 12. Height?\text{Triangle area 78, base 12. Height?}

Problem 24

Rhombus area 60, one diagonal 10. Other?\text{Rhombus area 60, one diagonal 10. Other?}

Problem 25

Trap area 84, h=7,b1=10. Find b2.\text{Trap area 84, } h=7, b_1 = 10. \text{ Find } b_2.

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