MATHEMATICS CLASS NOTES: GRADE 5

TOPIC: MEASUREMENT – Perimeter, Area, and Volume of Rectangles, Squares, Triangles, and Irregular Shapes

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Welcome, Young Mathematicians!

Have you ever wondered how a tailor knows exactly how much lace is needed to border a beautiful Agbada or Iro and Buba? Or how a builder knows how many bags of cement are needed to floor a living room? How about how much water fits inside a plastic container or a water tank in your compound?

All of these real-life tasks rely on three powerful mathematical tools: Perimeter, Area, and Volume. In this comprehensive set of class notes, we are going to explore these concepts deeply, step by step, using shapes we see every day around our homes and schools in Nigeria.

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SECTION 1: Comprehensive Core Concepts

1. Understanding 2D vs. 3D Shapes

Before we start measuring, we must understand the difference between flat shapes and solid objects:

• Two-Dimensional (2D) Shapes: These are flat shapes. They only have two dimensions: length (how long) and breadth/width (how wide). Examples include a sheet of paper, a football pitch, or a drawing of a triangle on a blackboard. We measure Perimeter and Area on 2D shapes.
• Three-Dimensional (3D) Shapes: These are solid objects that occupy physical space. They have three dimensions: length, breadth, and height/depth. Examples include a Milo tin, a brick, a classroom cupboard, or a cardboard box. We measure Volume in 3D shapes.

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CONCEPT A: PERIMETER (The Outer Boundary)

What is Perimeter?

Imagine you are an ant standing at one corner of your mathematics exercise book. If you crawl all the way around the outer edge until you return to your starting point, the total distance you travelled is the perimeter.

Definition: The Perimeter is the total distance around the outside edge of a flat, two-dimensional shape.

• Unit of Measurement: Because perimeter is a measure of length, we use standard linear units such as millimeters (mm), centimeters (cm), meters (m), or kilometers (km).

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1. Perimeter of a Square

A square is a flat shape with four equal sides. It also has four right angles ($90^\circ$). Because all four sides are of equal length, we don't need to measure every single side individually if we already know the length of one side.

• Formula: $\text{Perimeter} = \text{Side} + \text{Side} + \text{Side} + \text{Side}$ $\text{Perimeter} = 4 \times \text{Side} \quad (\text{or } P = 4s)$

Worked Example:

A square-shaped window in a classroom has a side length of $45\text{ cm}$. What is the perimeter of the window?

• Step 1: Identify the shape and its given dimensions. $\text{Shape} = \text{Square}; \quad \text{Side length } (s) = 45\text{ cm}$
• Step 2: Write down the formula. $P = 4 \times s$
• Step 3: Substitute the value and calculate. $P = 4 \times 45\text{ cm}$ $P = 45 + 45 + 45 + 45 = 180\text{ cm}$
• Answer: The perimeter of the classroom window is $180\text{ cm}$.

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2. Perimeter of a Rectangle

A rectangle has four sides, but unlike a square, only its opposite sides are equal in length. The longer side is called the Length ($L$) and the shorter side is called the Breadth ($B$) or Width ($W$).

• Formula: $\text{Perimeter} = \text{Length} + \text{Breadth} + \text{Length} + \text{Breadth}$ $\text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) \quad (\text{or } P = 2(L + B))$

Worked Example:

An agricultural plot behind a school compound is rectangular. Its length is $12\text{ meters}$ and its breadth is $8\text{ meters}$. Find the perimeter of the plot so the students know how much wire mesh to buy for a fence.

• Step 1: Identify the dimensions. $\text{Length } (L) = 12\text{ m}, \quad \text{Breadth } (B) = 8\text{ m}$
• Step 2: Write down the formula. $P = 2(L + B)$
• Step 3: Add the length and breadth first (following BODMAS), then multiply by 2. $P = 2 \times (12\text{ m} + 8\text{ m})$ $P = 2 \times 20\text{ m}$ $P = 40\text{ m}$
• Answer: The perimeter of the plot is $40\text{ m}$.

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3. Perimeter of a Triangle

A triangle is a three-sided flat shape. It can have equal sides (equilateral), two equal sides (isosceles), or three unequal sides (scalene). To find its perimeter, we simply add the lengths of its three sides.

• Formula: $\text{Perimeter} = a + b + c \quad (\text{where } a, b, \text{ and } c \text{ are the lengths of the three sides})$

Worked Example:

A triangular-shaped traffic sign has side lengths of $30\text{ cm}$, $30\text{ cm}$, and $30\text{ cm}$ (an equilateral triangle). What is its perimeter?

• Step 1: Identify the side lengths. $a = 30\text{ cm}, \quad b = 30\text{ cm}, \quad c = 30\text{ cm}$
• Step 2: Write the formula. $P = a + b + c$
• Step 3: Add the values together. $P = 30\text{ cm} + 30\text{ cm} + 30\text{ cm} = 90\text{ cm}$
• Answer: The perimeter of the traffic sign is $90\text{ cm}$.

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4. Perimeter of Irregular Shapes

An irregular shape is a flat shape whose sides and angles are not all equal. For example, an L-shaped room floor or a compound shape made of joined rectangles. To find the perimeter, you must find the length of every single outer side and add them all together. If some side lengths are missing, you must calculate them using the lengths of parallel sides.

Worked Example:

Look at this L-shaped floor plan:

      6 m
  +---------+
  |         | 3 m
  |   +-----+
  | 4 m | 2 m
  |     |
  +-----+ 5 m
    2 m

• Step 1: Ensure all outer sides are known.
  • Top horizontal side = $6\text{ m}$
  • Right upper vertical side = $3\text{ m}$
  • Middle horizontal side = $4\text{ m}$
  • Right lower vertical side = $5\text{ m}$
  • Bottom horizontal side = $2\text{ m}$
  • Left vertical side = (The sum of the right vertical sides: $3\text{ m} + 5\text{ m} = 8\text{ m}$)
• Step 2: Add all outer sides together. $P = 6\text{ m} + 3\text{ m} + 4\text{ m} + 5\text{ m} + 2\text{ m} + 8\text{ m}$ $P = 28\text{ m}$
• Answer: The perimeter of the L-shaped floor is $28\text{ m}$.

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CONCEPT B: AREA (The Flat Surface Inside)

What is Area?

If perimeter is the fence around a yard, Area is the grass growing inside that fence.

Definition: Area is the amount of flat surface covered by a two-dimensional shape.

• Unit of Measurement: Area is measured in square units. This is because we count how many unit squares can fit inside the shape. We write them with a small superscript 2, such as square centimeters ($\text{cm}^2$) or square meters ($\text{m}^2$).

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1. Area of a Square

Since a square has equal sides, the space inside is calculated by multiplying its side length by itself.

• Formula: $\text{Area} = \text{Side} \times \text{Side}$ $\text{Area} = s^2 \quad (\text{or } A = L \times L)$

Worked Example:

A square ceramic floor tile has a side length of $30\text{ cm}$. What is the area of the tile?

• Step 1: Identify the side length. $\text{Side } (s) = 30\text{ cm}$
• Step 2: Write down the formula. $A = s \times s$
• Step 3: Multiply. $A = 30\text{ cm} \times 30\text{ cm} = 900\text{ cm}^2$
• Answer: The area of one floor tile is $900\text{ cm}^2$ (900 square centimeters).

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2. Area of a Rectangle

To find the area of a rectangle, we multiply its length by its breadth. This calculates how many unit squares fit in grid-like rows and columns.

• Formula: $\text{Area} = \text{Length} \times \text{Breadth}$ $\text{Area} = L \times B$

Worked Example:

A classroom teacher's table has a length of $1.5\text{ m}$ and a breadth of $1\text{ m}$. What is the surface area of the tabletop?

• Step 1: Identify the dimensions. $L = 1.5\text{ m}, \quad B = 1\text{ m}$
• Step 2: Use the formula. $A = L \times B$
• Step 3: Calculate. $A = 1.5\text{ m} \times 1\text{ m} = 1.5\text{ m}^2$
• Answer: The surface area of the tabletop is $1.5\text{ m}^2$.

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3. Area of a Triangle

To understand why the formula for the area of a triangle works, imagine a rectangle cut in half diagonally from corner to corner. You get two identical triangles! Therefore, the area of a triangle is exactly half of the area of a rectangle or parallelogram that has the same base and height.

• Key Terms:
  • Base ($b$): The bottom side on which the triangle stands.
  • Height ($h$): The vertical distance perpendicular ($90^\circ$) from the base to the opposite highest point.
• Formula: $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}$ $\text{Area} = \frac{b \times h}{2}$

Worked Example:

Find the area of a triangle with a base of $8\text{ cm}$ and a perpendicular height of $6\text{ cm}$.

• Step 1: Identify the measurements. $\text{Base } (b) = 8\text{ cm}, \quad \text{Height } (h) = 6\text{ cm}$
• Step 2: Write down the formula. $A = \frac{1}{2} \times b \times h$
• Step 3: Compute. $A = \frac{1}{2} \times 8\text{ cm} \times 6\text{ cm}$ $A = 4\text{ cm} \times 6\text{ cm} = 24\text{ cm}^2$
• Answer: The area of the triangle is $24\text{ cm}^2$.

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4. Area of Irregular Shapes (Grid-Square Method)

For completely irregular shapes with curved edges (like a leaf or a handprint), we estimate the area using a grid square method.

1. Place the irregular shape on a grid of $1\text{ cm} \times 1\text{ cm}$ square blocks.
2. Trace the outline.
3. Count all the fully completed squares inside the shape.
4. Count any square that is half-filled or more than half-filled as $1$ whole square.
5. Ignore any square that is less than half-filled.
6. Add the counted squares together to get an estimated area in $\text{cm}^2$.

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CONCEPT C: VOLUME (The Space Inside Solid Objects)

What is Volume?

While perimeter and area are about flat 2D surfaces, Volume is about 3D objects. If you have an empty lunchbox, its volume is the amount of food space inside it.

Definition: Volume is the amount of three-dimensional space occupied by a solid object, or the capacity of a container to hold a substance.

• Unit of Measurement: Volume is measured in cubic units because we are multiplying three spatial dimensions (Length $\times$ Breadth $\times$ Height). We write them with a small superscript 3, such as cubic centimeters ($\text{cm}^3$) or cubic meters ($\text{m}^3$). Litres ($\text{L}$) and milliliters ($\text{mL}$) are also used for liquids.

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1. Volume of a Cube (3D Square)

A cube is a 3D solid where all six faces are equal squares. This means its length, breadth, and height are exactly the same.

• Formula: $\text{Volume} = \text{Side} \times \text{Side} \times \text{Side}$ $\text{Volume} = s^3 \quad (\text{or } V = L \times L \times L)$

Worked Example:

A cubic rubik's cube toy has a side length of $6\text{ cm}$. What is its volume?

• Step 1: Identify side length. $\text{Side } (s) = 6\text{ cm}$
• Step 2: Write down the formula. $V = s \times s \times s$
• Step 3: Calculate. $V = 6\text{ cm} \times 6\text{ cm} \times 6\text{ cm}$ $V = 36 \times 6 = 216\text{ cm}^3$
• Answer: The volume of the rubik's cube is $216\text{ cm}^3$.

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2. Volume of a Cuboid / Rectangular Prism (3D Rectangle)

A cuboid is a solid box-shaped object. Its six faces are rectangular, meaning its length, breadth, and height can all be different.

• Formula: $\text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height}$ $\text{Volume} = L \times B \times H$

Worked Example:

A rectangular shipping carton used for importing electronics through Lagos port measures $50\text{ cm}$ long, $30\text{ cm}$ wide, and $20\text{ cm}$ high. Find its volume.

• Step 1: Identify dimensions. $L = 50\text{ cm}, \quad B = 30\text{ cm}, \quad H = 20\text{ cm}$
• Step 2: Write down formula. $V = L \times B \times H$
• Step 3: Calculate. $V = 50\text{ cm} \times 30\text{ cm} \times 20\text{ cm}$ $V = 1,500 \times 20 = 30,000\text{ cm}^3$
• Answer: The volume of the carton is $30,000\text{ cm}^3$.

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3. Volume of a Triangular Prism (3D Triangle)

A triangular prism is a 3D shape with two triangular bases at each end, joined by three rectangular sides. Think of a Toblerone chocolate box or a tent.

• Formula: $\text{Volume} = \text{Area of the Triangular Base} \times \text{Height (or Length) of the Prism}$ $\text{Volume} = \left(\frac{1}{2} \times \text{base} \times \text{height}\right) \times \text{Length of Prism}$

Worked Example:

A triangular prism has a triangular face with a base of $4\text{ cm}$ and a height of $3\text{ cm}$. The length of the entire prism is $10\text{ cm}$. What is its volume?

• Step 1: Calculate the area of the triangular face first. $\text{Area of Triangle} = \frac{1}{2} \times 4\text{ cm} \times 3\text{ cm} = 6\text{ cm}^2$
• Step 2: Multiply this area by the length of the prism. $\text{Volume} = 6\text{ cm}^2 \times 10\text{ cm} = 60\text{ cm}^3$
• Answer: The volume of the triangular prism is $60\text{ cm}^3$.

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SECTION 2: Real-World Examples (Nigerian Contexts)

Scenario A: Building a Goat Pen in Abeokuta

Baba Chidi wants to build a small rectangular outdoor pen for his goats.

• The Problem: He has a yard and needs to fence a space of $10\text{ meters}$ in length and $6\text{ meters}$ in width. He needs to buy wire fencing from the market.
• The Math: To buy the right amount of fence, he calculates the perimeter: $P = 2(L + B) = 2(10\text{ m} + 6\text{ m}) = 2(16\text{ m}) = 32\text{ meters}$ He walks into the local hardware shop confidently asking for exactly $32\text{ meters}$ of fencing wire.

Scenario B: Tiling a Parlour Floor in Enugu

Amina’s parents want to lay new tiles in their rectangular living room (parlour).

• The Problem: The parlour floor is $8\text{ meters}$ long and $5\text{ meters}$ wide. Each tile box covers $2\text{ m}^2$ of floor area. How many boxes of tiles do they need?
• The Math: First, find the area of the room: $A = L \times B = 8\text{ m} \times 5\text{ m} = 40\text{ m}^2$ Next, divide the total area by the coverage of one tile box: $\text{Number of boxes} = \frac{40\text{ m}^2}{2\text{ m}^2/\text{box}} = 20\text{ boxes}$ Amina helps her parents save money by calculating that they need exactly $20$ boxes, avoiding buying too much.

Scenario C: Supplying Water to a School in Kano

During a dry week, a primary school in Kano needs to buy water from a tanker.

• The Problem: The school's underground water storage tank is a cuboid shape, measuring $3\text{ meters}$ long, $2\text{ meters}$ wide, and $2\text{ meters}$ deep (height). How much water can it hold?
• The Math: Calculate the volume (capacity) of the tank: $V = L \times B \times H = 3\text{ m} \times 2\text{ m} \times 2\text{ m} = 12\text{ m}^3$ Since $1\text{ cubic meter } (\text{m}^3)$ is equal to $1,000\text{ litres}$ of water, the tank can hold: $12 \times 1,000 = 12,000\text{ litres of water}$

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SECTION 3: Practical Applications (Step-by-Step Guides)

Guide 1: Finding the Area of an Irregular Leaf

How can we measure the area of an irregular shape like a leaf plucked from a mango tree in the schoolyard?

Materials Needed:

• A mango leaf (or any leaf)
• A sheet of $1\text{ cm}$ square grid paper (or graph paper)
• A sharp pencil

Step-by-Step Procedure:

1. Place and Hold: Lay your grid paper flat on a table. Place the mango leaf flat in the center of the paper. Hold it still with one hand.
2. Trace: Use your pencil to trace carefully around the entire outer edge of the leaf. Remove the leaf.
3. Count Whole Squares ($\checkmark$): Look inside the boundary you drew. Identify and put a tick mark ($\checkmark$) inside every single square that is completely inside the leaf line. Write down this number (e.g., Whole Squares = $14$).
4. Count Half-or-More Squares ($\bullet$): Look at the squares along the edge. If more than half of a square is inside the boundary, mark it with a dot ($\bullet$) and count it as $1$ whole square. Write this down (e.g., Half-or-More Squares = $8$).
5. Discard Tiny Portions: If a square is barely touched by the leaf (less than half is inside), ignore it completely.
6. Calculate Total Area: $\text{Estimated Area} = \text{Whole Squares} + \text{Half-or-More Squares}$ $\text{Estimated Area} = 14 + 8 = 22\text{ cm}^2$

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SECTION 4: Suggested Home Projects (Project-Based Learning)

PROJECT 1: Become an Interior Designer

Objective: Apply perimeter and area concepts to map out a room in your house.

Materials Needed:

• A measuring tape or a long ruler ($30\text{ cm}$)
• An exercise book and a pencil
• Colored crayons

Step-by-Step Instructions:

1. Choose a Room: Select your bedroom or your family’s parlour/living room.
2. Draw a Rough Sketch: In your notebook, sketch a 2D floor shape of the room (it will likely be a rectangle or an L-shape).
3. Measure the Walls: With help from an adult or sibling, use the measuring tape to measure the length of each wall in meters. Write down the numbers next to each wall on your drawing.
4. Calculate Perimeter: Add the lengths of all walls to find how much skirting board runs around the base of the room.
5. Calculate Area: Use the area formula ($L \times B$) to find the total floor area. If the room is an irregular shape (like an L-shape), split it into two simpler rectangles, find the area of each, and add them together.
6. Decorate: Use crayons to color your floor plan. Write down the final Perimeter and Area calculations at the bottom of your project page.

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PROJECT 2: The Upcycled Cardboard Storage Box

Objective: Calculate the volume of a box and determine the surface area of wrapping paper needed to cover it.

Materials Needed:

• An empty rectangular cardboard box (like an empty noodle box or shoebox)
• A ruler
• Wrapping paper or old newspapers
• Scissors and glue

       +-------------------+
      /                   /|
     /    Shoebox        / | Height (H)
    +-------------------+  |
    |                   |  +
    |    Length (L)     | / Breadth (B)
    +-------------------+/

Step-by-Step Instructions:

1. Measure Dimensions: Measure the Length ($L$), Breadth ($B$), and Height ($H$) of your box in centimeters. Round the measurements to the nearest whole number.
2. Calculate Volume: Multiply $L \times B \times H$. This tells you how much space is inside your box for storage. Write down your volume in $\text{cm}^3$.
3. Find Surface Area (Paper Needed): To wrap the box, you must cover all 6 flat faces. Calculate:
   • Top & Bottom faces Area: $2 \times (L \times B)$
   • Front & Back faces Area: $2 \times (L \times H)$
   • Left & Right faces Area: $2 \times (B \times H)$
   • Total Area needed: Add these three results together.
4. Wrap It: Cut out wrapping paper based on your area calculations and glue it neatly around the box. You now have a beautiful storage box!

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SECTION 5: Life Skills Integration & Career Connections

Why do we spend time learning these mathematical operations? They are essential tools for many rewarding career paths!

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SECTION 6: Assessment Through Application

Put on your thinking cap! Solve these practical problems to test your understanding. Show all step-by-step workings.

Exercises

1. The Fenced Garden: A school yard has a triangular garden space with sides of length $7\text{ m}$, $9\text{ m}$, and $12\text{ m}$.

   • a) What is the total length of fencing required to secure the garden?
   • b) If fencing wire costs $\text{N}500$ per meter, how much will the school spend in total?

2. Tiling Chinedu's Shop: Chinedu has a square shop in Balogun market. The side length of the shop floor is $6\text{ meters}$.

   • a) Calculate the total floor area of Chinedu’s shop.
   • b) If he wants to use square floor tiles of side length $1\text{ meter}$, how many tiles will he buy?

3. The Water Tank: A family in Benin City has a rectangular overhead plastic water tank that measures $2\text{ m}$ long, $1.5\text{ m}$ wide, and $2\text{ m}$ high.

   • a) What is the volume of the water tank in $\text{m}^3$?
   • b) How many litres of water can it hold when full? (Hint: $1\text{ m}^3 = 1,000\text{ litres}$).

4. The Triangular Sign Post: A metal worker is cutting a triangular piece of sheet metal to make a warning road sign. The base of the triangle is $40\text{ cm}$ and the height is $30\text{ cm}$. Find the area of the metal plate.

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Step-by-Step Solutions (Teacher's Guide / Self-Correction)

1. Garden Fence Solution:

   • a) $\text{Perimeter} = a + b + c = 7\text{ m} + 9\text{ m} + 12\text{ m} = \mathbf{28\text{ m}}$.
   • b) $\text{Cost} = 28\text{ m} \times \text{N}500 = \mathbf{\text{N}14,000}$.

2. Tiling Shop Solution:

   • a) $\text{Area of Shop} = \text{Side} \times \text{Side} = 6\text{ m} \times 6\text{ m} = \mathbf{36\text{ m}^2}$.
   • b) $\text{Area of one tile} = 1\text{ m} \times 1\text{ m} = 1\text{ m}^2$. $\text{Tiles needed} = \frac{36\text{ m}^2}{1\text{ m}^2} = \mathbf{36\text{ tiles}}$

3. Water Tank Solution:

   • a) $\text{Volume} = L \times B \times H = 2\text{ m} \times 1.5\text{ m} \times 2\text{ m} = \mathbf{6\text{ m}^3}$.
   • b) $\text{Capacity in litres} = 6 \times 1,000 = \mathbf{6,000\text{ litres}}$.

4. Triangular Sign Solution:

   • $\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 40\text{ cm} \times 30\text{ cm}$
   • $\text{Area} = 20\text{ cm} \times 30\text{ cm} = \mathbf{600\text{ cm}^2}$.

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SECTION 7: Student Reflection Questions

Think about these questions and write down your answers in your homework journal:

1. If two rectangles have the exact same perimeter, do they also have to have the exact same area? Tip: Draw a rectangle of $6\text{ cm} \times 2\text{ cm}$ and another of $5\text{ cm} \times 3\text{ cm}$. Calculate both their perimeters and areas to see!
2. Why do we write the units of perimeter as $\text{cm}$, the units of area as $\text{cm}^2$, and the units of volume as $\text{cm}^3$? Explain this in your own words using what you know about dimensions.
3. Look around your home. Find three objects that have volume. Why is volume a useful measurement for those three objects?