Perimeter and area

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课程笔记

Perimeter Basics

Perimeter is the total distance around the outside of a 2D shape.
To find the perimeter, add up the lengths of all the sides.
For a rectangle with length l and width w, the perimeter is P = 2(l + w).
For a square with side length s, the perimeter is P = 4s.
Perimeter is measured in units of length, such as metres (m) or centimetres (cm).

Area Basics

Area is the measure of the surface a 2D shape covers.
Area is measured in square units, such as square metres (m²) or square centimetres (cm²).
A square with sides of 1 metre has an area of 1 square metre (1 m²).
The area of a shape can be found by counting how many unit squares fit inside it.

Area of Rectangles

The area of a rectangle is given by the formula A = l × w, where l is the length and w is the width.
For a square with side length s, the area is A = s².
Example: A rectangle of length 5 cm and width 3 cm has area 5 × 3 = 15 cm².

Area of Triangles

The area of a triangle is given by the formula A = 1/2 × b × h, where b is the base and h is the perpendicular height.
The height must be measured at right angles to the base.
Example: A triangle with base 6 m and height 4 m has area 1/2 × 6 × 4 = 12 m².

Area of Parallelograms

The area of a parallelogram is given by the formula A = b × h, where b is the base and h is the perpendicular height.
The perpendicular height is the vertical distance between the base and the opposite side.
Example: A parallelogram with base 8 cm and height 5 cm has area 8 × 5 = 40 cm².

Area of Trapezia

A trapezium has one pair of parallel sides.
The area of a trapezium is given by the formula A = 1/2 × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular height between them.
Example: A trapezium with parallel sides 6 m and 10 m and height 4 m has area 1/2 × (6 + 10) × 4 = 32 m².

Area of Compound Shapes

A compound shape is made by combining simpler shapes.
To find the area of a compound shape, split it into rectangles, triangles, or other known shapes.
Calculate the area of each part and add them together.
Sometimes it is easier to subtract the area of a missing part from a larger shape.

Working Backwards from Area

If you know the area and all but one length, you can find the missing length by rearranging the area formula.
For a rectangle, if A and w are known, then l = A / w.
For a triangle, if A and b are known, then h = 2A / b.
For a parallelogram, if A and b are known, then h = A / b.
For a trapezium, if A, a, and h are known, then b = (2A / h) − a.

Choosing Correct Units

Use linear units (e.g., cm, m) for perimeter.
Use square units (e.g., cm², m²) for area.
Convert between units of area by squaring the length conversion factor.
Example: Since 1 m = 100 cm, 1 m² = 100² cm² = 10,000 cm².
Common metric area units include square millimetres (mm²), square centimetres (cm²), square metres (m²), and square kilometres (km²).

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练习题

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1.Which unit is the standard SI unit of area?
Easy
- Asquare metre
- Bmetre
- Ccubic metre
- Dkilogram
2.Area is measured in square units.
Easy
True or false?
3.A rectangle has length 8 cm and width 5 cm. What is its area?
Medium
- A40 cm²
- B26 cm²
- C13 cm²
- D80 cm²
4.A rectangle has length 8 cm and width 5 cm. What is its perimeter?
Medium
- A26 cm
- B40 cm
- C13 cm
- D21 cm
5.Which formula gives the area of a triangle?
Medium
- AA = \frac{1}{2} \times b \times h
- BA = b × h
- CA = 2(b + h)
- DA = b + h
6.A parallelogram has base 10 cm and perpendicular height 6 cm. What is its area?
Medium
- A60 cm²
- B30 cm²
- C16 cm²
- D32 cm²
7.A trapezium has parallel sides of length 5 cm and 9 cm, and the perpendicular distance between them is 4 cm. What is its area?
Hard
- A28 cm²
- B56 cm²
- C18 cm²
- D45 cm²
8.Which of the following are units of area? (select all that apply)
Medium
- Acm²
- Bm²
- Cm
- Dkg
- Ehectare

复习笔记

## Perimeter and Area — KS3 Revision **Perimeter** is the total distance around the outside of a 2D shape. Add up all the side lengths. Units: mm, cm, m, km (length units). **Area** is the measure of a region's size on a surface — think of the amount of paint needed to cover it with a single coat. It is measured by comparing the shape to squares of a fixed size. Units: mm^2, cm^2, m^2, km^2 (square units). ## Key formulae - Rectangle: A = l × w (l = length, w = width) - Triangle: A = 1/2 × b × h (b = base, h = perpendicular height) - Parallelogram: A = b × h (b = base, h = perpendicular height) - Trapezium: A = 1/2 × (a + b) × h (a and b = the two parallel sides, h = perpendicular height between them) For a triangle and a parallelogram, h must be the **perpendicular** height, not a slanted side. ## Compound shapes Split a compound shape into rectangles, triangles, parallelograms or trapezia. Find each area, then add them. Sometimes it is easier to subtract a missing piece from a larger shape. ## Working backwards Substitute the known values into the formula and solve for the missing length. - Rectangle: l = A / w - Triangle: b = 2A / h - Parallelogram: b = A / h - Trapezium: a + b = 2A / h ## Units Every unit of length has a corresponding unit of area — the area of a square with that side length. Algebraically, area units are the squares of the length units. - 1 m = 100 cm, so 1 m^2 = 10,000 cm^2 - 1 cm = 10 mm, so 1 cm^2 = 100 mm^2 - 1 km = 1000 m, so 1 km^2 = 1,000,000 m^2 - 1 m^2 = 1,000,000 mm^2 In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. For example, 1 foot = 12 inches, so 1 square foot = 144 square inches (144 = 12^2). ## Exam tips - Check the unit asked for before writing your answer. - Perimeter is a length; area is measured in square units. - For compound shapes, label each part clearly. - Never mix units in one calculation — convert first.

Perimeter: The total distance around the outside of a 2D shape, found by adding all its side lengths.
Area: The measure of a region's size on a surface; the amount of material of a given thickness needed to fashion a model of the shape, or the paint needed to cover it with a single coat.
Square metre (m^2): The SI unit of area: the area of a square whose sides are one metre long.
Unit square: A square defined to have area one; the area of any other shape is a dimensionless real number found by comparing it to unit squares.
Perpendicular height: The height measured at right angles to the base; used as h in the triangle, parallelogram and trapezium area formulae.
Compound shape: A shape made from two or more simpler shapes, such as rectangles, triangles, parallelograms or trapezia.
Surface area: The area of the boundary surface of a solid shape such as a sphere, cone or cylinder.
Hectare: A unit of area commonly used to measure land: 1 hectare = 10,000 square metres = 0.01 square kilometres.

闪卡

What is perimeter?

The perimeter is the total distance around the outside of a 2D shape. It is a length, so it is measured in units such as cm or m.

What is area?

Area is the measure of the size of a region on a surface. It is measured in square units, such as cm² or m².

How do you find the perimeter of a rectangle?

Add the lengths of all four sides, or use P = 2(l + w), where l is length and w is width.

What is the formula for the area of a rectangle?

A = l × w, where l is length and w is width. For a square, A = s², where s is the side length.

What is the formula for the area of a triangle?

A = 1/2 × b × h, where b is the base and h is the perpendicular height.

What is the formula for the area of a parallelogram?

A = b × h, where b is the base and h is the perpendicular height.

What is the formula for the area of a trapezium?

A = 1/2 × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular height between them.

How do you find the area of a compound shape?

Split the shape into simpler shapes such as rectangles and triangles, find the area of each part, then add them together.

How can you work backwards from an area to find a missing length?

Substitute the known values into the area formula and solve for the unknown length. For example, if A = l × w and A and w are known, then l = A / w.

What is the standard unit of area in the International System of Units (SI)?

The square metre, written as m². It is the area of a square with sides of one metre.

Convert 1 square metre to square centimetres.

1 m² = 10,000 cm².

Convert 1 square centimetre to square millimetres.

1 cm² = 100 mm².

Convert 1 square kilometre to square metres.

1 km² = 1,000,000 m².

How do you convert between square units in non-metric systems?

Square the conversion factor for the corresponding length units. For example, since 1 foot = 12 inches, 1 square foot = 12² = 144 square inches.

What is the difference between area and surface area?

Area usually refers to the size of a flat 2D region. Surface area is the total area of the boundary surface of a 3D object.

What is a hectare in square metres?

1 hectare = 10,000 m². It is equal to 100 ares.

What is an acre in square yards and square feet?

1 acre = 4,840 square yards = 43,560 square feet. It is approximately 40% of a hectare.

What is the area of a square with side length 5 cm?

A = s² = 5² = 25 cm².

What is the perimeter of a rectangle with length 8 m and width 3 m?

P = 2(l + w) = 2(8 + 3) = 2 × 11 = 22 m.

What is the area of a triangle with base 10 cm and perpendicular height 6 cm?

A = 1/2 × b × h = 1/2 × 10 × 6 = 30 cm².

What is the area of a parallelogram with base 7 m and perpendicular height 4 m?

A = b × h = 7 × 4 = 28 m².

What is the area of a trapezium with parallel sides 5 cm and 9 cm, and height 4 cm?

A = 1/2 × (a + b) × h = 1/2 × (5 + 9) × 4 = 1/2 × 14 × 4 = 28 cm².

A rectangle has area 36 cm² and width 4 cm. What is its length?

l = A / w = 36 / 4 = 9 cm.

How do you find the perimeter of a compound shape?

Add the lengths of all the outer sides. Be careful not to include any internal edges.

What is the area of a square with side length 1 m in square centimetres?

1 m = 100 cm, so area = 100² = 10,000 cm².

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