Perimeter and Area of Polygons
Calculate the perimeter and area of rectangles, triangles, parallelograms, trapezoids, and regular polygons — with real-world applications for each.
Every time you tile a floor, fence a yard, paint a wall, or order carpet, you're calculating area and perimeter. Perimeter tells you how much edging, fencing, or trim you need. Area tells you how much material covers the surface. These two measurements appear on every home improvement project, real estate listing, and landscaping quote — and getting them wrong means buying too much or too little material.
- Calculate the perimeter of any polygon by summing its side lengths
- Apply area formulas for rectangles, triangles, parallelograms, and trapezoids
- Solve real-world problems involving material estimation and cost
\(A = l \times w\)
P = 2l + 2w
\(A = \frac{1}{2}bh\)
P = a + b + c
\(A = b \times h\)
P = 2a + 2b
\(A = \frac{1}{2}(b_1+b_2)h\)
P = sum of all sides
A rectangular room is 14 ft long and 11 ft wide. Find the perimeter (for baseboards) and area (for flooring).
- Perimeter: \(P = 2(14) + 2(11) = 28 + 22 = 50\) ft of baseboard
- Area: \(A = 14 \times 11 = 154\) ft² of flooring
You need 50 linear feet of baseboard and 154 square feet of flooring material.
A trapezoidal deck has parallel sides of 12 ft and 20 ft, and a height of 8 ft. Find the area.
\[ A = \frac{1}{2}(12 + 20)(8) = \frac{1}{2}(32)(8) = 128 \text{ ft}^2 \]The deck area is 128 ft². If decking boards cost $4.50/ft², the material cost is \(128 \times 4.50 = \$576\).
A yard is shaped like an irregular polygon with sides of 25 ft, 18 ft, 30 ft, and 22 ft. Fencing costs $12 per linear foot. What's the total fencing cost?
- Perimeter: \(25 + 18 + 30 + 22 = 95\) ft
- Cost: \(95 \times 12 = \$1{,}140\)
The total fencing cost is $1,140.
- A parallelogram has base 15 cm and height 6 cm. Find its area.
- A trapezoid has bases 9 m and 15 m, with height 7 m. Find its area.
- A square has a perimeter of 52 inches. What is its area?
▶ Show Answers
- \(A = 15 \times 6 =\) 90 cm².
- \(A = \frac{1}{2}(9+15)(7) = \frac{1}{2}(24)(7) =\) 84 m².
- Side = 52 ÷ 4 = 13 in.; \(A = 13^2 =\) 169 in².
- Confusing perimeter and area units: Perimeter is in linear units (ft, m); area is in square units (ft², m²). Never mix them.
- Using a slant side as the height for parallelograms/trapezoids: Height must be perpendicular to the base. The slant side of a parallelogram is longer than the true height.
- Forgetting the ½ in the trapezoid formula: The trapezoid area is half the sum of both bases multiplied by the height — don't skip the ½.
- Perimeter = sum of all sides (linear units); area = space inside (square units).
- Height is always perpendicular to the base — not the slant side.
- Trapezoid area averages the two parallel bases, then multiplies by height: \(\frac{b_1+b_2}{2} \times h\).
- Real-world translation: perimeter = edging/fencing/trim; area = coverage/material.
Real estate agents calculate square footage to list properties. Interior designers compute area to order paint, carpet, tile, and wallpaper. Landscape architects calculate area for sod, mulch, and plantings. In every case, the formula is the same — but the stakes vary from a few extra gallons of paint to a $50,000 landscaping budget error.
Calculator Connection
The Rectangle Calculator handles both perimeter and area for rectangles and squares. The Trapezoid Calculator computes area and perimeter for trapezoid shapes. The Polygon Angle Calculator works with regular polygons (equal sides and angles).
Try it with the Calculator
Apply what you've learned with these tools.
Perimeter and Area of Polygons — Quiz
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