Perimeter and area of basic shapes
Introduced| English | Meaning |
|---|---|
| area/ˈeərɪə/ | The measure of a two-dimensional region, expressed in square units. |
| perimeter/pəˈrɪmɪtə/ | perimeter |
| perpendicular height/ˌpɜːpənˈdɪkjʊlə haɪt/ | The distance from a base to the opposite point or parallel face, measured at a right angle to the base. |
How big is your garden?
- Estate agents quote floor area 面积 in square metres. A "50 m² flat" is one whose floor covers 50 squares, each 1 m on a side.
- Finding area is one of the most practical things mathematics does — and it starts with a few key formulas.
Area scaling lab
area = side^2
Change side length and see why area grows quadratically.
Perimeter 周长
- Perimeter = the distance all the way round the outside of a shape.
- Rectangle: $P = 2(\text{length} + \text{width})$.
A rectangle $8\text{ cm} \times 5\text{ cm}$: $P = 2(8 + 5) = 26\text{ cm}$.
Area of the basic shapes; $b$ is the base, $h$ the perpendicular height 垂直高度, and $a$, $b$ the parallel sides of a trapezium
A rectangle is 8 cm by 5 cm. Find its perimeter (cm).
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Perimeter = 2(8 + 5) = 26 cm.
Match each shape to its area formula.
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Each shape has its own area rule; the trapezium averages the parallel sides.
Area formulas
| Shape | Area | Notes |
|---|---|---|
| rectangle | $\text{length} \times \text{width}$ | |
| triangle | $\dfrac{1}{2} \times b \times h$ | $h$ is perpendicular height |
| parallelogram | $b \times h$ | $h$ is perpendicular height |
| trapezium | $\dfrac{1}{2}(a+b)h$ | $a, b$ are the parallel sides |
Perpendicular height, not slant height. For triangles and parallelograms, $h$ must be measured at right angles to the base. Using the slant side gives the wrong answer.
Common solids and the dimensions used in their volume and surface-area formulas
A triangle has base 10 cm and height 6 cm. Find its area (cm²).
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½ × 10 × 6 = 30 cm².
A parallelogram has base 12 cm and perpendicular height 7 cm. Find its area (cm²).
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Area = b × h = 12 × 7 = 84 cm².
In the triangle area formula, h is the slant height of the triangle.
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h must be the perpendicular height (at 90° to the base), not the slant side.
A rectangle is 8 cm by 5 cm. Find its area (cm²).
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Area = length × width = 8 × 5 = 40 cm².
Worked example — trapezium
- Parallel sides $6\text{ cm}$ and $10\text{ cm}$, height $4\text{ cm}$.
- Area $= \dfrac{1}{2}(6+10) \times 4 = \dfrac{1}{2} \times 16 \times 4 = 32\text{ cm}^2$.
A triangle's area uses the perpendicular height — the height at $90^{\circ}$ to the base, not the slant side.
A trapezium has parallel sides 6 cm and 10 cm and height 4 cm. Find its area (cm²).
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½(6 + 10) × 4 = ½ × 16 × 4 = 32 cm².
Why does the triangle formula work?
- Any triangle is exactly half of a parallelogram with the same base and height.
- That's why the formula has the $\dfrac{1}{2}$ — it's half the parallelogram's area.
You've got it
- rectangle area $= l \times w$; triangle $= \dfrac{1}{2}bh$; parallelogram $= bh$
- trapezium $= \dfrac{1}{2}(a+b)h$ ($a, b$ are the parallel sides)
- $h$ is always the perpendicular height
Write from an area of 20
- A triangle has base 8 cm and height 5 cm.
- Find the area.
A triangle has base 8 cm and height 5 cm. What is its area, in square centimetres?
Half of 40 is 20.
What is the area?
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Half base times height is 20. Other options — 30 (the lesson triangle); 40 (the surrounding rectangle).