Area of Rectangles and Triangles
Use $A = lw$ and $A = \frac{1}{2}bh$ — and always measure perpendicular height.
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A rectangle is 9 cm long and 4 cm wide. How many 1 cm × 1 cm squares fit inside it? Now think: if you cut the rectangle along its diagonal, what fraction of the original area does each triangle have?
Area measures the amount of 2D space inside a shape. For rectangles: $A = l \times w$. For triangles: $A = \frac{1}{2} \times b \times h$, where $h$ is the perpendicular height.
Area counts the unit squares that fit inside the boundary. A rectangle $l \times w$ tiles perfectly with $l \times w$ unit squares. A triangle is exactly half the rectangle that surrounds it — which is why the formula has a $\frac{1}{2}$.
Know
- Rectangle area: $A = l \times w$
- Triangle area: $A = \frac{1}{2} \times b \times h$
- Area units are always squared (cm², m²)
Understand
- Why $h$ must be perpendicular (not slant) height
- How triangle area is always half the surrounding rectangle
Can Do
- Calculate areas of rectangles and triangles from diagrams or word problems
- Find the area of composite shapes (add or subtract simpler areas)
- Solve real-world problems involving area and cost
Wrong: For a triangle with base 8 cm and slant side 10 cm, using $A = \frac{1}{2} \times 8 \times 10 = 40$ cm².
Right: Use only the perpendicular height — the height that forms a right angle with the base.
Wrong: Writing area units as cm instead of cm² (area is 2D — always squared).
Right: $A = 30$ cm² — always include the "²" superscript for area.
For any rectangle: $A = l \times w$. For a square with side $s$: $A = s^2$.
Key points:
- $l$ and $w$ must be in the same units before multiplying
- $A = 6 \times 4 = 24$ cm² — the rectangle fits $24$ unit squares
- Square shortcut: $A = s^2 = 7^2 = 49$ cm²
- Finding a side: $l = A \div w$ (rearrange the formula)
For a right-angled triangle, the two legs are automatically perpendicular — use them as base and height directly.
If the right angle is between sides $a$ and $b$:
- Base $= a$, perpendicular height $= b$ (or vice versa)
- $A = \frac{1}{2} \times a \times b$
- Example: legs 10 cm and 7 cm: $A = \frac{1}{2} \times 10 \times 7 = 35$ cm²
The hypotenuse is never used as a height in the area formula.
For any triangle, choose any side as base. Draw (or imagine) a vertical line from the opposite vertex to that base — that perpendicular length is $h$.
The perpendicular height $h$ may fall outside the triangle for obtuse triangles — that's fine. The formula $A = \frac{1}{2}bh$ always works as long as $h \perp b$.
- Mark the right angle where $h$ meets the base
- Ignore all slant sides when finding area
- Example: base 14 m, perpendicular height 9 m: $A = \frac{1}{2} \times 14 \times 9 = 63$ m²
Composite figures can be split into rectangles and triangles. Either add the parts, or find the total shape's area and subtract the removed piece.
Strategy:
- Identify simpler shapes within the composite
- Find each area separately
- Add (if combining) or subtract (if a piece is removed)
Example: L-shape $10 \times 8$ minus a $4 \times 3$ cut-out:
$A = (10 \times 8) - (4 \times 3) = 80 - 12 = 68$ cm²
Rectangle / Square
- $A = l \times w$
- $A = s^2$ (square)
- Find side: $l = A \div w$
Triangle
- $A = \frac{1}{2} \times b \times h$
- $h$ must be perpendicular to $b$
- Right triangle: use the two legs
Composite Shapes
- Split into rectangles + triangles
- Add parts (or total − removed)
- Keep units consistent throughout
Check
- Did you use perpendicular height?
- Did you include ½ for triangles?
- Are your units squared?
How are you completing this lesson?
Watch Me Solve It · 3 examples
- 1Write the formula$A = l \times w$Both dimensions are lengths — multiply to get area.
- 2Substitute values$A = 12.5 \times 4.8$
- 3Calculate$A = 60$ m²$12.5 \times 4.8 = 12.5 \times 4 + 12.5 \times 0.8 = 50 + 10 = 60$ m².
- 1Write the formula$A = \frac{1}{2} \times b \times h$Always use perpendicular height — the height at right angles to the base.
- 2Substitute values$A = \frac{1}{2} \times 10 \times 7$
- 3Calculate$A = 35$ cm²$\frac{1}{2} \times 70 = 35$ cm². The triangle is half the $10 \times 7$ rectangle.
- 1Find total rectangle area$A_{\text{total}} = 10 \times 8 = 80$ cm²Treat the whole shape as a single rectangle first.
- 2Find the removed piece$A_{\text{cut}} = 4 \times 3 = 12$ cm²
- 3Subtract$A = 80 - 12 = 68$ cm²Total area minus the piece that was taken away.
Brain Trainer · 4 problems
Calculate the area for each shape. Work it, then reveal the answer.
1 Rectangle $9$ cm $\times$ $5$ cm. Area?
$A = 9 \times 5 =$ 45 cm²2 Triangle base $12$ cm, perpendicular height $5$ cm. Area?
$A = \frac{1}{2} \times 12 \times 5 =$ 30 cm²3 Square side $8$ m. Area?
$A = 8^2 =$ 64 m²4 Rectangle $A = 72$ cm², $l = 9$ cm. Find $w$.
$w = 72 \div 9 =$ 8 cm
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. A triangular sail has a base of 3.6 m and a perpendicular height of 4.5 m. Sailcloth costs $12/m². Find the area of the sail and the total cost of the sailcloth.
Q7. A 6 cm × 6 cm square has a 4 cm × 4 cm square cut from one corner. Find the remaining area.
Q8. An L-shaped floor has an overall rectangle of 10 m × 8 m with a 4 m × 3 m corner removed. Tiles cost $35/m². (a) Find the area of the floor. (b) Calculate the total tiling cost.
Quick Check
1. B — $A = \frac{1}{2} \times 9 \times 12 = 54$ cm².
2. C — $A = 6 \times 9 = 54$ cm².
3. A — $h$ is the perpendicular height — at right angles to the base.
4. B — $w = 84 \div 7 = 12$ cm.
5. C — $A = 7 \times 6 - 2 \times 3 = 42 - 6 = 36$ cm².
Model Answers
Q6 (3 marks): $A = \frac{1}{2} \times 3.6 \times 4.5 = \frac{1}{2} \times 16.2 = 8.1$ m². Cost $= 8.1 \times \$12 = \$97.20$.
Q7 (2 marks): Total $= 6^2 = 36$ cm². Cut-out $= 4^2 = 16$ cm². Remaining $= 36 - 16 = 20$ cm².
Q8 (4 marks): (a) $A = 10 \times 8 - 4 \times 3 = 80 - 12 = 68$ m². (b) Cost $= 68 \times \$35 = \$2380$.
Triangle Garden Design
A triangular garden has a base of 8 m. (a) What is the minimum perpendicular height needed so the area is at least 20 m²? (b) A rectangular lawn of the same area has a length of 8 m — find its width. (c) If the triangular garden is isosceles with the two equal sides equal to the slant from the apex to each base corner, and the perpendicular height equals your answer to (a), find the length of the equal sides to 2 decimal places.
Reveal solution
(a) $A = \frac{1}{2} \times 8 \times h \geq 20$, so $4h \geq 20$, giving $h \geq 5$ m. Minimum $h = 5$ m. (b) $A = l \times w$: $20 = 8 \times w$, so $w = 2.5$ m. (c) Each equal side is the hypotenuse of a right triangle with legs $h = 5$ m and half-base $= 4$ m. $s = \sqrt{5^2 + 4^2} = \sqrt{41} \approx 6.40$ m.
$A = l \times w$
Rectangle area — multiply length by width. Units are squared.
$A = s^2$
Square area — side length squared. The square is a special rectangle.
$A = \frac{1}{2}bh$
Triangle area — always use perpendicular height, not slant side.
Right triangle shortcut
Use the two legs directly as base and height: $A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$.
Composite shapes
Split into simpler shapes. Add areas (combining) or subtract (cutting away).
Area and cost
Total cost $=$ area $\times$ cost per unit area. Check units match.
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