Perimeter and area

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Apuntes de la lección

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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Preguntas de práctica

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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

Más preguntas

9.Match each shape to the correct area formula.
Medium
- Rectangle
- Triangle
- Parallelogram
- Trapezium
- A = b × h
- A = 1/2 × b × h
- A = 1/2 × (a + b) × h
- A = l × w
10.Put these areas in order from smallest to largest.
Medium
- 500 cm²
- 0.05 m²
- 5000 mm²
- 0.5 m²
11.Which of the following statements are true about perimeter and area? (select all that apply)
Medium
- APerimeter is measured in linear units such as cm.
- BArea is measured in square units such as cm².
- CA shape with a larger perimeter always has a larger area.
- DTwo different shapes can have the same area but different perimeters.
12.A rectangle has an area of 36 cm² and a length of 9 cm. What is its width?
Medium
- A3 cm
- B4 cm
- C6 cm
- D27 cm
13.The area of a shape is measured in square units.
Easy
True or false?
14.A square has a perimeter of 20 cm. What is its area?
Medium
- A16 cm²
- B20 cm²
- C25 cm²
- D100 cm²
15.Match each area to its equivalent in another unit.
Medium
- 1 m²
- 1 cm²
- 1 hectare
- 1 km²
- 10,000 cm²
- 100 mm²
- 10,000 m²
- 1,000,000 m²
16.Put these steps in the correct order to find the area of a trapezium.
Medium
- Add the lengths of the parallel sides
- Multiply by the perpendicular height
- Divide by 2
- Write down the final answer with units
17.A rectangle has length 3 m and width 2 m. What is its area?
Easy
- A6 m²
- B5 m²
- C10 m²
- D6 m
18.A trapezium has parallel sides of length 6 cm and 10 cm, and the perpendicular height is 4 cm. What is its area?
Medium
- A32 cm²
- B64 cm²
- C40 cm²
- D80 cm²
19.Which of the following are equal to 1 m²? (select all that apply)
Medium
- A10,000 cm²
- B1,000,000 mm²
- C100 cm²
- D0.0001 km²
20.The area of a triangle is calculated using the perpendicular height, not the slant height.
Easy
True or false?
21.Match each shape to its area formula.
Medium
- Rectangle
- Triangle
- Parallelogram
- Trapezium
- l × w
- 1/2 × b × h
- b × h
- 1/2 × (a + b) × h
22.A rectangular lawn measures 12 m by 7 m. A square flower bed of side 3 m is cut out of the lawn. What area of lawn remains?
Medium
- A75 m²
- B84 m²
- C72 m²
- D63 m²
23.A parallelogram has base 14 cm and perpendicular height 5 cm. A triangle is drawn inside it with the same base and the same perpendicular height. What is the area of the triangle?
Hard
- A35 cm²
- B70 cm²
- C17.5 cm²
- D140 cm²
24.Which of the following statements are true about the area of a triangle? (select all that apply)
Medium
- AIt is half the area of a parallelogram with the same base and perpendicular height.
- BIt is found using the perpendicular height, not a slanted side.
- CDoubling the base always doubles the area.
- DIt is measured in cubic units.
- EThe formula is base × height.
25.To find the area of a parallelogram you multiply the base by the perpendicular height.
Easy
True or false?
26.Put these steps in the correct order to find the area of a compound shape made from a rectangle and a triangle joined together.
Medium
- Identify the rectangle and the triangle that make up the compound shape.
- Find the area of the rectangle.
- Find the area of the triangle.
- Add the two areas together.
27.A rectangle has an area of 60 cm². Its length is 12 cm. What is its perimeter?
Hard
- A34 cm
- B17 cm
- C5 cm
- D60 cm
28.A square has an area of 49 cm². What is its perimeter?
Medium
- A28 cm
- B14 cm
- C49 cm
- D7 cm
29.A rectangular garden is 15 m long and 10 m wide. A path of width 2 m runs along the inside of all four sides. What is the area of the path?
Hard
- A84 m²
- B150 m²
- C66 m²
- D100 m²
30.Which of these is the SI unit of area?
Easy
- Asquare metre (m²)
- Bmetre (m)
- Ccubic metre (m³)
- Dkilogram (kg)
31.How many square centimetres are there in one square metre?
Easy
- A100
- B1000
- C10000
- D1000000
32.How many square millimetres are there in one square centimetre?
Medium
- A10
- B100
- C1000
- D10000
33.A square has sides of length 1 m. What is its area in square millimetres?
Medium
- A1000 mm²
- B10000 mm²
- C100000 mm²
- D1000000 mm²
34.A rectangle is 4 m long and 3 m wide. What is its area?
Easy
- A7 m²
- B12 m²
- C14 m²
- D24 m²
35.A rectangle has length 5 cm and width 4 cm. Which statement is correct?
Medium
- AIts area is 20 cm² and its perimeter is 18 cm.
- BIts area is 18 cm² and its perimeter is 20 cm.
- CIts area is 20 cm and its perimeter is 18 cm².
- DIts area is 9 cm² and its perimeter is 20 cm.
36.A rectangle has an area of 24 cm² and a length of 6 cm. What is its perimeter?
Medium
- A4 cm
- B10 cm
- C18 cm
- D20 cm
37.A square has an area of 64 cm². What is its perimeter?
Hard
- A8 cm
- B16 cm
- C32 cm
- D64 cm
38.Which of the following statements are true? (select all that apply)
Medium
- AArea is measured in square units.
- BPerimeter is measured in square units.
- CA rectangle 3 m by 2 m has an area of 6 m².
- D1 m² = 10000 cm².
- E1 hectare = 1000 m².
39.Perimeter is measured in square units.
Easy
True or false?
40.1 hectare is equal to 10000 square metres.
Easy
True or false?
41.How many square centimetres are there in 1 square metre?
Easy
42.How many square millimetres are there in 1 square centimetre?
Easy
43.Complete the sentence with the correct word.
Medium
The area of a rectangle is the ____ of its length and width.
44.Complete the formula for the area of a trapezium.
Medium
A = \frac{1}{2} \times (a + b) \times ____
45.A triangle has a base of 12 cm and a perpendicular height of 7 cm. What is its area in cm²?
Medium
46.Complete the formula for the area of a parallelogram.
Medium
A = ____ × ____
47.A triangle has base 8 cm and perpendicular height 5 cm. Calculate its area in cm².
Medium
48.A rectangle has an area of 48 cm² and a width of 6 cm. Find its length in cm.
Medium
49.Write the formula for the area of a triangle using b for base and h for perpendicular height.
Easy
50.A parallelogram has an area of 72 cm² and a base of 9 cm. Find its perpendicular height in cm.
Hard
51.A rectangle has an area of 6 m². Give this area in square millimetres.
Medium
52.A square has a perimeter of 20 cm. Calculate the area of the square in cm².
Medium
53.Explain the difference between perimeter and area.
Medium
54.Explain how to find the area of a compound shape made from a rectangle and a triangle.
Hard
55.Describe how to find the area of a compound shape made from a rectangle and a triangle.
Medium
56.How many square centimetres are there in 1 square metre?
Easy
- A10000 cm²
- B11000 cm²
- C9000 cm²
- D12000 cm²
57.How many square millimetres are there in 1 square centimetre?
Easy
- A100 mm²
- B110 mm²
- C120 mm²
- D90 mm²
58.A triangle has a base of 12 cm and a perpendicular height of 7 cm. What is its area in cm²?
Medium
- A50 cm²
- B38 cm²
- C42 cm²
- D46 cm²
59.A triangle has base 8 cm and perpendicular height 5 cm. Calculate its area in cm².
Medium
- A20 cm²
- B24 cm²
- C22 cm²
- D18 cm²
60.A rectangle has an area of 48 cm² and a width of 6 cm. Find its length in cm.
Medium
- A7 cm
- B8 cm
- C10 cm
- D9 cm
61.A parallelogram has an area of 72 cm² and a base of 9 cm. Find its perpendicular height in cm.
Hard
- A9 cm
- B8 cm
- C10 cm
- D7 cm
62.A rectangle has an area of 6 m². Give this area in square millimetres.
Medium
- A5400000 mm²
- B7200000 mm²
- C6600000 mm²
- D6000000 mm²
63.A square has a perimeter of 20 cm. Calculate the area of the square in cm².
Medium
- A31 cm²
- B22 cm²
- C25 cm²
- D28 cm²

Apuntes de repaso

## 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.

Tarjetas

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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