Length, perimeter, area & circles

Year 7 core

By the end of this topic you should be able to:

convert between metric length units (mm, cm, m, km),
calculate the perimeter of any polygon,
find the area of a rectangle, a triangle, and a parallelogram,
name the parts of a circle (centre, radius, diameter, circumference),
use $\pi$ to relate the circumference of a circle to its radius and diameter.

1. Metric length units

Length conversions

Metric ladder (each step is x10 or /10 between adjacent units)

1 \text{ cm} = 10 \text{ mm}, \quad 1 \text{ m} = 100 \text{ cm}, \quad 1 \text{ km} = 1000 \text{ m}.

Converting

Bigger -> smaller unit: multiply. Smaller -> bigger unit: divide.

$3.5$ m $= 3.5 \times 100 = 350$ cm
$8400$ mm $= 8400 \div 10 = 840$ cm

2. Perimeter

The perimeter is the total distance around the outside of a shape. Add up every side length.

Worked example 1 Perimeter of an irregular polygon

A pentagon has sides $4$ cm, $6$ cm, $5$ cm, $7$ cm, $4$ cm. Find the perimeter.

P = 4 + 6 + 5 + 7 + 4 = 26 \text{ cm}.

3. Area of standard shapes

Area measures the amount of surface a shape covers. Units are squared: mm^2, cm^2, m^2, km^2.

Area formulas (Year 7 core)

Rectangle

A = L \times W.

Triangle

A = \tfrac{1}{2} \times b \times h,

where $b$ is the base and $h$ is the perpendicular height from the base to the opposite vertex.

Parallelogram

A = b \times h,

where $h$ is the perpendicular height - the same as for a triangle on the same base.

Square (special case of rectangle)

A = s^{2}.

Triangle and parallelogram with base b and perpendicular height h.

Worked example 2 Triangle area

A triangle has base $8$ cm and perpendicular height $5$ cm. Find its area.

A = \tfrac{1}{2} \times 8 \times 5 = 20 \text{ cm}^{2}.

Worked example 3 Parallelogram area

A parallelogram has base $12$ m and perpendicular height $7$ m. Find its area.

A = 12 \times 7 = 84 \text{ m}^2.

4. Circles: parts and the role of pi

Parts of a circle: centre, radius r, diameter d, circumference C.

The centre is the point in the middle.
The radius ( $r$ ) is the distance from the centre to any point on the circle.
The diameter ( $d$ ) is the distance straight across, through the centre; it equals $2r$ .
The circumference ( $C$ ) is the total distance around the circle.

Circle relationships

Diameter and radius

d = 2r.

Circumference in terms of pi

C = \pi \times d \;=\; 2 \pi r.

The symbol $\pi$ (pi) is the constant ratio of circumference to diameter. $\pi \approx 3.14159 \approx \tfrac{22}{7}$ .

Worked example 4 Circumference from a radius

A circle has radius $5$ cm. Find its circumference, using $\pi \approx 3.14$ .

C = 2 \pi r = 2 \times 3.14 \times 5 = 31.4 \text{ cm}.

Worked example 5 Diameter from a circumference

A circle has circumference $44$ m. Find its diameter, using $\pi \approx \tfrac{22}{7}$ .

d = \frac{C}{\pi} = 44 \div \frac{22}{7} = 44 \times \frac{7}{22} = 14 \text{ m}.

Practice: Year 7 core

Fluency

Length and perimeter

Convert $250$ cm to metres.
Convert $3.4$ km to metres.
Convert $85$ mm to cm.
Convert $2500$ m to km.
Find the perimeter of a rectangle $12$ cm by $7$ cm.
Find the perimeter of a square with side $9$ cm.
A triangle has sides $5$ cm, $7$ cm, $9$ cm. Find its perimeter.
A regular hexagon has side $4$ cm. Find its perimeter.

Fluency

Area of rectangles, triangles, parallelograms

Find the area of a $15$ cm by $4$ cm rectangle.
Find the area of a square with side $6$ cm.
Find the area of a triangle with base $10$ cm and height $8$ cm.
Find the area of a triangle with base $12$ m and height $5$ m.
Find the area of a parallelogram with base $9$ cm and height $6$ cm.
A rectangle has area $48$ cm^2 and length $8$ cm. Find its width.
A triangle has area $36$ cm^2 and base $9$ cm. Find its height.
A parallelogram has area $60$ m^2 and height $5$ m. Find its base.

Fluency

Circles

Use $\pi \approx 3.14$ unless another value is given.

A circle has radius $4$ cm. Find its diameter.
A circle has diameter $18$ m. Find its radius.
Find the circumference of a circle with radius $10$ cm.
Find the circumference of a circle with diameter $14$ cm, using $\pi \approx \tfrac{22}{7}$ .
A circle has circumference $31.4$ cm. Find its diameter.
True or false: for every circle, circumference $\div$ diameter gives about the same number.

Reasoning

Explain and reason

Two rectangles have the same perimeter. Does it follow that they have the same area? Justify with a numerical example.
Pete writes the area of a triangle with base $6$ and slant side $5$ as $\tfrac{1}{2} \times 6 \times 5 = 15$ . Explain what is wrong.
Is it possible for one shape to have a larger perimeter but a smaller area than another? Give an example.
A pizza of diameter $30$ cm is cut exactly in half. What is the perimeter of each half-pizza (the crust plus the straight cut)?
Without calculating, decide which has the larger area: a square with side $10$ cm or a rectangle $12$ cm by $8$ cm. Explain.

Problem solving

Real-world problems

A rectangular paddock is $80$ m by $45$ m. What is the cost of fencing it at $18 per metre?
A triangular sail has base $3.5$ m and height $4$ m. What is its area in square metres?
A circular garden pond has diameter $3.5$ m. How long is a rope needed to go once around the edge? (Use $\pi \approx \tfrac{22}{7}$ .)
A rectangular garden is $8$ m by $12$ m. Convert the area to square centimetres.
A bicycle wheel has diameter $70$ cm. How far (to the nearest metre) does the bike travel in $20$ turns of the wheel? (Use $\pi \approx \tfrac{22}{7}$ .)

Extension

More area formulas (extension)

Trapezium

A = \tfrac{1}{2}(a + b) \times h,

where $a$ and $b$ are the parallel sides and $h$ is the perpendicular distance between them.

Composite shapes

Break the shape into rectangles and triangles (or start with a big rectangle and subtract the missing piece).

Practice: Extension

Reasoning

Trapezium and composite areas

Find the area of a trapezium with parallel sides $4$ cm and $10$ cm and height $3$ cm.
Find the area of a trapezium with parallel sides $6$ m and $10$ m and height $4$ m.
An L-shape is made of an $8$ m by $5$ m rectangle with a $3$ m by $2$ m rectangle removed from one corner. Find its area.
A path $1$ m wide runs around a $10$ m by $6$ m garden, on the outside. Find the area of the path.
A rectangular piece of cardboard is $40$ cm by $30$ cm. A $5$ cm square is cut from each corner. What is the remaining area?