Area And Perimeter

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课程笔记

Perimeter

Perimeter is the total distance around the outside of a 2D shape; for a circle it is called circumference.
Perimeter is a length measured in one dimension, in units such as mm, cm, m.
Add together the lengths of all the sides of the shape.
For any regular shape, the perimeter is the number of sides multiplied by the length of one side (e.g., square side x cm has perimeter 4x cm).
For compound shapes, split into rectangles, triangles or parts of circles, and use properties of shapes (equal sides, Pythagoras' theorem) to find missing lengths.
In an L-shape, the sum of the lengths of two shorter sides equals the length of the longer side opposite.

Perimeter of a Compound Shape

Worked Example: Compound Shape Perimeter

The shape splits into a triangle and two rectangles; the dashes on the triangle show its two slanted sides are equal, so the missing side is 6 cm.
The horizontal lengths of the two rectangles add up to the longest horizontal line: 15 + missing = 18, so the missing horizontal length is 3 cm.
Add all the sides: 6 + 6 + 18 + 2 + 3 + 4 + 15 = 54 cm.

Compound Shape Perimeter

Area

Area is the amount of space within the perimeter of a 2D shape, measured in square units (mm², cm², m²).
Rectangle area = length × width (A = lw).
Triangle area = ½ × base × perpendicular height (A = ½bh) — this formula is given in the exam.
Trapezium area = ½ × (sum of parallel sides) × perpendicular height (A = ½(a+b)h).
Parallelogram area = base × perpendicular height (A = bh).
The perpendicular height is the distance between the base and the opposite side, not necessarily a side length.

Area Formulas

Worked Example: Trapezium Area

Parallel sides a = 30 cm, b = 15 cm, perpendicular height h = 20 cm.
Area = ½ × (30 + 15) × 20 = 450 cm².

Area of a Trapezium

Worked Example: Parallelogram Area

Base = 15 cm, perpendicular height = 12 cm.
Area = 15 × 12 = 180 cm².

Area of a Parallelogram

Worked Example: Triangle Area

Base = 8 cm, perpendicular height = 7 cm.
Area = ½ × 8 × 7 = 28 cm².

Area of a Triangle

Adding & Subtracting Areas

Compound shapes can be split into standard shapes (rectangles, triangles, etc.) and their areas added together.
Alternatively, complete the shape into a larger standard shape, then subtract the area of the extra piece added.
For the shape shown, the area is the rectangle's area minus the triangle's area.
Always check for missing lengths using the given dimensions and the properties of the shapes involved.

Compound Shape by Subtraction

Worked Example: Compound Shape Area (Pentagon)

Split the pentagon into a rectangle (12 cm by 4 cm) and a triangle (base 7 cm, height 5 cm), using the given lengths to find the triangle's base and height.
Rectangle area = 12 × 4 = 48 cm²; triangle area = ½ × 7 × 5 = 17.5 cm².
Total area = 48 + 17.5 = 65.5 cm².

Compound Shape Area (Pentagon)

Problem Solving with Areas

Real-life problems often involve area and cost (e.g., carpet, paint, tiles) — read the context carefully for key words like area, cost, minimum, maximum.
Compound units (e.g. £/m²) hint at which calculations are needed.
Annotate diagrams with known and missing lengths as you work through the problem.
Compare different options (e.g., companies) by calculating the total cost for each using the area.
Show your working even if you don't reach a final answer — marks are given for correct steps.

Splitting a Compound Shape for a Real Problem

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练习题

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1.What is the perimeter of a rectangle with length 8 cm and width 5 cm?
Easy
- A26 cm
- B13 cm
- C40 cm
- D18 cm
2.The area of a triangle with base 10 cm and perpendicular height 6 cm is
Easy
- A30 cm²
- B60 cm²
- C16 cm²
- D20 cm²
3.A square has side length 7 cm. Its perimeter is
Easy
- A28 cm
- B49 cm
- C14 cm
- D21 cm
4.The area of a trapezium with parallel sides 12 cm and 8 cm and perpendicular height 5 cm is
Easy
- A50 cm²
- B100 cm²
- C40 cm²
- D60 cm²
5.The area of triangle ABC is 27 cm² and AB = 6 cm. Calculate the perpendicular height h from AB to C.
Easy
- A9 cm
- B4.5 cm
- C18 cm
- D3 cm
6.Calculate the area of the trapezium with parallel sides 10 cm and 6 cm and height 4 cm.
Easy
- A32 cm²
- B40 cm²
- C24 cm²
- D16 cm²
7.A right-angled triangle has legs 3 cm and 4 cm. What is its area?
Easy
- A6 cm²
- B12 cm²
- C7 cm²
- D5 cm²
8.The vertices of a square ABCD lie on the circumference of a circle of radius 8 cm. Calculate the area of the square.
Hard
- A128 cm²
- B64 cm²
- C256 cm²
- D32 cm²

复习笔记

## Area & Perimeter – Quick Revision ### Perimeter - **Perimeter** = total distance around a 2D shape (circle: circumference). - Add all side lengths. For regular shapes, multiply number of sides by side length. - For compound shapes, split into rectangles/triangles; use properties (equal sides, parallel sides) to find missing lengths. ### Area Formulas - **Rectangle**: \(A = l \times w\) - **Triangle**: \(A = \frac{1}{2} b h\) (given in exam) - **Trapezium**: \(A = \frac{1}{2}(a+b)h\) (a, b parallel sides, h perpendicular height) - **Parallelogram**: \(A = b h\) (b base, h perpendicular height) - **Square**: \(A = \text{side}^2\) ### Compound Shapes - Split into standard shapes (rectangles, triangles, etc.). - Add areas of parts, or subtract area of an extra shape (e.g., rectangle minus triangle). ### Problem Solving - Read context: area often links to cost (e.g., £/m²). - Find missing lengths using subtraction, Pythagoras, or properties. - For bounds: side length correct to nearest cm → lower bound = value - 0.5, upper bound = value + 0.5; perimeter bounds = 4 × side bounds. ### Key Tips - Always check units (convert if needed). - Show working – even partial steps earn marks. - For tiling: divide total area by tile area, not just length/width separately.

Perimeter: The total distance around the outside of a 2D shape.
Area: The amount of space within the perimeter of a 2D shape, measured in square units.
Compound shape: A shape made up of two or more simpler 2D shapes (e.g., rectangles, triangles).
Trapezium: A quadrilateral with one pair of parallel sides. Area = ½(a+b)h.
Perpendicular height: The height measured at right angles to the base.
Upper bound: The maximum possible value of a measurement given its degree of accuracy (e.g., 15 cm to nearest cm → upper bound = 15.5 cm).
Lower bound: The minimum possible value of a measurement given its degree of accuracy (e.g., 15 cm to nearest cm → lower bound = 14.5 cm).

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