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Area and Perimeter Formulas

Area and perimeter are the two most fundamental measurements in geometry. Area measures the space inside a shape. Perimeter measures the distance around it. Every shape has its own formulas, but the underlying logic is the same for all of them.

In this lesson

  1. Area vs perimeter — what each measures
  2. Squares and rectangles
  3. Triangles
  4. Circles
  5. Composite shapes

1 Area vs Perimeter — What Each Measures

Area is the amount of two-dimensional space inside a shape. It is measured in square units — square inches, square meters, square feet. If you're buying carpet, painting a wall, or fencing a garden, you need area.

Perimeter is the total distance around the outside of a shape. It is measured in linear units — inches, meters, feet. If you're fencing a yard, framing a picture, or running around a track, you need perimeter.

A helpful distinction

Think of area as how much paint you need to cover a surface, and perimeter as how much tape you'd need to outline its edges. Same shape — completely different measurements for completely different purposes.

2 Squares and Rectangles

Rectangle: Area = length × width. Perimeter = 2(length + width). A rectangle 8cm by 5cm has area 40cm² and perimeter 26cm.

Square: Area = side². Perimeter = 4 × side. A square is just a rectangle where all four sides are equal.

Rectangle Problem
A room is 12 feet long and 10 feet wide. How much carpet is needed? How much baseboard trim?
1Carpet = area = length × width = 12 × 10 = 120 sq ft
2Trim = perimeter = 2(12 + 10) = 2(22) = 44 linear feet
Answer: 120 sq ft of carpet, 44 linear feet of trim

3 Triangles

Area = ½ × base × height. The height must be perpendicular to the base — it's the straight-up distance from the base to the opposite vertex, not the length of a slanted side.

Perimeter = sum of all three sides (a + b + c).

Triangle Area
A triangle has base 10cm and perpendicular height 6cm.
1Area = ½ × base × height = ½ × 10 × 6 = 30 cm²
Answer: Area = 30 cm²

For equilateral triangles (all sides equal, all angles 60°): Area = (√3/4) × side². A simpler approach: use the standard formula with the calculated height. For a side of length s, the height is s√3/2.

4 Circles

Area = πr² where r is the radius. Circumference (perimeter) = 2πr = πd where d is the diameter.

Circle Problem
A circular garden has radius 7 meters. Find the area and circumference.
1Area = π × 7² = π × 49 ≈ 153.94 m²
2Circumference = 2 × π × 7 = 14π ≈ 43.98 m
Answer: Area ≈ 153.94 m², circumference ≈ 43.98 m

π (pi) ≈ 3.14159. The area formula πr² comes from the fact that a circle can be cut into infinitely many thin slices and rearranged into an approximate rectangle with width r and height πr, giving area πr².

5 Composite Shapes

Composite shapes are made from combinations of basic shapes. To find the area, break the composite into familiar shapes, calculate each area separately, then add or subtract.

L-Shape Area
An L-shaped room: the full rectangle would be 10×8=80, but a 4×3 corner is cut out.
1Approach 1 — Subtraction: Full rectangle 10×8 = 80. Cut corner 4×3 = 12. Area = 80 − 12 = 68 sq units
2Approach 2 — Addition: Split into two rectangles. Top: 10×5 = 50. Bottom right: 6×3 = 18. Area = 50 + 18 = 68 sq units
Answer: 68 square units — both methods give the same answer
Height ≠ slant side in triangles

The height in the triangle area formula must be the perpendicular height, not the length of a slanted side. For a right triangle the legs are perpendicular, so either leg can be the base with the other as the height. For other triangles, you may need to calculate the perpendicular height separately.

Practice Problems

A rectangle has area 48 cm² and width 6 cm. What is the length?
Area = length × width. 48 = length × 6. Length = 48/6 = 8 cm.
A triangle has base 14 and area 49. What is the height?
Area = ½ × base × height. 49 = ½ × 14 × h = 7h. h = 49/7 = 7.
A circle has circumference 31.4m. What is the radius? (Use π ≈ 3.14)
Circumference = 2πr. 31.4 = 2 × 3.14 × r = 6.28r. r = 31.4/6.28 = 5 m.
Find the area of a shape made of a 6×4 rectangle with a semicircle on one end (diameter 4).
Rectangle area = 6 × 4 = 24. Semicircle radius = 2. Semicircle area = ½π(2²) = ½ × 3.14159 × 4 ≈ 6.28. Total ≈ 30.28 square units.