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Parallelograms, Trapezoids and Regular Polygons

Find the perimeter and area of parallelograms, trapezoids and regular polygons, and work backwards to a missing length.

Practice this skill 30 min Polygons: area and perimeter Leer en español
Key idea

What you'll learn

This lesson takes what you know about rectangles and triangles and stretches it to other flat shapes:

  • parallelograms (a "pushed-over" rectangle),
  • trapezoids (one pair of parallel sides), and
  • regular polygons (all sides equal, like a stop sign or a honeycomb cell).

You will find their perimeter and area, and you will work backwards to a missing base, height or side.

Why it matters on the GED: The formulas for parallelograms and trapezoids are on the formula sheet, so the GED tests whether you can pick the right measurements — especially the height, which is often a dashed line while a slanted side is labeled as a distraction. Regular polygons show up mostly as perimeter questions (fencing a hexagonal gazebo, the edge of a stop sign).

In real life: decks, lots and garden beds are often trapezoids; floor tiles and patio stones are hexagons; stop signs are octagons.

Words to know

  • Polygon — a closed flat shape made of straight sides (triangles, rectangles and hexagons are all polygons).
  • Parallel — two lines that run side by side and never meet (railroad tracks).
  • Parallelogram — a four-sided shape with two pairs of parallel sides; opposite sides are equal (a rectangle pushed sideways).
  • Rhombus — a parallelogram with all four sides equal (a diamond).
  • Trapezoid — a four-sided shape with exactly one pair of parallel sides.
  • Bases (b_1, b_2) — the two parallel sides of a trapezoid (top 6 in, bottom 10 in).
  • Legs — the two non-parallel sides of a trapezoid.
  • Height (h) — the straight-across distance between the parallel sides, measured at a right angle; often a dashed line.
  • Regular polygon — a polygon with all sides equal and all angles equal (a stop sign).
  • Pentagon / hexagon / octagon — polygons with 5 / 6 / 8 sides.
  • Apothem (a) — in a regular polygon, the distance from the center straight to the middle of a side.

All Math words in the glossary

Parallelograms: area = base × height

Cut the triangle off one end of a parallelogram and slide it to the other end — you get a rectangle with the same base and the same height. So

A = bh
  • Base 9 cm, height 5 cm, slanted side 6 cm: A = 9 \times 5 = 45\text{ cm}^2.
Left: a parallelogram with base 9 cm, slanted side 6 cm and dashed height 5 cm; the red right triangle at the left end is cut off. Right: the triangle slid to the other end makes a 9 by 5 rectangle, so the area is base × height = 45 square centimetres. b = 9 cm h = 5 cm 6 cm cut off the triangle… b = 9 cm h = 5 cm …slide it over: 9 × 5 = 45 cm²

The slanted side is NOT the height. The height is the dashed line that meets the base at a right angle. Using the slanted side (6 cm) would give 54 cm², which is too big. The height is always shorter than the slanted side.

Perimeter: opposite sides are equal, so add the base and the slanted side and double: P = 2(9 + 6) = 30\text{ cm}. Here the slanted side is used and the height is not — the opposite of area.

There is no \tfrac{1}{2} in the parallelogram formula. A triangle is half of a parallelogram; a parallelogram is a whole one.

Rhombus: it is a parallelogram, so A = bh works; its perimeter is 4s.

Trapezoids: average the bases

A trapezoid's two bases have different lengths. Average them, then multiply by the height:

A = \tfrac{1}{2}h(b_1 + b_2)
  • Bases 6 in and 10 in, height 4 in: A = \tfrac{1}{2}(4)(6 + 10) = \tfrac{1}{2}(4)(16) = 2 \times 16 = 32\text{ in}^2.
  • Same thing by averaging: (6 + 10) \div 2 = 8, and 8 \times 4 = 32 ✓.
A trapezoid with parallel bases 6 in (top) and 10 in (bottom) and dashed height 4 in. A green dashed line halfway up has the average length, 8 in. Area = ½ × 4 × (6 + 10) = 32 square inches, the same as 8 × 4. b₂ = 10 in b₁ = 6 in h = 4 in average = 8 in A = ½(4)(6 + 10) = 2 × 16 = 32 in² (or 8 × 4)

Order of steps: add the bases first (they are in parentheses), then multiply by the height, then take half.

Which numbers are the bases? The two parallel sides — usually the top and bottom. The slanted legs are not bases and are not the height.

Right trapezoid: if one leg meets the bases at a right angle, that leg is the height.

Perimeter: add all four sides: bases 7 and 11 with legs 5 and 4.1 → P = 7 + 11 + 5 + 4.1 = 27.1.

A trapezoid with bases 7 (top) and 11 (bottom) and slanted legs 5 and 4.1, drawn to scale. The perimeter adds all four sides: 7 + 11 + 5 + 4.1 = 27.1. 11 7 5 4.1 P = 7 + 11 + 5 + 4.1 = 27.1
Three regular polygons: a pentagon (5 equal sides), a hexagon (6 equal sides), and an octagon (8 equal sides) pentagon: 5 sides hexagon: 6 sides octagon: 8 sides

Regular polygons have equal sides. Count the sides to get the perimeter: P = n \times s.

Regular polygons: perimeter and area

Perimeter. All n sides are equal, so

P = n \times s
NameSides nExample: s = 10
Pentagon5P = 50
Hexagon6P = 60
Octagon8P = 80

Working backwards: a regular pentagon with perimeter 65 m has s = 65 \div 5 = 13 m.

Area. Draw lines from the center to every corner. A regular polygon with n sides splits into n identical triangles. Each triangle has base s and height equal to the apothem a.

A = n \times \tfrac{1}{2}sa = \tfrac{1}{2}aP
  • Regular hexagon, side 10 in, apothem 8.66 in: one triangle = \tfrac{1}{2}(10)(8.66) = 43.3; six triangles = 6 \times 43.3 = 259.8\text{ in}^2.
A regular hexagon with side 10 in, split by dashed lines from the center into 6 identical triangles. One triangle is shaded; its base is a side, 10 in, and its height is the red apothem, 8.66 in. One triangle has area 43.3, so the hexagon has 6 × 43.3 = 259.8 square inches. s = 10 in a apothem a = 8.66 in one triangle: ½(10)(8.66) = 43.3 6 triangles: 6 × 43.3 = 259.8 in²
  • Same with \tfrac{1}{2}aP: P = 60, so A = \tfrac{1}{2}(8.66)(60) = 259.8\text{ in}^2 ✓.

The GED will give you the apothem (or the triangle's height) if it asks for this area.

Regular hexagon with each side 10 in. Lines from the center to two neighboring corners form one of six identical triangles; the dashed height of that triangle (the apothem) is 8.66 in 10 in apothem 8.66 in

One of the six identical triangles in a regular hexagon. Its base is a side (10 in) and its height is the apothem (8.66 in).

Formula

Formulas

A_{\text{parallelogram}} = bh \qquad A_{\text{trapezoid}} = \tfrac{1}{2}h(b_1 + b_2) P_{\text{regular polygon}} = ns \qquad A_{\text{regular polygon}} = \tfrac{1}{2}aP
  • b = base; h = height (at a right angle to the base); b_1, b_2 = the two parallel sides of a trapezoid.
  • n = number of sides; s = length of one side; a = apothem; P = perimeter.

On the GED formula sheet: the parallelogram and trapezoid area formulas are on the sheet. The regular-polygon formulas are not — but you don't need to memorize them: perimeter is just "add the equal sides," and area is "split into triangles," using \tfrac{1}{2}bh (which is on the sheet).

Worked example

Example 1 (easy): a parallelogram

A parallelogram has a base of 15 cm, a slanted side of 10 cm and a height of 8 cm. Find its area and perimeter.

A parallelogram with base 15 cm, slanted sides 10 cm and a dashed height of 8 cm at a right angle to the base. h = 8 cm b = 15 cm 15 cm 10 cm 10 cm
  1. Area uses base × height: A = 15 \times 8 = 120.
  2. Perimeter uses the sides: P = 2(15 + 10) = 2(25) = 50.

Answer: area 120 cm²; perimeter 50 cm

Check: the height (8) is shorter than the slanted side (10), as it must be ✓.

Worked example

Example 2 (easy): a trapezoid

A trapezoid has bases of 9 in and 15 in and a height of 6 in. Find its area.

A trapezoid with bases 9 in and 15 in and a dashed height of 6 in. 15 in 9 in h = 6 in
  1. Add the bases: 9 + 15 = 24.
  2. Multiply by the height: 24 \times 6 = 144.
  3. Take half: 144 \div 2 = 72.

Answer: 72 in²

Check: average base = 12; 12 \times 6 = 72 ✓.

Worked example

Example 3 (word problem): railing for a gazebo

A gazebo floor is a regular octagon with sides of 5 ft. A railing will go along every side except one, which is left open as the entrance. How many feet of railing are needed?

A regular octagon gazebo floor with 5 ft sides. Seven sides have a thick blue railing; the bottom side is a red dashed opening for the entrance. entrance (open), 5 ft 5 ft railing on 7 sides
  1. An octagon has 8 sides; one is open, so the railing covers 8 - 1 = 7 sides.
  2. Railing: 7 \times 5 = 35.

Answer: 35 ft

Check: the full perimeter is 8 \times 5 = 40 ft; minus the 5 ft entrance gives 35 ft ✓.

Worked example

Example 4 (work backwards): height of a trapezoid lot

A trapezoid-shaped lot has an area of 120 ft². Its parallel sides are 10 ft and 14 ft. How far apart are the parallel sides?

A trapezoid-shaped lot with parallel sides 10 ft and 14 ft and area 120 square feet. The dashed height h between the parallel sides is unknown. 14 ft 10 ft h = ? A = 120 ft²
  1. "How far apart the parallel sides are" is the height.
  2. 120 = \tfrac{1}{2}h(10 + 14) = \tfrac{1}{2}h(24) = 12h.
  3. Divide by 12: h = 120 \div 12 = 10.

Answer: 10 ft

Check: \tfrac{1}{2}(10)(24) = 120 ✓.

Worked example

Example 5 (GED-hard): area of a hexagon tile

A floor tile is a regular hexagon with 6 in sides. The distance from the center to the middle of a side (the apothem) is about 5.2 in. What is the area of one tile? A box of 50 tiles covers how many square inches?

A regular hexagon floor tile with 6 in sides split into 6 identical triangles from the center. The shaded triangle has base 6 in and height equal to the apothem, 5.2 in. s = 6 in a apothem a = 5.2 in
  1. Split the hexagon into 6 triangles, each with base 6 and height 5.2.
  2. One triangle: \tfrac{1}{2}(6)(5.2) = 15.6.
  3. Six triangles: 6 \times 15.6 = 93.6 in².
  4. A box of 50 tiles: 50 \times 93.6 = 4{,}680 in².

Answer: 93.6 in² per tile; 4,680 in² per box

Check: \tfrac{1}{2}aP = \tfrac{1}{2}(5.2)(36) = 93.6 ✓.

Calculator tip

Calculator tip: TI-30XS MultiView

  • Trapezoid: use parentheses so the bases are added first: 0.5 × 4 × ( 6 + 10 ) enter → 32. Without the parentheses the calculator multiplies before adding and gives a wrong answer.
  • Regular polygon area: 6 × 0.5 × 10 × 8.66 enter → 259.8.
  • Working backwards: for the trapezoid lot, 2 × 120 ÷ ( 10 + 14 ) enter → 10.
  • Fractions: the n/d key enters fractions; toggle switches between fraction and decimal answers.

When not to use it: perimeter of a regular polygon (6 \times 8) and most parallelogram areas are quick mental math — save calculator time for decimals.

Common mistake

Common mistakes

  • Using the slanted side as the height. Parallelogram base 9, side 6, height 5: area is 9 \times 5 = 45, not 9 \times 6 = 54.
  • Halving a parallelogram. \tfrac{1}{2}bh is for triangles. Parallelogram base 9, height 5 → 45, not 22.5.
  • Forgetting the \tfrac{1}{2} for a trapezoid. Bases 12 and 18, height 10: \tfrac{1}{2}(10)(30) = 150, not 300.
  • Multiplying the two bases. In a trapezoid the bases are added: \tfrac{1}{2}(4)(7 + 11) = 36, not \tfrac{1}{2}(4)(7)(11) = 154.
  • Miscounting the sides. Pentagon 5, hexagon 6, octagon 8. A stop sign (octagon) with 12.4 in sides has P = 8 \times 12.4 = 99.2 in.
  • Squaring the side for perimeter. A hexagon with 8 in sides has P = 6 \times 8 = 48 in, not 8^2 = 64.
Test strategy

GED test strategy

  • Find the right angle first. Look for the little square: the segment it sits on is the height. Cross out slanted lengths when finding area.
  • Use the formula sheet for parallelograms and trapezoids — but write the numbers in carefully, bases inside the parentheses.
  • Estimate: a trapezoid's area is between (short base × height) and (long base × height). Bases 12 and 18, height 10 → between 120 and 180. Answers of 300 or 40 are out.
  • Plug in on missing-length questions: test each answer choice in the formula.
  • Round up when buying materials sold in whole units (buckets, boxes, rolls).

Check yourself

Polygons: area and perimeterNo calculator

What is the area of the parallelogram?

Parallelogram with base 9 cm, slanted side 6 cm, and a dashed height of 5 cm 9 cm 5 cm 6 cm
Answer choices
Hint

Which labeled length meets the base at a right angle? Only that one is used with the base.

Check yourself

Polygons: area and perimeterNo calculatorFill in

Find the area of the trapezoid.

Trapezoid with parallel bases 6 in (top) and 10 in (bottom) and a dashed height of 4 in 10 in 6 in 4 in

Area = in²

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Use the trapezoid formula from the formula sheet. Which two sides are parallel?

Summary

  • Parallelogram: A = bh with the perpendicular height (no \tfrac{1}{2}); perimeter uses the slanted sides.
  • Trapezoid: A = \tfrac{1}{2}h(b_1 + b_2) — add the parallel bases, times height, then halve.
  • Regular polygon: P = ns; area = n triangles = \tfrac{1}{2}aP.
  • Work backwards by undoing: h = A \div b (parallelogram); h = 2A \div (b_1 + b_2) (trapezoid); s = P \div n.
  • The height is never a slanted side — look for the right-angle mark.
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