Area And Perimeter

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Notes

Perimeter

**Perimeter** is the total distance around the outside of a 2D shape.
For a circle, perimeter is called **circumference**.
Add all side lengths; for regular shapes, multiply side length by number of sides.
For **compound shapes**, split into rectangles, triangles, or parts of circles.
Use properties (e.g., equal sides, parallel sides) to find missing lengths.
Units: mm, cm, m, etc.

Area on a Grid

Count whole squares inside the shape.
Pair up half squares or parts to make whole squares.
Multiply the total squares by the area each square represents.
Example: 14 whole squares on $a 1 cm^{2}$ grid gives area $14 cm^{2}$ .

Area Formulas

**Rectangle**: Area = length × width.
**Triangle**: Area = ½ × base × perpendicular height.
**Trapezium**: Area = ½ $\times (sum$ of parallel $sides) \times$ height.
**Parallelogram**: Area = base × perpendicular height.
The triangle formula is given in exams; others should be memorised.

Adding & Subtracting Areas

**Compound shapes** can be split into standard shapes (rectangles, triangles).
Find area of each part and add them.
Alternatively, add an extra shape to form a standard shape, then subtract its area.
Choose the simplest split to minimise calculations.

Problem Solving with Areas

Real-life problems often involve area and cost (e.g., carpet, paint).
Read carefully: key words like 'area', 'cost per $m^{2}'$ hint at calculations.
Annotate diagrams with known and missing lengths.
Compare options (e.g., different companies) to find minimum cost.
Show working even if final answer is not reached – marks for correct steps.

Worked Example: Compound Shape Perimeter

Split shape into triangle and rectangles.
Use equal side marks to find missing lengths (e.g., triangle side $= 6 cm)$ .
Horizontal lengths: sum of shorter = longer opposite (e.g., 15 + ? $= 18$ → ? $= 3 cm)$ .
Add all sides: $6+6+18+2+3+4+15 = 54$ cm.

Worked Example: Trapezium Area

Parallel sides $a=30$ cm, $b=15$ cm, height $h=20$ cm.
Area = ½ $\times (30+15) \times 20 = 450 cm^{2}$ .

Worked Example: Parallelogram Area

Base $= 15$ cm, perpendicular height $= 12$ cm.
Area $= 15 \times 12 = 180 cm^{2}$ .

Worked Example: Triangle Area

Base $= 8$ cm, height $= 7$ cm.
Area = ½ $\times 8 \times 7 = 28 cm^{2}$ .

Worked Example: Compound Shape Area (Pentagon)

Split into rectangle $(12\times 4)$ and triangle (base 7, height 5).
Rectangle area $= 48 cm^{2}$ , triangle area = ½×7×5 $= 17.5 cm^{2}$ .
Total area $= 65.5 cm^{2}$ .

Perimeter of a Compound Shape

Area of a Trapezium

Compound Shape Split into Rectangles

Area of a Triangle

Practice questions

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1.What is the perimeter of a square with side length 5 cm?
Easy
- A10 cm
- B15 cm
- C20 cm
- D25 cm
2.The area of a square is $64 cm^{2}$ . What is the length of one side?
Easy
- A8 cm
- B16 cm
- C32 cm
- D4 cm
3.A rectangle has length 7 cm and width 3 cm. What is its area?
Easy
- A $10 cm^{2}$
- B $21 cm^{2}$
- C $20 cm^{2}$
- D $24 cm^{2}$
4.What is the area of a triangle with base 6 cm and perpendicular height 4 cm?
Easy
- A $12 cm^{2}$
- B $24 cm^{2}$
- C $10 cm^{2}$
- D $20 cm^{2}$
5.The length of a rectangle is 3 cm longer than its width. The perimeter is 26 cm. What is the width?
Medium
- A5 cm
- B6 cm
- C7 cm
- D8 cm
6.A trapezium has parallel sides of lengths 12 cm and 8 cm, and perpendicular height 5 cm. What is its area?
Medium
- A $50 cm^{2}$
- B $40 cm^{2}$
- C $100 cm^{2}$
- D $30 cm^{2}$
7.A parallelogram has base 15 cm and perpendicular height 12 cm. What is its area?
Medium
- A $180 cm^{2}$
- B $90 cm^{2}$
- C $150 cm^{2}$
- D $30 cm^{2}$
8.The area of triangle ABC is $27 cm^{2}$ and $AB = 6$ cm. What is the perpendicular height from C to AB?
Hard
- A9 cm
- B4.5 cm
- C3 cm
- D12 cm

Summary

## Area & Perimeter – Quick Revision ### Perimeter - **Perimeter** = total distance around a 2D shape. For a circle, it's called 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, Pythagoras) to find missing lengths. ### Area - **Area** = space inside a 2D shape. Units: cm², m², etc. - **Counting squares**: count whole squares, pair up half squares. #### Key Formulas - **Rectangle**: Area = length × width A = l × w - **Triangle**: Area = ½ × base × perpendicular height A = ½ b h *(given in exam)* - **Trapezium**: Area = ½ × (sum of parallel sides) × perpendicular height A = ½ (a + b) h - **Parallelogram**: Area = base × perpendicular height A = b h ### Adding & Subtracting Areas - For compound shapes: split into standard shapes, find each area, then **add**. - Alternatively, add an extra shape to make a larger standard shape, then **subtract** the extra area. ### Problem Solving with Areas - Read carefully: real-life scenarios (carpet, paint, tiles). - Look for key words: area, cost, minimum, maximum. - Annotate diagrams with missing lengths. - Combine area with other topics (e.g., cost per m²). - Show working – even partial work earns marks. **Tip:** Always check units and convert if needed.

Key terms

Perimeter: The total distance around the outside of a 2D shape.
Area: The amount of space within the perimeter of a 2D shape.
Compound shape: A shape made up of two or more simpler 2D shapes (e.g., rectangles, triangles).
Perpendicular height: The height measured at right angles to the base.
Trapezium: A quadrilateral with one pair of parallel sides.
Parallelogram: A quadrilateral with opposite sides parallel and equal in length.

Slides

Perimeter

Perimeter is the total distance around the outside of a 2D shape.
For a circle, it is called the circumference.
To find perimeter, add the lengths of all sides.
For a regular shape, multiply number of sides by side length.

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

10 cm
15 cm
20 cm
25 cm

Area of Rectangles and Triangles

Area of a rectangle = length × width.
Area of a triangle = ½ × base × perpendicular height.
The triangle formula is given in the exam.

The area of a rectangle with length 8 cm and width 3 cm $is ____ cm^{2}$ .

Area of Trapezium and Parallelogram

Area of a trapezium = ½ $\times (a + b) \times h$ , where a and b are parallel sides and h is perpendicular height.
Area of a parallelogram = base × perpendicular height.
The perpendicular height is the distance between the base and the opposite side.

A trapezium has parallel sides 6 cm and 10 cm, and height 4 cm. What is its area?

$32 cm^{2}$
$40 cm^{2}$
$64 cm^{2}$
$16 cm^{2}$

Area on a Grid

Count whole squares inside the shape.
Pair up half squares to make whole squares.
Multiply the total number of squares by the area each square represents.

Which of these are true about finding area on a grid?

You only count whole squares.
Half squares can be paired to make whole squares.
The scale tells you the area of one square.

Compound Shapes

Compound shapes can be split into standard shapes (rectangles, triangles, etc.).
Find the area of each part and add them together.
Sometimes it is easier to subtract an extra shape from a larger rectangle.

A compound shape is made of a rectangle 5 cm by 3 cm and a triangle with base 5 cm and height 2 cm. Total area $= ____ cm^{2}$ .

Problem Solving with Area

Real-life problems often involve area and cost.
Read the question carefully and identify what you need to find.
Break the problem into steps: find area, then cost, etc.

A carpet costs £5 per $m^{2}$ . A room has area $12 m^{2}$ . How much does the carpet cost?

£50
£60
£70
£120

Perimeter of Compound Shapes

Add all the outer side lengths.
Use properties of shapes to find missing lengths.
For example, opposite sides of a rectangle are equal.

A compound shape has sides: 4 cm, 5 cm, 4 cm, 5 cm, and two missing sides of 2 cm each. Perimeter $= ____ cm$ .

Area of a Right-Angled Triangle

Use the formula: area = ½ × base × height.
The base and height are the two sides that meet at the right angle.

A right-angled triangle has base 6 cm and height 8 cm. What is its area?

$24 cm^{2}$
$48 cm^{2}$
$14 cm^{2}$
$28 cm^{2}$

Using Algebra in Area and Perimeter

You may need to form expressions for perimeter or area.
For example, perimeter of a rectangle with length l and width w is $2(l + w)$ .
Area of a square with side x is $x^{2}$ .

A rectangle has length (x+2) and width x. Its perimeter $is ____$ .

Summary

Perimeter is the distance around a shape.
Area formulas: rectangle, triangle, trapezium, parallelogram.
Compound shapes: split or subtract.
Always check units and read the problem carefully.

A parallelogram has base 10 cm and perpendicular height 5 cm. What is its area?

$50 cm^{2}$
$25 cm^{2}$
$15 cm^{2}$
$30 cm^{2}$

Flashcards

What is perimeter?

Perimeter is the total distance around the outside of a 2D shape. For a circle, it is called the circumference.

How do you find the perimeter of a regular polygon?

Multiply the number of sides by the length of one side. For example, a square of side x cm has perimeter 4x cm.

What is area?

Area is the amount of space within the perimeter of a 2D shape, measured in square units $(e.g., cm^{2}, m^{2})$ .

How do you find the area of a shape on a square grid?

Count the total number of whole squares inside the shape. Pair up half squares or parts of squares to make whole ones. Multiply the number of squares by the area each square represents.

What is the formula for the area of a rectangle?

Area = length × width $(A = l \times w)$ .

What is the formula for the area of a triangle?

Area = ½ × base × perpendicular height $(A =$ ½ bh).

What is the formula for the area of a trapezium?

Area = ½ $\times (sum$ of parallel $sides) \times$ perpendicular height $(A =$ ½ $(a + b) h)$ .

What is the formula for the area of a parallelogram?

Area = base × perpendicular height $(A = b \times h)$ .

How do you find the area of a compound shape?

Split the compound shape into standard shapes (e.g., rectangles, triangles), find the area of each, and add them together. Alternatively, add an extra shape and subtract its area.

A square has area $64 cm^{2}$ . What is the length of one side?

Side length $= \sqrt{64} = 8$ cm.

A rectangle has length 7a and width 2a. Write an expression for its perimeter in simplest form.

Perimeter $= 2(7a + 2a) = 2(9a) = 18a$ .

A rectangle has length 7a and width 2a. Write an expression for its area in simplest form.

Area $= 7a \times 2a = 14a^{2}$ .

A rectangle has length l cm longer than width w. Its perimeter is 26 cm. Find l and w if $l = w + 3$ .

Perimeter $= 2(l + w) = 26$ , so $l + w = 13$ . Substitute $l = w + 3: (w + 3) + w = 13$ → $2w = 10$ → $w = 5, l = 8$ .

A rectangular patio is 6 m by 8 m. What is its perimeter?

Perimeter $= 2(6 + 8) = 2 \times 14 = 28 m$ .

A rectangular patio is 6 m by 8 m. How many $0.5 m \times 0.5 m$ tiles are needed to cover it?

Area of patio $= 6 \times 8 = 48 m^{2}$ . Area of one tile $= 0.5 \times 0.5 = 0.25 m^{2}$ . Number of tiles $= 48 \div 0.25 = 192$ .

A triangle has area $27 cm^{2}$ and base $AB = 6$ cm. Find the perpendicular height h.

Area = ½ × base × height → $27 =$ ½ $\times 6 \times h$ → $27 = 3h$ → $h = 9$ cm.

A square has perimeter 12x. Find an expression for its area in terms of x.

Side length $= 12x \div 4 = 3x$ . Area $= (3x)^{2} = 9x^{2}$ .

A right-angled triangle has legs 8 cm and 7 cm. What is its area?

Area = ½ $\times 8 \times 7 = 28 cm^{2}$ .

A trapezium has parallel sides 30 cm and 15 cm, and height 20 cm. Find its area.

Area = ½ $\times (30 + 15) \times 20 =$ ½ $\times 45 \times 20 = 450 cm^{2}$ .

A parallelogram has base 15 cm and perpendicular height 12 cm. Find its area.

Area $= 15 \times 12 = 180 cm^{2}$ .

How do you find the perimeter of a compound shape?

Add together the lengths of all sides. Use properties of shapes (e.g., equal sides) to find missing lengths.

A shape on a 1 cm grid has 14 whole squares and some half squares that pair to make 2 whole squares. What is its area?

Total area $= 14 + 2 = 16 cm^{2}$ .

A rectangle is 3 cm by 8 cm. Five congruent rectangles are joined to make a shape. What is the perimeter?

The shape's perimeter depends on arrangement. For $a 3\times 8$ rectangle, perimeter of one is 22 cm. For five joined, the total perimeter is not simply 5×22; you must add only outer edges. (Answer not uniquely given without diagram.)

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