Properties and Perimeters of Plane Shapes for JSS 1

Plane Shapes

Performance Objectives

By the end of the lesson, students should be able to:

1. state the similarities and differences between shapes;
2. find the perimeter of regular polygons;
3. find the area of regular polygons;
4. find the area of real plane objects by physical measurement.

Introduction to Plane Shapes

A plane shape is a flat, two-dimensional figure that lies entirely on a single flat surface.

Plane shapes have length and width but do not have thickness or depth. Examples of these shapes are found on flat surfaces such as walls, floors, paper sheets, and table tops.

Observe the diagram below to see the common plane shapes studied in this unit.

Properties of Plane Shapes

Each plane shape has specific properties that distinguish it from others. These properties include the number of sides, the number of vertices (corners), the relationship between the sides, and the angles between them.

The Square

A square is a four-sided plane shape with equal sides and equal angles.

The properties of a square include:

1. It has four sides, and all four sides are equal in length.
2. It has four interior angles, and each angle is a right angle measuring $90^\circ$.
3. The opposite sides are parallel to each other.
4. It has two diagonals of equal length that bisect each other at right angles ($90^\circ$).

The Rectangle

A rectangle is a four-sided plane shape with opposite sides equal and parallel, and all interior angles equal to $90^\circ$.

The properties of a rectangle include:

1. It has four sides, with opposite sides equal in length.
2. It has four interior angles, and each angle is a right angle measuring $90^\circ$.
3. The opposite sides are parallel to each other.
4. It has two diagonals of equal length that bisect each other.

The Triangle

A triangle is a three-sided plane shape with three interior angles.

The properties of a triangle include:

1. It has three straight sides.
2. It has three vertices and three interior angles.
3. The sum of the three interior angles is always $180^\circ$.

The Trapezium

A trapezium is a four-sided plane shape with only one pair of parallel sides.

The properties of a trapezium include:

1. It has four straight sides.
2. It has one pair of opposite sides that are parallel to each other.
3. The non-parallel sides may or may not be equal in length.

The Parallelogram

A parallelogram is a four-sided plane shape with two pairs of parallel and equal opposite sides.

The properties of a parallelogram include:

1. It has four sides, with opposite sides equal in length.
2. Both pairs of opposite sides are parallel to each other.
3. Opposite interior angles are equal in size.
4. The diagonals bisect each other, but they are not equal in length.

The Circle

A circle is a round plane shape bounded by a single curved line called the circumference, where every point on the line is equidistant from a fixed central point.

The properties of a circle include:

1. It has no straight sides and no corners (vertices).
2. It has a center point from which all points on the boundary are at an equal distance.
3. The distance from the center to any point on the boundary is called the radius ($r$).
4. The distance across the circle through the center is called the diameter ($d$), which is twice the length of the radius ($d = 2r$).

Similarities and Differences Between Plane Shapes

Comparing plane shapes helps in identifying their unique features and understanding how they relate to one another.

The table below shows the key similarities and differences among the six plane shapes:

Key comparisons from the table include:

1. Similarities between a Square and a Rectangle: Both shapes have four sides, opposite sides that are parallel, and four interior angles that are each $90^\circ$.
2. Differences between a Square and a Rectangle: A square must have all four sides equal in length, whereas a rectangle only requires opposite sides to be equal in length.
3. Similarities between a Parallelogram and a Rectangle: Both shapes have four sides, with opposite sides being equal and parallel.
4. Differences between a Parallelogram and a Rectangle: A rectangle must have four right angles ($90^\circ$), while a parallelogram does not require right angles; only its opposite angles must be equal.
5. Differences between a Trapezium and a Parallelogram: A trapezium has only one pair of parallel sides, while a parallelogram has two pairs of parallel sides.

Perimeter of Plane Shapes

Perimeter is the total distance around the outer boundary of a closed plane shape.

To find the perimeter of any shape with straight sides, the lengths of all its outer sides are added together. The standard units of measurement for perimeter are millimeters ($\text{mm}$), centimeters ($\text{cm}$), and meters ($\text{m}$).

Observe the diagram below to see how perimeter represents the boundary path of a shape.

Perimeter of a Square

Since a square has four equal sides, its perimeter is calculated by multiplying the length of one side by four.

The formula for the perimeter ($P$) of a square with side length $s$ is:

$$P = 4s$$

Worked Example 1

Calculate the perimeter of a square with a side length of $8\text{ cm}$.

Solution:

State the formula:

$$P = 4s$$

Substitute the given value into the formula:

$$P = 4 \times 8\text{ cm}$$
$$P = 32\text{ cm}$$

Worked Example 2

A square floor tile has a side length of $30\text{ cm}$. Find its perimeter.

Solution:

State the formula:

$$P = 4s$$

Substitute the given value into the formula:

$$P = 4 \times 30\text{ cm}$$
$$P = 120\text{ cm}$$

Perimeter of a Rectangle

A rectangle has two equal lengths and two equal widths. Its perimeter is calculated by adding the length and the width together, then multiplying the sum by two.

The formula for the perimeter ($P$) of a rectangle with length $l$ and width $w$ is:

$$P = 2(l + w)$$

Worked Example 1

Find the perimeter of a rectangle that is $12\text{ cm}$ long and $5\text{ cm}$ wide.

Solution:

State the formula:

$$P = 2(l + w)$$

Substitute the given values into the formula:

$$P = 2(12\text{ cm} + 5\text{ cm})$$

Add the values inside the bracket first:

$$P = 2(17\text{ cm})$$

Multiply by 2:

$$P = 34\text{ cm}$$

Worked Example 2

A rectangular school farm has a length of $45\text{ m}$ and a width of $20\text{ m}$. Calculate the perimeter of the farm.

Solution:

State the formula:

$$P = 2(l + w)$$

Substitute the given values into the formula:

$$P = 2(45\text{ m} + 20\text{ m})$$

Add the values inside the bracket first:

$$P = 2(65\text{ m})$$

Multiply by 2:

$$P = 130\text{ m}$$

Perimeter of a Triangle

The perimeter of a triangle is calculated by adding the lengths of its three sides together.

The formula for the perimeter ($P$) of a triangle with sides $a$, $b$, and $c$ is:

$$P = a + b + c$$

Worked Example 1

Find the perimeter of a triangle with side lengths of $6\text{ cm}$, $8\text{ cm}$, and $10\text{ cm}$.

Solution:

State the formula:

$$P = a + b + c$$

Substitute the given values into the formula:

$$P = 6\text{ cm} + 8\text{ cm} + 10\text{ cm}$$
$$P = 24\text{ cm}$$

Worked Example 2

An equilateral triangle has all three sides equal in length. Calculate the perimeter of an equilateral triangle with a side length of $15\text{ cm}$.

Solution:

Since all three sides are equal, $a = 15\text{ cm}$, $b = 15\text{ cm}$, and $c = 15\text{ cm}$.

State the formula:

$$P = a + b + c$$

Substitute the values:

$$P = 15\text{ cm} + 15\text{ cm} + 15\text{ cm}$$
$$P = 45\text{ cm}$$

Perimeter of a Trapezium

The perimeter of a trapezium is calculated by adding the lengths of all its four outer sides together.

The formula for the perimeter ($P$) of a trapezium with sides $a$, $b$, $c$, and $d$ is:

$$P = a + b + c + d$$

Worked Example 1

Find the perimeter of a trapezium with side lengths of $5\text{ cm}$, $7\text{ cm}$, $12\text{ cm}$, and $8\text{ cm}$.

Solution:

State the formula:

$$P = a + b + c + d$$

Substitute the given values into the formula:

$$P = 5\text{ cm} + 7\text{ cm} + 12\text{ cm} + 8\text{ cm}$$
$$P = 32\text{ cm}$$

Worked Example 2

Calculate the perimeter of the trapezium shown below, where the parallel sides are $14\text{ m}$ and $20\text{ m}$, and the non-parallel sides are $10\text{ m}$ and $12\text{ m}$.

Solution:

State the formula:

$$P = a + b + c + d$$

Substitute the given values into the formula:

$$P = 14\text{ m} + 20\text{ m} + 10\text{ m} + 12\text{ m}$$
$$P = 56\text{ m}$$

Perimeter of a Parallelogram

A parallelogram has opposite sides that are equal in length. Its perimeter is calculated in the same way as a rectangle, by adding the lengths of two adjacent sides and multiplying the sum by two.

The formula for the perimeter ($P$) of a parallelogram with adjacent side lengths $a$ and $b$ is:

$$P = 2(a + b)$$

Worked Example 1

Find the perimeter of a parallelogram with adjacent sides of length $9\text{ cm}$ and $6\text{ cm}$.

Solution:

State the formula:

$$P = 2(a + b)$$

Substitute the given values into the formula:

$$P = 2(9\text{ cm} + 6\text{ cm})$$

Add the values inside the bracket first:

$$P = 2(15\text{ cm})$$

Multiply by 2:

$$P = 30\text{ cm}$$

Worked Example 2

A parallelogram-shaped garden has side lengths of $25\text{ m}$ and $18\text{ m}$. Calculate its perimeter.

Solution:

State the formula:

$$P = 2(a + b)$$

Substitute the given values into the formula:

$$P = 2(25\text{ m} + 18\text{ m})$$

Add the values inside the bracket first:

$$P = 2(43\text{ m})$$

Multiply by 2:

$$P = 86\text{ m}$$

Perimeter of a Circle (Circumference)

The perimeter of a circle is called its circumference. It is calculated using the radius or the diameter of the circle, along with the mathematical constant pi ($\pi$), which is approximately equal to $\frac{22}{7}$ or $3.142$.

The formula for the circumference ($C$) of a circle with radius $r$ is:

$$C = 2\pi r$$

The formula for the circumference ($C$) of a circle with diameter $d$ is:

$$C = \pi d$$

Worked Example 1

Find the circumference of a circle with a radius of $7\text{ cm}$. (Take $\pi = \frac{22}{7}$)

Solution:

State the formula:

$$C = 2\pi r$$

Substitute the given values into the formula:

$$C = 2 \times \frac{22}{7} \times 7\text{ cm}$$

Simplify the expression by canceling out the $7$ in the denominator and numerator:

$$C = 2 \times 22 \times 1\text{ cm}$$
$$C = 44\text{ cm}$$

Worked Example 2

Calculate the circumference of a circular plate with a diameter of $20\text{ cm}$. (Take $\pi = 3.14$)

Solution:

State the formula using diameter:

$$C = \pi d$$

Substitute the given values into the formula:

$$C = 3.14 \times 20\text{ cm}$$
$$C = 62.8\text{ cm}$$

Area of Plane Shapes

Area is the measure of the total surface space enclosed within the boundary of a closed plane shape.

Area is measured in square units, such as square millimeters ($\text{mm}^2$), square centimeters ($\text{cm}^2$), and square meters ($\text{m}^2$).

Observe the diagram below to see how area represents the flat space inside a shape, measured in square units.

Area of a Square

The area of a square is calculated by multiplying the length of its side by itself.

The formula for the area ($A$) of a square with side length $s$ is:

$$A = s^2$$

Worked Example 1

Find the area of a square with a side length of $6\text{ cm}$.

Solution:

State the formula:

$$A = s^2$$

Substitute the given value into the formula:

$$A = 6\text{ cm} \times 6\text{ cm}$$
$$A = 36\text{ cm}^2$$

Worked Example 2

A square courtyard has a side length of $12\text{ m}$. Calculate its area.

Solution:

State the formula:

$$A = s^2$$

Substitute the given value into the formula:

$$A = 12\text{ m} \times 12\text{ m}$$
$$A = 144\text{ m}^2$$

Area of a Rectangle

The area of a rectangle is calculated by multiplying its length by its width.

The formula for the area ($A$) of a rectangle with length $l$ and width $w$ is:

$$A = l \times w$$

Worked Example 1

Calculate the area of a rectangle with a length of $10\text{ cm}$ and a width of $4\text{ cm}$.

Solution:

State the formula:

$$A = l \times w$$

Substitute the given values into the formula:

$$A = 10\text{ cm} \times 4\text{ cm}$$
$$A = 40\text{ cm}^2$$

Worked Example 2

A classroom floor is in the shape of a rectangle. It has a length of $8\text{ m}$ and a width of $6\text{ m}$. Find the area of the floor.

Solution:

State the formula:

$$A = l \times w$$

Substitute the given values into the formula:

$$A = 8\text{ m} \times 6\text{ m}$$
$$A = 48\text{ m}^2$$

Area of a Parallelogram

The area of a parallelogram is calculated by multiplying the length of its base by its perpendicular height.

The formula for the area ($A$) of a parallelogram with base $b$ and perpendicular height $h$ is:

$$A = b \times h$$

Worked Example 1

Find the area of a parallelogram with a base of $15\text{ cm}$ and a perpendicular height of $8\text{ cm}$.

Solution:

State the formula:

$$A = b \times h$$

Substitute the given values into the formula:

$$A = 15\text{ cm} \times 8\text{ cm}$$
$$A = 120\text{ cm}^2$$

Worked Example 2

A parallelogram-shaped metal sheet has a base of $1.2\text{ m}$ and a height of $0.5\text{ m}$. Calculate its area.

Solution:

State the formula:

$$A = b \times h$$

Substitute the given values into the formula:

$$A = 1.2\text{ m} \times 0.5\text{ m}$$
$$A = 0.6\text{ m}^2$$

Area of a Circle

The area of a circle is calculated by multiplying pi ($\pi$) by the square of the radius.

The formula for the area ($A$) of a circle with radius $r$ is:

$$A = \pi r^2$$

Worked Example 1

Find the area of a circle with a radius of $14\text{ cm}$. (Take $\pi = \frac{22}{7}$)

Solution:

State the formula:

$$A = \pi r^2$$

Substitute the given values into the formula:

$$A = \frac{22}{7} \times (14\text{ cm})^2$$
$$A = \frac{22}{7} \times 14\text{ cm} \times 14\text{ cm}$$

Simplify the expression by dividing $14$ by $7$:

$$A = 22 \times 2\text{ cm} \times 14\text{ cm}$$
$$A = 44 \times 14\text{ cm}^2$$
$$A = 616\text{ cm}^2$$

Worked Example 2

Calculate the area of a circular pond with a radius of $3.5\text{ m}$. (Take $\pi = \frac{22}{7}$)

Solution:

State the formula:

$$A = \pi r^2$$

Substitute the given values into the formula:

$$A = \frac{22}{7} \times (3.5\text{ m})^2$$
$$A = \frac{22}{7} \times 3.5\text{ m} \times 3.5\text{ m}$$

Simplify the expression by dividing $3.5$ by $7$ (which gives $0.5$):

$$A = 22 \times 0.5\text{ m} \times 3.5\text{ m}$$
$$A = 11 \times 3.5\text{ m}^2$$
$$A = 38.5\text{ m}^2$$

How to Measure and Calculate the Area of Real Plane Objects

Physical measurement involves using measuring instruments, such as rulers and measuring tapes, to find the dimensions of real-world objects. These dimensions are then used to calculate the perimeter and area of the objects.

Observe the image below to see how a measuring tape is used to find the dimensions of a real object.

The required materials for this activity are:

1. A standard metric ruler (for small objects)
2. A flexible measuring tape (for larger objects)
3. A notebook and a pen to record measurements
4. Real plane objects (such as a textbook cover, a classroom desk, or a square floor tile)

The procedure involves the following steps:

1. Identify the shape of the object to be measured. For example, a textbook cover is a rectangle, and a floor tile is usually a square.
2. Place the zero mark of the ruler or measuring tape at one corner of the object.
3. Extend the measuring tape along the edge of the object to the next corner to measure the length. Record this value in centimeters ($\text{cm}$) or meters ($\text{m}$).
4. Measure the adjacent edge to find the width of the object. Record this value.
5. Use the recorded measurements to calculate the perimeter and area of the object using the appropriate formulas.

Practical Example

A student measures the top surface of a classroom desk. The shape of the desk top is a rectangle. The measured length is $120\text{ cm}$ and the measured width is $50\text{ cm}$. Calculate the perimeter and the area of the desk top.

Solution:

First, calculate the perimeter of the desk top:

$$P = 2(l + w)$$
$$P = 2(120\text{ cm} + 50\text{ cm})$$
$$P = 2(170\text{ cm})$$
$$P = 340\text{ cm}$$

Next, calculate the area of the desk top:

$$A = l \times w$$
$$A = 120\text{ cm} \times 50\text{ cm}$$
$$A = 6000\text{ cm}^2$$

The perimeter of the desk top is $340\text{ cm}$ and the area is $6000\text{ cm}^2$.

Exercises

Multiple Choice Questions

Choose the correct option for each of the following questions:

1. Which of the following plane shapes has only one pair of parallel sides?
   1. Square
   2. Rectangle
   3. Trapezium
   4. Parallelogram
2. What is the perimeter of a square with a side length of $15\text{ cm}$?
   1. $30\text{ cm}$
   2. $60\text{ cm}$
   3. $90\text{ cm}$
   4. $225\text{ cm}$
3. A rectangle has a length of $14\text{ cm}$ and a width of $6\text{ cm}$. What is its area?
   1. $20\text{ cm}^2$
   2. $40\text{ cm}^2$
   3. $84\text{ cm}^2$
   4. $120\text{ cm}^2$
4. Find the circumference of a circle with a diameter of $14\text{ m}$. (Take $\pi = \frac{22}{7}$)
   1. $22\text{ m}$
   2. $44\text{ m}$
   3. $88\text{ m}$
   4. $154\text{ m}$
5. Which property is shared by both a square and a rectangle?
   1. All four sides are equal in length.
   2. The diagonals are perpendicular to each other.
   3. All four interior angles are $90^\circ$.
   4. There are no parallel sides.

Theory Questions

Answer the following questions showing all steps clearly:

1. State two similarities and two differences between a parallelogram and a rectangle.
2. A triangle has sides of length $7\text{ cm}$, $24\text{ cm}$, and $25\text{ cm}$. Calculate its perimeter.
3. A circle has a radius of $21\text{ cm}$. Calculate:
   1. Its circumference.
   2. Its area.

   (Take $\pi = \frac{22}{7}$)

4. A parallelogram has a base of $18\text{ cm}$ and a perpendicular height of $10\text{ cm}$. Find its area.
5. A student measured a square floor tile and found the side length to be $40\text{ cm}$. Calculate the perimeter and the area of the tile.