# Plane Shapes : Area of Regular and Irregular Shapes

**SUBJECT: MATHEMATICS**

**CLASS: BASIC FIVE / PRIMARY 5**

TERM : SECOND TERM

WEEK : WEEK 9

### TOPIC : Plane Shapes : Area of Regular and Irregular Shapes

- Meaning of Area
- Area of regular plane shapes like rectangle, triangle, square and circle
- Area of irregular plane shapes
- Area of right angled triangle
- Real life problems on Area

**Importance**

- Farmers use it to know the number of seedlings to plant on a piece of land
- Horticulturist use it to know what quantity of flowers to plant or carpet grasses to cultivate on a field
- Painters use it to calculate the number of paint buckets to paint a room

### Learning Objectives :

Pupils should be able to

- Find the area of Regular and Irregular Shapes
- Calculate the area of right angled triangle
- Solve real life problems on area of Regular, irregular and right angled triangle
- Relate area to real life problem and solve them

### Learning Activities :

Pupils in pairs

- Use classroom in unlocking the concept of area. They count the number of column in the class and multiply it with the number of rows, the resulting figures gives a rough estimate of area of the class
- Draw diagonal on a rectangular plane sheet top identify the right angle. Use a pair of scissor to cut out the triangle and use a ruler to measure the base and height. Then use the information to calculate the area

### Embedded Core Skills

- Critical thinking and problem solving skills
- Communication and Collaboration
- Student Leadership skills and Personal Development

### Learning Resources

- Flash cards
- Formula of perimeters of shapes
- Cardboards to cut to different shapes
- Textbooks and Workbook with examples on area of Regular, irregular and right angled triangle.

### Content

### Area of Regular Shapes

The area of a regular shape is the measure of the two-dimensional space inside the shape. The formula for finding the area of a regular shape depends on the shape in question.

For example:

- The area of a square is found by multiplying the length of one of its sides by itself (A = s^2)
- The area of a rectangle is found by multiplying the length by the width (A = l * w)
- The area of a triangle is found by multiplying the base by the height, and then dividing by 2 (A = (b * h) / 2)
- The area of a circle is found by multiplying pi (approximately 3.14) by the radius squared (A = πr^2).

It’s important to note that the formulas above are for regular shapes, where all the sides or angles are equal. The area of an irregular shape can be found using different methods, such as breaking the shape into smaller regular shapes and finding the area of each, or using calculus.

- The area of a square football field is measured in square meters, where the length of one side of the field is 100m. The formula for finding the area of a square is A = s^2, so in this case, the area of the field would be A = 100m x 100m = 10,000 square meters.
- The area of a rectangular swimming pool is measured in square meters, where the length is 25m and the width is 10m. The formula for finding the area of a rectangle is A = l x w, so in this case, the area of the pool would be A = 25m x 10m = 250 square meters.
- The area of a circular lake is measured in square kilometers, where the radius of the lake is 2km. The formula for finding the area of a circle is A = πr^2, so in this case, the area of the lake would be A = 3.14 x (2km)^2 = 12.56 square kilometers.
- The area of a triangular park is measured in square meters, where the base of the triangle is 100m and the height is 50m. The formula for finding the area of a triangle is A = (b x h)/2, so in this case, the area of the park would be A = (100m x 50m)/2 = 2500 square meters.
- The area of a hexagonal flower garden is measured in square meters, where the length of one side of the hexagon is 5m. The formula for finding the area of a hexagon is A = (3√3/2) x a^2, so in this case, the area of the garden would be A = (3√3/2) x (5m)^2 = 64.95 square meters.
- The area of a octagonal playground is measured in square meters, where the length of one side of the octagon is 8m. The formula for finding the area of an octagon is A = (2 + √2) x a^2, so in this case, the area of the playground would be A = (2 + √2) x (8m)^2 = 332.79 square meters.

Shape | Formula for Area | Example (unit: m^2) |
---|---|---|

Square | A = s^2 | A = 100m x 100m = 10,000 |

Rectangle | A = l x w | A = 25m x 10m = 250 |

Triangle | A = (b x h)/2 | A = (100m x 50m)/2 = 2500 |

Circle | A = πr^2 | A = 3.14 x (2m)^2 = 12.56 |

Hexagon | A = (3√3/2) x a^2 | A = (3√3/2) x (5m)^2 = 64.95 |

Octagon | A = (2 + √2) x a^2 | A = (2 + √2) x (8m)^2 = 332.79 |

**Evaluation**

Q 1 – If the length and width of a rectangle are 12 feet and 11 feet, then find its area.

A – 136 square feet

B – 132 square feet

C – 130 square feet

D – 112 square feet

Q 2 – If the side of a square is 7 m, then find its area.

A – 49 square meter

B – 48 square meter

C – 45 square meter

D – 42 square meter

Q 3 – If the length and width of a rectangle are 10 m and 8 m, then find its area.

A – 80 square cm

B – 86 square meter

C – 80 square meter

D – 84 square meter

Q 4 – If the side of a square is 11 cm, then find its area.

A – 110 square cm

B – 132 square cm

C – 120 square cm

D – 121 square cm

Q 5 – If the length and width of a rectangle are 8 cm and 6 cm, then find its area.

A – 48 square cm

B – 48 square mm

C – 46 square cm

D – 42 square cm

Q 6 – If the side of a square is 14 m, then find its area.

A – 192 square meter

B – 198 square meter

C – 196 square meter

D – 196 square units

Q 7 – If the side of a square is 19 inches, then find its area.

A – 361 square feet

B – 361 square inches

C – 360 square inches

D – 324 square inches

Q 8 – If the length and width of a rectangle are 15 cm and 13 cm, then find its area.

A – 192 square cm

B – 195 square m

C – 196 square cm

D – 195 square cm

Q 9 – If the length and width of a rectangle are 18 mm and 16 mm, then find its area.

A – 288 square mm

B – 289 square mm

C – 280 square mm

D – 268 square mm

Q 10 – If the side of a square is 23 feet, then find its area.

A – 528 square feet

B – 529 square feet

C – 525 square feet

D – 576 square feet

Q 11 – If the side of a square is 6 yards, then find its area.

A – 36 square yards

B – 34 square yards

C – 32 square yards

D – 30 square yards

Q 12 – If the length and width of a rectangle are 7 inches and 8 inches, then find its area.

A – 52 square inches

B – 56 square inches

C – 54 square inches

D – 50 square inches

Q 13 – If the radius of a circle is 5 cm, then find its area.

A – 25π square cm

B – 20π square cm

C – 19π square cm

D – 22π square cm

Q 14 – If the base of a triangle is 9 cm and the height is 6 cm, then find its area.

A – 27 square cm

B – 24 square cm

C – 21 square cm

D – 23 square cm

Q 15 – If the side of a square is 8 cm, then find its area.

A – 64 square cm

B – 68 square cm

C – 62 square cm

D – 66 square cm

Q 16 – If the side of a square is 10 inches, then find its area.

A – 100 square inches

B – 120 square inches

C – 110 square inches

D – 115 square inches

Q 17 – If the length and width of a rectangle are 12 m and 15 m, then find its area.

A – 180 square m

B – 170 square m

C – 165 square m

D – 175 square m

Q 18 – If the radius of a circle is 4 feet, then find its area.

A – 16π square feet

B – 12π square feet

C – 14π square feet

D – 15π square feet

Q 19 – If the base of a triangle is 7 cm and the height is 8 cm, then find its area.

A – 28 square cm

B – 25 square cm

C – 24 square cm

D – 26 square cm

Q 20 – If the side of a square is 9 cm, then find its area.

A – 81 square cm

B – 83 square cm

C – 82 square cm

D – 80 square cm

### Area of Irregular Shapes

The area of an irregular shape is the measure of the two-dimensional space inside the shape. Unlike regular shapes, where the formula for finding the area is fixed and based on the shape’s dimensions, irregular shapes can have more complex shapes and therefore the calculation of their area can be more difficult. There are several methods that can be used to find the area of an irregular shape:

- Break the shape into smaller regular shapes and find the area of each one. Then add all the areas together to find the total area of the irregular shape.
- Use geometric shapes like triangles, rectangles, and circles to approximate the area of the irregular shape.
- Use calculus by finding the definite integral of the function that describes the shape’s boundary.
- Use the method of Quadrature which is a method of approximating the area of an irregular shape by dividing it into a large number of small shapes and summing their areas.
- Use image processing to convert the image of the irregular shape into a set of points, and then use numerical integration to estimate the area of the irregular shape.

- The area of a irregular shaped lake can be found by using the method of breaking the shape into smaller regular shapes and finding the area of each one. The lake has a circular section with a radius of 2 km and a triangular section with a base of 3km and a height of 4km. The area of the circular section is A = πr^2 = 3.14 x (2km)^2 = 12.56 square kilometers and the area of the triangular section is A = (b * h) / 2 = (3km * 4km) / 2 = 6 square kilometers. Therefore, the total area of the lake is 12.56 + 6 = 18.56 square kilometers.
- The area of an irregular shaped field can be found by using the method of geometric shapes to approximate the area. The field can be approximated as a rectangle with a length of 3km and a width of 2km and a triangle with a base of 1km and a height of 1km. The area of the rectangle is A = l
*w = 3km * 2km = 6 square kilometers and the area of the triangle is A = (b*h)/2 = (1km*1km)/2 = 0.5 square kilometers. Therefore, the total area of the field is 6 + 0.5 = 6.5 square kilometers. - The area of an irregular shaped building can be found by using image processing to convert the image of the building into a set of points, and then use numerical integration to estimate the area of the building. Based on the image the building has an area of approximately 800 square meters
- The area of an irregular shaped terrain can be found by using the method of Quadrature, which is a method of approximating the area by dividing it into a large number of small shapes and summing their areas. For example, the terrain can be divided into rectangles, triangles and circles and their areas can be calculated using their respective formulas and then added together to find the total area of the terrain.
- The area of an irregular shaped garden can be found by using calculus by finding the definite integral of the function that describes the garden’s boundary. For example, if the garden’s boundary is defined by a mathematical function, we can use definite integral to find its area.
- The area of an irregular shaped pool can be found by using a combination of methods. For example, if the pool has a circular section and a rectangular section, the area of the circular section can be found using the formula for a circle, and the area of the rectangular section can be found using the formula for a rectangle. Then we can add the area of both sections to find the total area of the pool.

Shape | Method used | Formula or Approximation | Example (unit: m^2) |
---|---|---|---|

Irregular shaped lake | Breaking into regular shapes | Add areas of each section | 18.56 square kilometers |

Irregular shaped field | Geometric shapes | rectangle: lw, triangle: (bh)/2 |
6.5 square kilometers |

Irregular shaped building | Image processing and numerical integration | – | 800 square meters |

Irregular shaped terrain | Quadrature | – | – |

Irregular shaped garden | Calculus | Definite integral of the function | – |

Irregular shaped pool | Combination of methods | Circle: πr^2, rectangle: l*w | – |

Please note that the table above is just an example, you can add or remove shapes and their methods as you desire. Also, you can change the unit of measurement from m^2 to any other unit you want. It is important to note that finding the area of irregular shapes is not always easy and many times it requires a combination of different methods and it is also important to mention that the accuracy of the result will depend on the method used and the level of precision required.

### Area of Right-angle triangle

The area of a right-angled triangle can be found using the formula: A = (base x height) / 2. In this formula, the base refers to one of the legs of the right-angled triangle and the height refers to the other leg. To find the area of a right-angled triangle, we need to know the lengths of both legs.

For example, if the base of a right-angled triangle is 5 meters and the height is 8 meters, the area of the triangle would be:

A = (5 x 8) / 2 = 20 square meters

Another example, if the base of a right-angled triangle is 3 feet and the height is 4 feet, the area of the triangle would be:

A = (3 x 4) / 2 = 6 square feet

It’s important to note that the area of a right-angled triangle is always half of the product of the base and the height, regardless of the unit of measurement.

Also, In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and it’s the longest side of the triangle and the two other sides are known as legs or catheti and one of them is the base and the other is the height and the area of the right angled triangle is always half of the product of the base and the height.

### Pythagorean Theory

Pythagoras theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

We can use this theorem to find the area of a right-angled triangle by first finding the length of the hypotenuse, and then using that value in the area formula A = (base x height) / 2.

- For example, if we know that one leg of a right-angled triangle is 6 cm and the other leg is 8 cm, we can use Pythagoras theorem to find the length of the hypotenuse. We can use the formula c^2 = a^2 + b^2, where c is the length of the hypotenuse. c^2 = 6^2 + 8^2 = 36 + 64 = 100. The length of the hypotenuse is √100 = 10 cm. Now that we know the hypotenuse we can use the formula A = (6 x 8) / 2 = 24 cm^2 to find the area of the right-angled triangle.
- Another example, if we know that one leg of a right-angled triangle is 3 feet and the other leg is 4 feet, we can use Pythagoras theorem to find the length of the hypotenuse. We can use the formula c^2 = a^2 + b^2, where c is the length of the hypotenuse. c^2 = 3^2 + 4^2 = 9 + 16 = 25. The length of the hypotenuse is √25 = 5 feet. Now that we know the hypotenuse we can use the formula A = (3 x 4) / 2 = 6 ft^2 to find the area of the right-angled triangle.
- If we know that one leg of a right-angled triangle is 4 m and the other leg is 5 m, we can use Pythagoras theorem to find the length of the hypotenuse. We can use the formula c^2 = a^2 + b^2, where c is the length of the hypotenuse. c^2 = 4^2 + 5^2 = 16 + 25 = 41. The length of the hypotenuse is √41 = 6.4 m. Now that we know the hypotenuse we can use the formula A = (4 x 5) / 2 = 10 m^2 to find the area of the right-angled triangle.
- If we know that one leg of a right-angled triangle is 5 cm and the other leg is 12 cm, we can use Pythagoras theorem to find the length of the hypotenuse. We can use the formula c^2 = a^2 + b^2, where c is the length of the hypotenuse. c^2 = 5^2 + 12^2 = 25 + 144 = 169. The length of the hypotenuse is √169 = 13 cm. Now that we know the hypotenuse we can use the formula A = (5 x 12) / 2 = 30 cm^2 to find the area of the right-angled triangle.
- If we know that one leg of a right-angled triangle is 8 inches and the other leg is 15 inches, we can use Pythagoras theorem to find the length of the hypotenuse. We can use the formula

**Evaluation**

Q 1 – What is the area of a right-angled triangle with legs measuring 6 cm and 8 cm?

A – 12 cm^2

B – 24 cm^2

C – 36 cm^2

D – 48 cm^2

Q 2 – What is the area of a square with side length of 7 m?

A – 49 square meter

B – 48 square meter

C – 45 square meter

D – 42 square meter

Q 3 – What is the area of a rectangle with length of 10 m and width of 8 m?

A – 80 square cm

B – 86 square meter

C – 80 square meter

D – 84 square meter

Q 4 – What is the area of an irregular shape that can be approximated as a circle with a radius of 5 cm and a triangle with base of 3 cm and a height of 4 cm?

A – 32.5 square cm

B – 35 square cm

C – 37.5 square cm

D – 40 square cm

Q 5 – What is the area of a right-angled triangle with legs measuring 3 feet and 4 feet?

A – 6 square feet

B – 8 square feet

C – 10 square feet

D – 12 square feet

Q 6 – What is the area of a regular hexagon with side length of 6 cm?

A – 63 square cm

B – 54 square cm

C – 72 square cm

D – 81 square cm

Q 7 – What is the area of an irregular shape that can be approximated as a rectangle with length of 5 m and width of 7 m and a triangle with base of 4 m and a height of 6 m?

A – 35 square meters

B – 42 square meters

C – 49 square meters

D – 56 square meters

Q 8 – What is the area of a right-angled triangle with legs measuring 7 cm and 24 cm?

A – 84 square cm

B – 98 square cm

C – 112 square cm

D – 126 square cm

Q 9 – What is the area of an irregular shape that can be approximated as a circle with a radius of 8 cm and a square with side length of 10 cm?

A – 201 square cm

B – 212 square cm

C – 224 square cm

D – 236 square cm

Q 10 – What is the area of a right-angled triangle with legs measuring 5 inches and 12 inches?

A – 30 square inches

B – 35 square inches

C – 40 square inches

D – 45 square inches

**Lesson Presentation**

Previous Lesson : The subject teacher revises the previous lesson which was Plane Shapes : Perimeter of Regular and Irregular Shapes

**Step 1**: Review the concept of area and how it is measured in square units. Introduce the concept of regular shapes and their area formulas (e.g. square: side^2, rectangle: length x width). Introduce the concept of irregular shapes and the methods used to approximate their area. Introduce the concept of right-angled triangles and the Pythagorean Theorem.

**Step 2**: Provide examples of regular shapes and have students calculate their areas using the appropriate formulas. Then, provide examples of irregular shapes and have students approximate their areas using the methods discussed earlier. Finally, provide examples of right-angled triangles and have students calculate their areas using the Pythagorean Theorem.

**Step** 3: Provide students with worksheets that include a variety of regular, irregular, and right-angled triangles. Have students work in pairs or small groups to solve the problems on the worksheet. Monitor students’ progress and provide feedback as necessary.

**Step 4**: Provide students with additional practice problems to complete as homework. Encourage students to apply the concepts and methods learned in the lesson to real-world scenarios.

**Conclusion**: Review the main concepts of the lesson and provide an opportunity for students to share their understanding. Summarize the key points of the lesson and provide a preview of the next lesson.

**Assessment:**

- Formative assessment can be done through observation during the independent practice and class discussion.
- Summative assessment can be done through a quiz or a test