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\[ P,\ A,\ V,\ SA \]
Welcome to Grade 8 Measurement! In this lesson we explore perimeter, area, volume and surface area — the four pillars of measurement. Use the Back and Next buttons to work through each concept step by step.
\[\text{Perimeter} = \text{total distance around a shape}\]
Perimeter is the distance around the outside of a 2D shape. Think of it as the length of a fence that would enclose the shape. Measured in units: mm, cm, m, km.
\[\begin{aligned} \text{Rectangle:} &\quad P = 2(l+w) \\ \text{Square:} &\quad P = 4s \\ \text{Triangle:} &\quad P = a+b+c \end{aligned}\]
These are the three main perimeter formulas. For a rectangle, add length and width, then multiply by 2. For a square all sides are equal so multiply one side by 4. For a triangle add all three sides.
\[ C = 2\pi r = \pi d \]
The perimeter of a circle is called the circumference. C = 2πr where r is the radius, or C = πd where d is the diameter. Use π ≈ 3.14 unless told otherwise.
\[\begin{aligned} P &= 2(l+w) \\ &= 2(12+7) \\ &= 2(19) \\ &= 38\text{ cm} \end{aligned}\]
Worked example: Rectangle with l = 12 cm, w = 7 cm. Substitute into the formula and simplify. The perimeter is 38 cm.
\[\text{Area} = \text{space INSIDE a 2D shape}\]
Area measures the flat space enclosed by a shape. Area is always measured in square units: mm², cm², m².
\[ A = l \times w \]
Area of a rectangle = length × width. Every unit of length times every unit of width gives a square unit of area.
\[ A = s^2 \]
A square is a special rectangle where all sides are equal. Its area is side squared — s².
\[\begin{aligned} A &= l \times w \\ &= 15 \times 8 \\ &= 120\text{ cm}^2 \end{aligned}\]
Worked example: Rectangle with l = 15 cm, w = 8 cm. Area = 15 × 8 = 120 cm². Note the squared unit.
\[ A = \dfrac{1}{2} \times b \times h \]
The area of a triangle is half the base times the perpendicular height. The height MUST be perpendicular (at 90°) to the base — it is NOT the slant side.
\[ \text{Triangle} = \dfrac{1}{2} \times \text{Rectangle} \]
A triangle is exactly half of a rectangle with the same base and height. That is why we multiply by ½.
\[\begin{aligned} A &= \dfrac{1}{2} \times 9 \times 6 \\ &= \dfrac{1}{2} \times 54 \\ &= 27\text{ cm}^2 \end{aligned}\]
Worked example: base = 9 cm, perpendicular height = 6 cm.
\[ h \perp b \]
Key tip: The height h must always be at 90° to the base b. In an obtuse triangle the height may fall OUTSIDE the triangle — but the formula stays the same.
\[ A = \pi r^2 \]
The area of a circle = π times radius squared. Remember r is the radius (half the diameter).
\[ r = \dfrac{d}{2} \]
If you are given the diameter, halve it to get the radius before substituting. Always substitute r (radius) into A = πr².
\[\begin{aligned} A &= \pi r^2 \\ &= \pi \times 5^2 \\ &= \pi \times 25 \\ &\approx 78.54\text{ cm}^2 \end{aligned}\]
Worked example: circle with radius r = 5 cm. A = π × 25 ≈ 78.54 cm².
\[ C = 2\pi r \quad A = \pi r^2 \]
Remember: circumference uses r to the power 1, area uses r squared. Don’t mix them up!
\[ V = \text{space inside a 3D solid} \]
Volume is the amount of three-dimensional space inside a solid. Measured in cubic units: mm³, cm³, m³.
\[ V = l \times w \times h \]
The volume of a rectangular prism (cuboid) = length × width × height. Also written as V = A_base × h, where A_base is the area of the base face.
\[\begin{aligned} V &= l \times w \times h \\ &= 8 \times 4 \times 5 \\ &= 160\text{ cm}^3 \end{aligned}\]
Worked example: l = 8 cm, w = 4 cm, h = 5 cm. V = 8 × 4 × 5 = 160 cm³.
\[ V = \dfrac{1}{2}bh \times l \]
For a triangular prism: Volume = area of triangular face × length of prism = ½bh × l.
\[ 1\text{ m}^3 = 1\,000\,000\text{ cm}^3 \]
Important unit conversion: 1 m³ = 1 000 000 cm³. Also: 1 litre = 1 000 cm³, so 1 m³ = 1 000 litres.
\[ SA = \text{total area of ALL faces} \]
Surface area is the total area of every face of a 3D solid. Imagine unfolding the solid into a flat net — the surface area is the total area of that net.
\[ SA = 2(lw + lh + wh) \]
A rectangular prism has 3 pairs of identical rectangular faces. Surface area = 2 × (length×width + length×height + width×height).
\[\begin{aligned} SA &= 2(lw+lh+wh) \\ &= 2[(8)(4)+(8)(5)+(4)(5)] \\ &= 2[32+40+20] \\ &= 2(92) \\ &= 184\text{ cm}^2 \end{aligned}\]
Worked example using l = 8, w = 4, h = 5 cm. Surface area = 184 cm².
\[ SA_{\text{cube}} = 6s^2 \]
A cube has 6 identical square faces. Since each face has area s², the total surface area = 6s².
\[\begin{aligned} \text{Prism:} &\quad SA = 2(lw+lh+wh) \\ \text{Cube:} &\quad SA = 6s^2 \end{aligned}\]
Recap: rectangular prism uses three pairs of faces; a cube uses six identical faces. Always answer in cm² (or m²).
\[\begin{aligned} P_{\text{rect}} &= 2(l+w) & A_{\text{rect}} &= lw \\ A_{\triangle} &= \tfrac{1}{2}bh & A_{\odot} &= \pi r^2 \\ V &= lwh & SA &= 2(lw+lh+wh) \end{aligned}\]
You have completed Grade 8 Measurement! Here is the full formula sheet. Perimeter is in linear units (cm), area in cm², volume in cm³. Now test yourself with the quiz below.