Loading…
Calculate perimeter and area of common shapes
Learn step-by-step with practice exercises built right in.
How do we measure around a shape and the space inside it? Perimeter and area are fundamental concepts for measuring two-dimensional figures!
Perimeter is the distance around the outside of a shape.
Think: Walking around the edge of a shape - how far would you walk?
Units: Linear units (inches, feet, meters, cm, etc.)
Formula idea: Add up all the side lengths!
Area is the amount of space inside a shape.
Think: How many square tiles would cover the shape?
Units: Square units (square inches, square feet, square meters, cm², etc.)
Formula idea: Length × Width for rectangles, variations for other shapes!
Perimeter:
A rectangle has length 8 cm and width 5 cm. Find the perimeter and area.
Step 1: Find the perimeter. Perimeter = 2l + 2w P = 2(8) + 2(5) P = 16 + 10 = 26 cm
Step 2: Find the area. Area = l × w A = 8 × 5 = 40 cm²
Answer: Perimeter = 26 cm, Area = 40 cm²
A square has sides of length 6 inches. What is its perimeter and area?
Avoid these 3 frequent errors
See how this math is used in the real world
Solve .
Review key concepts with our flashcard system
Explore more Pre-Algebra topics
Area:
Different measurements for different purposes!
Perimeter of rectangle = 2 × length + 2 × width
Or: P = 2l + 2w
Or: P = 2(l + w)
Example: Rectangle with length 8 cm, width 5 cm
P = 2(8) + 2(5) P = 16 + 10 P = 26 cm
Or: P = 2(8 + 5) = 2(13) = 26 cm
All sides added: 8 + 5 + 8 + 5 = 26 cm
Area of rectangle = length × width
A = l × w
Example: Rectangle with length 8 cm, width 5 cm
A = 8 × 5 A = 40 cm²
Read as "40 square centimeters"
Think: 8 rows of 5 square centimeters = 40 squares total
A square has all four sides equal.
Perimeter of square = 4 × side
P = 4s
Example: Square with side 6 inches
P = 4 × 6 P = 24 inches
Simple: just multiply side length by 4!
Area of square = side × side
A = s²
Example: Square with side 6 inches
A = 6 × 6 A = 36 square inches A = 36 in²
This is why we call it "squared"!
Perimeter of triangle = sum of all three sides
P = a + b + c
Example: Triangle with sides 5 cm, 7 cm, 8 cm
P = 5 + 7 + 8 P = 20 cm
Just add all three sides!
Area of triangle = 1/2 × base × height
A = (1/2)bh or A = bh/2
Important: Height must be PERPENDICULAR to the base!
Example: Triangle with base 10 cm, height 6 cm
A = (1/2) × 10 × 6 A = (1/2) × 60 A = 30 cm²
Think: A triangle is half a rectangle!
Height (altitude):
Any side can be the base!
Area of parallelogram = base × height
A = bh
Example: Parallelogram with base 9 m, height 4 m
A = 9 × 4 A = 36 m²
Note: Height is perpendicular to base, not the slanted side!
Perimeter: Add all four sides (not just base × 2!)
Area of trapezoid = 1/2 × (base₁ + base₂) × height
A = (1/2)(b₁ + b₂)h
Trapezoid: Quadrilateral with exactly one pair of parallel sides
Example: Trapezoid with bases 8 ft and 12 ft, height 5 ft
A = (1/2)(8 + 12) × 5 A = (1/2)(20) × 5 A = 10 × 5 A = 50 ft²
Think: Average of the two bases, times the height!
Circumference is the perimeter of a circle.
C = 2πr (using radius) C = πd (using diameter)
π (pi) ≈ 3.14 or π ≈ 22/7
Example: Circle with radius 7 cm
C = 2π(7) C = 14π C ≈ 14 × 3.14 C ≈ 43.96 cm
Or: C = πd = π(14) ≈ 43.96 cm
Area of circle = πr²
A = πr²
Example: Circle with radius 5 inches
A = π(5)² A = π(25) A = 25π A ≈ 25 × 3.14 A ≈ 78.5 in²
Remember: Radius squared, then multiply by π!
Diameter = 2 × radius Radius = diameter ÷ 2
Example: Circle with diameter 20 m
Radius = 20 ÷ 2 = 10 m
Then: C = πd = π(20) ≈ 62.8 m A = πr² = π(10)² = 100π ≈ 314 m²
Composite figure: Made of multiple shapes combined.
Strategy:
Example: L-shaped figure
Can be: Two rectangles added Or: Large rectangle minus small rectangle
Both methods give same answer!
L-shape:
Or:
Same answer both ways!
Figure: Rectangle 8 × 6 with semicircle on top (diameter 8)
Rectangle area: A = 8 × 6 = 48 units²
Semicircle area: Radius = 8 ÷ 2 = 4 Full circle: A = π(4)² = 16π Semicircle: A = 16π ÷ 2 = 8π ≈ 25.12 units²
Total area: 48 + 25.12 ≈ 73.12 units²
Perimeter: Linear units
Area: Square units
Converting units:
Always include units in your answer!
Perimeter:
Area:
Different uses, different measurements!
If you know perimeter:
Rectangle perimeter = 40 cm, length = 12 cm, find width
P = 2l + 2w 40 = 2(12) + 2w 40 = 24 + 2w 16 = 2w w = 8 cm
If you know area:
Rectangle area = 63 m², length = 9 m, find width
A = l × w 63 = 9 × w w = 7 m
If you double the dimensions, area quadruples!
Original square: side = 2, area = 4 Doubled square: side = 4, area = 16
Why? Area = s²
If you triple dimensions, area is multiplied by 9! (3²)
Different shapes can have same perimeter but different areas!
Example: Both have perimeter 24 units
Rectangle 8 × 4:
Rectangle 6 × 6 (square):
Square has larger area with same perimeter!
Different shapes can have same area but different perimeters!
Example: Both have area 36 units²
Rectangle 9 × 4:
Square 6 × 6:
Square has smaller perimeter with same area!
For shapes on a grid:
Method 1: Count whole squares and partial squares
Method 2: Enclose in rectangle, subtract uncovered parts
Estimate partial squares when needed!
Rectangle:
Square:
Triangle:
Parallelogram:
Trapezoid:
Circle:
❌ Mistake 1: Confusing perimeter and area
❌ Mistake 2: Wrong units
❌ Mistake 3: Using slant height instead of perpendicular height
❌ Mistake 4: Forgetting to square the radius in circle area
❌ Mistake 5: Adding areas when you should subtract
To find perimeter:
To find area:
For composite figures:
Estimate before calculating:
Perimeter: About how many sides added? Area: About how many squares fit inside?
Rounding:
Check: Does final answer match estimate?
Perimeter:
Area:
Key formulas:
Remember:
Tip 1: Draw and label figures
Tip 2: Write formula first
Tip 3: Check units
Tip 4: Estimate first
Tip 5: Practice with real objects
Perimeter and area measure different aspects of shapes:
Perimeter:
Area:
Key concepts:
Essential formulas:
Master these and you can measure any two-dimensional space!
Step 1: Find the perimeter. Perimeter = 4s P = 4(6) = 24 inches
Step 2: Find the area. Area = s² A = 6² = 36 square inches
Answer: Perimeter = 24 inches, Area = 36 square inches
A triangle has sides of 5 ft, 7 ft, and 8 ft. The height to the 8 ft base is 4 ft. Find the perimeter and area.
Step 1: Find the perimeter. Perimeter = sum of all sides P = 5 + 7 + 8 = 20 ft
Step 2: Find the area. Area = (1/2) × base × height A = (1/2) × 8 × 4 A = (1/2) × 32 A = 16 square feet
Answer: Perimeter = 20 ft, Area = 16 ft²
A circle has a radius of 7 meters. Find the circumference and area. Use π ≈ 3.14.
Step 1: Find the circumference. Circumference = 2πr C = 2 × 3.14 × 7 C = 43.96 meters
Step 2: Find the area. Area = πr² A = 3.14 × 7² A = 3.14 × 49 A = 153.86 square meters
Answer: Circumference ≈ 43.96 m, Area ≈ 153.86 m²
A rectangular garden is 12 feet by 8 feet. You want to put a fence around it and cover the ground with mulch. Fencing costs 2 per square foot. What is the total cost?
Step 1: Find perimeter (for fence). Perimeter = 2l + 2w P = 2(12) + 2(8) P = 24 + 16 = 40 feet
Step 2: Calculate fence cost. Fence cost = 40 feet × 120
Step 3: Find area (for mulch). Area = l × w A = 12 × 8 = 96 square feet
Step 4: Calculate mulch cost. Mulch cost = 96 ft² × 192
Step 5: Find total cost. Total = 192 = $312
Answer: 120 for fence, $192 for mulch)