Learning goals
- Explain area as the amount of surface a shape covers, measured in square units
- Apply rectangle, triangle, and circle area formulas with matching length units
- Choose the right triangle formula when you know base and height versus three sides
What area measures
Area is how much flat surface a shape covers. If length is measured in meters, area is in square meters (m²). Mixing meters with centimeters without converting first is the most common mistake.
Most school and DIY problems reduce to a few building-block shapes: rectangles (and squares), triangles, and circles. Complex floor plans are often split into those pieces, then added.
Before you plug into a formula, label what you know. For a triangle, “height” must be perpendicular to the base you chose — not just any side length.
Rectangle (and square)
A = length × width
Both sides must use the same unit. A square is the special case length = width, so A = side². If you only know perimeter and one side, find the other side first, then multiply.
Triangle (base and height)
A = (1/2) × base × height
Height is the perpendicular distance from the chosen base to the opposite vertex. Any side can be the base if you use the matching perpendicular height.
Triangle from three sides (Heron’s formula)
s = (a+b+c)/2, then A = √[s(s−a)(s−b)(s−c)]
Use this when you know all three sides but not a perpendicular height. First compute the semi-perimeter s, then the product under the square root. The sides must be able to form a triangle (each side shorter than the sum of the other two).
Circle
A = π × r²
r is the radius (center to edge). If you are given diameter d, use r = d/2. Keep π as a symbol until the end, or use a consistent approximation such as 3.1416 when a decimal is required.
Example 1 — Rectangle room
Given
A rectangular room is 5.2 m long and 3.5 m wide. What is the floor area?
Steps
- Confirm both measurements are in meters.
- Apply A = length × width = 5.2 × 3.5.
- Compute 5.2 × 3.5 = 18.2.
Answer
The floor area is 18.2 m².
Tip: If one side were 350 cm, convert to 3.5 m first (or convert 5.2 m to 520 cm) so units match.
Example 2 — Triangle with base and height
Given
A triangular sail has base 8 ft and perpendicular height 5 ft. Find the area.
Steps
- Use A = (1/2) × base × height.
- Substitute: (1/2) × 8 × 5.
- Simplify: 4 × 5 = 20.
Answer
The sail area is 20 ft².
Tip: If the 5 ft measurement were a slanted side instead of height, this formula would not apply until you find the perpendicular height.
Example 3 — Circle from diameter
Given
A circular patio has diameter 6 m. Approximate the area using π ≈ 3.1416.
Steps
- Radius r = diameter ÷ 2 = 6 ÷ 2 = 3 m.
- Area A = π r² = 3.1416 × 3² = 3.1416 × 9.
- Compute 3.1416 × 9 = 28.2744.
Answer
The patio area is about 28.27 m².
Tip: Leaving the answer as 9π m² is exact; only convert to a decimal when the problem asks for one.
Example 4 — Heron’s formula
Given
A triangle has sides a = 5, b = 5, c = 6 (same units). Find the area.
Steps
- Semi-perimeter s = (5 + 5 + 6) ÷ 2 = 8.
- Compute s − a = 3, s − b = 3, s − c = 2.
- Area = √[8 × 3 × 3 × 2] = √144 = 12.
Answer
The area is 12 square units.
Using a slanted side as height
Common mistake
Taking any side length as “height” in A = (1/2)bh.
Better approach
Height must be perpendicular to the base. Drop an altitude or use Heron’s formula when you only know the three sides.
Forgetting to square the radius
Common mistake
Computing π × r or π × d instead of π × r².
Better approach
Area scales with the square of length. Doubling the radius multiplies area by four, not two.
Check your understanding
1.A square garden has side 9 m. What is its area?
Answer: A = 9² = 81 m².
2.Triangle base 12 cm, height 7 cm. Find the area.
Answer: A = (1/2) × 12 × 7 = 42 cm².
3.Circle radius 4 in. Leave the answer in terms of π.
Answer: A = π × 4² = 16π in².
4.Why must length and width use the same unit before multiplying?
Answer: Area is a product of two lengths; mixed units (e.g. m × cm) produce a meaningless or wrong square unit.
Key takeaways
- Area measures covering space in square units; convert lengths to one unit before multiplying.
- Rectangle: A = ℓw. Triangle: A = (1/2)bh with perpendicular height, or Heron’s formula from three sides.
- Circle: A = πr² (use r = d/2 when given diameter).
- Complex regions can often be split into rectangles, triangles, and circular parts, then added or subtracted.