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Triangles and Rectangles: Area and Perimeter

Find the perimeter and area of rectangles, squares and triangles, and work backwards to a missing side.

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

What you'll learn

In this lesson you will learn to find the perimeter (distance around) and the area (space inside) of rectangles, squares and triangles. You will also learn to work backwards: if you know the area or perimeter, you can find a missing side.

Why it matters on the GED: Almost every GED Math test has two or three questions about these shapes. They show up as plain figures ("find the area of the triangle"), as word problems (carpet, paint, fencing, a garden), and as "missing side" problems ("the area is 84 square feet and the width is 7 feet; how long is it?"). These same skills are the building blocks for circles, polygons, composite figures and 3-D volume later on.

In real life: You use area when you buy flooring, paint, sod or fabric. You use perimeter when you buy fencing, baseboard trim or a picture frame. Knowing which one a job needs saves real money.

Words to know

  • Rectangle — a four-sided shape with four right angles; opposite sides are equal (a door, a sheet of paper).
  • Square — a rectangle with all four sides equal (a 5 in by 5 in tile).
  • Triangle — a shape with three straight sides (a slice of pizza cut straight, a roof gable).
  • Right angle — a square corner of 90^\circ, marked with a small square in figures.
  • Right triangle — a triangle with one right angle.
  • Length (l) and width (w) — the two dimensions of a rectangle (a 12 ft by 8 ft room has l = 12, w = 8).
  • Base (b) — the side of a triangle you choose to "sit" the triangle on.
  • Height (h) — the straight-up distance from the base to the opposite corner, measured at a right angle to the base. It is often drawn as a dashed line.
  • Perimeter (P) — the total distance around a shape; add all the sides. Measured in plain units (ft, in, m).
  • Area (A) — the amount of flat surface inside a shape. Measured in square units (sq ft, ft², in²).
  • Square unit — a 1 \times 1 square; a 3 ft by 2 ft rug holds 6 of them, so its area is 6\text{ ft}^2.

All Math words in the glossary

Perimeter: walk around the outside

Perimeter is a length. Imagine walking along every edge of the shape and adding up the distance.

Any shape: add all the sides.

  • Triangle with sides 5, 7 and 9 in: P = 5 + 7 + 9 = 21\text{ in}.

Rectangle: there are two lengths and two widths, so

P = l + w + l + w = 2l + 2w
  • A 12 ft by 8 ft garden: P = 2(12) + 2(8) = 24 + 16 = 40\text{ ft}.

Square: all four sides are equal, so P = 4s.

  • A square tile with 6 in sides: P = 4 \times 6 = 24\text{ in}.
Perimeter means adding every side. A triangle with sides 5, 7 and 9 inches has perimeter 21 inches. A 12 ft by 8 ft rectangular garden has perimeter 2(12) + 2(8) = 40 ft. A square tile with 6 inch sides has perimeter 4 × 6 = 24 inches. 9 in 5 in 7 in P = 5 + 7 + 9 = 21 in 12 ft 12 ft 8 ft 8 ft P = 2(12) + 2(8) = 40 ft 6 in 6 in 6 in 6 in P = 4 × 6 = 24 in

Missing sides in a figure. Sometimes only some sides are labeled. Opposite sides of a rectangle are equal, so if the top is 15 ft, the bottom is also 15 ft.

Units: Perimeter is in ordinary units (ft, in, yd, m). If the sides are in different units, convert first: a rectangle 2 ft by 18 in is 24 in by 18 in, so P = 2(24) + 2(18) = 84\text{ in}.

Area: count the squares inside

Area tells you how many 1-by-1 squares cover the shape.

Rectangle: a 4 by 3 rectangle holds 4 squares in each of 3 rows, so 12 squares.

A = l \times w
  • A 12 ft by 8 ft garden: A = 12 \times 8 = 96\text{ ft}^2.
A 4 by 3 rectangle covered by 12 numbered unit squares: 4 squares in each of 3 rows, so the area is 4 × 3 = 12 square units. 1 5 9 2 6 10 3 7 11 4 8 12 4 units 3 units 4 × 3 = 12 square units

Square: A = s \times s = s^2.

  • A 6 in tile: A = 6^2 = 36\text{ in}^2.

Triangle: any triangle is exactly half of a rectangle (or parallelogram) with the same base and height. That is why there is a \tfrac{1}{2}:

A = \tfrac{1}{2} b h
  • Base 10 cm, height 7 cm: A = \tfrac{1}{2}(10)(7) = \tfrac{1}{2}(70) = 35\text{ cm}^2.
A triangle with base 10 cm and height 7 cm drawn inside a dashed 10 by 7 rectangle. The triangle fills exactly half of the rectangle: half of 70 is 35 square centimetres. b = 10 cm h = 7 cm Rectangle 10 × 7 = 70; triangle = ½ of 70 = 35 cm²

Finding the height of a triangle. The height must make a right angle with the base. It is not usually a slanted side.

  • In a right triangle, the two sides that make the square corner are the base and the height. The longest side (across from the right angle) is never the height.
  • In an obtuse triangle the height can fall outside the triangle; the base is extended with a dashed line.

Square units. When you multiply feet by feet you get square feet. Always label area with square units: \text{ft}^2, \text{in}^2, sq yd. A GED answer with the wrong unit type is a clue you used the wrong formula.

ShapePerimeterArea
Rectangle2l + 2wlw
Square4ss^2
Triangles_1 + s_2 + s_3\tfrac{1}{2}bh
Three triangles with base b and dashed height h. Left: height inside the triangle. Middle: right triangle where the height is the vertical side. Right: obtuse triangle where the height falls outside and the base is extended with a dashed line. b h Height inside b h Height is a side b h Height outside

Three ways the height of a triangle can look. In every case the height meets the base (or its extension) at a right angle.

Working backwards: find a missing side

Many GED questions give you the area or perimeter and ask for a side. Write the formula, fill in what you know, then undo the operations.

Rectangle, area known. A = lw, so divide: l = A \div w.

  • Area 84 ft², width 7 ft: l = 84 \div 7 = 12\text{ ft}. Check: 12 \times 7 = 84 ✓.

Rectangle, perimeter known. P = 2l + 2w. Subtract the two known sides, then divide by 2.

  • Perimeter 54 ft, length 15 ft: 54 - 2(15) = 54 - 30 = 24; w = 24 \div 2 = 12\text{ ft}. Shortcut: half the perimeter is l + w, so 27 - 15 = 12.

Square, area known. A = s^2, so take the square root: area 49 in² → s = \sqrt{49} = 7\text{ in}.

Triangle, area known. A = \tfrac{1}{2}bh. Multiply the area by 2 first, then divide by the side you know:

h = \frac{2A}{b} \qquad b = \frac{2A}{h}
  • Area 54 ft², base 12 ft: h = \dfrac{2(54)}{12} = \dfrac{108}{12} = 9\text{ ft}.
Working backwards. Left: a rectangle with area 84 square feet and width 7 ft; the missing length is 84 ÷ 7 = 12 ft. Right: a triangle with area 54 square feet and base 12 ft; the missing height is 2 × 54 ÷ 12 = 9 ft. A = 84 ft² l = ? (84 ÷ 7 = 12 ft) 7 ft Rectangle: l = A ÷ w b = 12 ft h = ? A = 54 ft² Triangle: h = 2A ÷ b = 108 ÷ 12 = 9 ft

When the sides are linked by words ("the length is 3 more than twice the width"), let w be the width, write the length as an expression (2w + 3), put both into the formula and solve the equation.

Formula

Formulas (all on the GED formula sheet)

P_{\text{rectangle}} = 2l + 2w \qquad A_{\text{rectangle}} = lw P_{\text{square}} = 4s \qquad A_{\text{square}} = s^2 P_{\text{triangle}} = s_1 + s_2 + s_3 \qquad A_{\text{triangle}} = \tfrac{1}{2}bh
  • l = length, w = width, s = side of a square, s_1, s_2, s_3 = the three sides of a triangle.
  • b = base of the triangle, h = height, measured at a right angle to the base.
  • P is in plain units (ft); A is in square units (ft²).

On the GED formula sheet: yes — all of these are printed on the formula sheet you get during the test. You still need to know which one to use and what each letter means, especially the height of a triangle.

Worked example

Example 1 (easy): a room

A bedroom is 15 ft long and 9 ft wide. Find its perimeter and its area.

A rectangular bedroom 15 ft long and 9 ft wide, with every side labelled. 15 ft 15 ft 9 ft 9 ft bedroom
  1. Perimeter: P = 2l + 2w = 2(15) + 2(9) = 30 + 18 = 48.
  2. Area: A = lw = 15 \times 9 = 135.

Answer: perimeter 48 ft; area 135 ft²

Check: half the perimeter, 24, should equal l + w = 15 + 9 = 24 ✓.

Worked example

Example 2 (easy): a triangle

A triangle has a base of 14 in and a height of 12 in. Its slanted sides are 13 in and 15 in. Find its area.

A triangle with base 14 in, slanted sides 13 in and 15 in, and a dashed height of 12 in meeting the base at a right angle. base 14 in 13 in 15 in h = 12 in
  1. Use only the base and the height: A = \tfrac{1}{2}bh.
  2. A = \tfrac{1}{2}(14)(12) = \tfrac{1}{2}(168) = 84.
  3. The slanted sides (13 in and 15 in) are not needed for area — they would only be used for perimeter.

Answer: 84 in²

Check: a 14 by 12 rectangle has area 168; the triangle is half of that, 84 ✓.

Worked example

Example 3 (word problem): fencing a yard

A rectangular yard is 40 ft long and 25 ft wide. Fencing costs $12 per foot. How much will it cost to fence the whole yard?

A rectangular yard 40 ft long and 25 ft wide with a fence drawn all the way around its edge. yard fence goes around 40 ft 40 ft 25 ft 25 ft
  1. Fencing goes around the yard, so you need the perimeter.
  2. P = 2(40) + 2(25) = 80 + 50 = 130\text{ ft}.
  3. Cost: 130 \times 12 = 1{,}560.

Answer: $1,560

Check: estimate 130 \times 10 = 1{,}300, a bit less than our answer, as expected since the price is $12 ✓.

Worked example

Example 4 (work backwards): a triangular sign

A triangular sign has an area of 60 cm². Its base is 15 cm. How tall is the sign?

A triangular sign with base 15 cm and area 60 square centimetres. The dashed height h is unknown. base 15 cm h = ? A = 60 cm²
  1. Write the formula: 60 = \tfrac{1}{2}(15)h.
  2. Multiply both sides by 2: 120 = 15h.
  3. Divide by 15: h = 120 \div 15 = 8.

Answer: 8 cm

Check: \tfrac{1}{2}(15)(8) = \tfrac{1}{2}(120) = 60 ✓.

Worked example

Example 5 (GED-hard): sides linked by words

The length of a rectangular patio is 4 ft more than its width. The perimeter is 52 ft. What is the area of the patio?

A rectangular patio with width w and length w + 4; its perimeter is 52 ft. length = w + 4 w P = 52 ft
  1. Let the width be w. Then the length is w + 4.
  2. Perimeter: 2(w + 4) + 2w = 52.
  3. Simplify: 2w + 8 + 2w = 52, so 4w + 8 = 52.
  4. Subtract 8: 4w = 44. Divide by 4: w = 11. Length = 11 + 4 = 15.
  5. Area: A = 15 \times 11 = 165.

Answer: 165 ft²

Check: perimeter 2(15) + 2(11) = 30 + 22 = 52 ✓, and 15 is 4 more than 11 ✓.

Calculator tip

Calculator tip: TI-30XS MultiView

Most rectangle and triangle questions use small whole numbers and can be done by hand — some appear in the no-calculator part of the test. When you do use the TI-30XS:

  • Triangle area: type 0.5 × 14 × 5 then enter → 35. Or type 14 × 5 ÷ 2. Either way works; the order of multiplying does not matter.
  • Perimeter: use parentheses: 2 × ( 15 + 9 ) enter → 48.
  • Square side from area: press 2nd then x² to get √, type 49, enter → 7.
  • Fractions: if a side is 3\tfrac{1}{2} ft, type it as 3.5, or use the n/d key for fractions. The toggle key switches an answer between fraction and decimal.

Don't over-use it: the calculator can't tell you whether to add or multiply. Decide first: around = perimeter (add), cover = area (multiply).

Common mistake

Common mistakes

  • Mixing up area and perimeter. For a 12 by 8 garden, 12 + 8 + 12 + 8 = 40 is the perimeter; the area is 12 \times 8 = 96. Fencing, trim, frame → perimeter. Carpet, paint, sod, tile → area.
  • Forgetting the \tfrac{1}{2} for a triangle. Base 10 and height 7 gives \tfrac{1}{2}(10)(7) = 35, not 70. If your triangle's area equals the full rectangle's area, you forgot to halve.
  • Using a slanted side as the height. The height must meet the base at a right angle. In a 9-12-15 right triangle the area is \tfrac{1}{2}(9)(12) = 54, not \tfrac{1}{2}(12)(15) = 90.
  • Adding only two sides of a rectangle. l + w is only half the perimeter. A 15 by 9 room has P = 48, not 24.
  • Working backwards with the wrong operation. If A = 84 and w = 7, then l = 84 \div 7 = 12 — not 84 \times 7 or 84 - 7.
  • Forgetting to double the area before dividing (triangles). If a triangle's area is 54 and its base is 12, the height is \tfrac{2 \times 54}{12} = 9, not 54 \div 12 = 4.5.
Test strategy

GED test strategy

  • Read the question twice and underline what it asks for. "How much fencing" = perimeter; "how much carpet" = area; "what is the width" = a side.
  • Look at the units in the answers. Square units (ft²) mean area. Plain units mean perimeter or a side.
  • Estimate first. A 14 by 11 room is roughly 14 \times 10 = 140 ft², so an answer of 50 or 500 is out.
  • Plug the answers back in on "missing side" questions: try each choice in the formula and see which one gives the stated area or perimeter.
  • Ignore extra numbers. GED figures often label slanted sides that you do not need for area. Use only the base and the height.
  • Round only at the end, and for buying materials (gallons of paint, boxes of tile) round up — you can't buy part of a can.

Check yourself

Triangles and rectangles: area and perimeterNo calculator

A rectangular vegetable garden is shown below.

Rectangle garden 12 ft long and 8 ft wide 12 ft 8 ft

What is the area of the garden?

Answer choices
Hint

Area measures the space inside the garden. Which operation combines the length and the width for a rectangle?

Check yourself

Triangles and rectangles: area and perimeterNo calculatorFill in

Find the area of the triangle.

Triangle with base 10 cm, a dashed height of 7 cm drawn from the top vertex straight down to the base, and slanted sides labeled 7.6 cm (left) and 9.9 cm (right) 10 cm 7 cm 7.6 cm 9.9 cm

Area = cm²

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

Hint

Only two of the labeled lengths belong in the area formula: the base and the length that meets the base at a right angle.

Summary

  • Perimeter = distance around (add the sides): rectangle 2l + 2w, square 4s, triangle s_1 + s_2 + s_3.
  • Area = surface inside, in square units: rectangle lw, square s^2, triangle \tfrac{1}{2}bh.
  • A triangle's height meets the base at a right angle; it may be a side (right triangle) or fall outside (obtuse triangle).
  • Work backwards by undoing: l = A \div w; w = (P - 2l) \div 2; triangle h = 2A \div b.
  • Match the job to the measure: fence/trim → perimeter; carpet/paint/sod → area. Round up when buying materials.
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