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Plane shapes in mathematics are any closed, flat, 2-dimensional shapes. A plane shape has length and breadth, but no thickness. Plane shapes have properties including sides, corners, and faces.
Many plane shapes are polygons, or any 2-dimensional shape with straight sides or lines that is closed and has no open sides. Polygons are plain figures or shapes made up of lines. Another type of plane shape is known as a quadrilateral, or a 2-dimensional shape with 4 straight sides that is closed and has no open sides.
The Circle
The circle is a shape that can be made by tracing a curve that is always the same distance from a point that we call the center. The distance around a circle is called the circumference of the circle.
The Triangle
The triangle is a shape that is formed by 3 straight lines that are called sides. There are different ways of classifying triangles, according to their sides or angles.
- According to their angles:
- Right triangle: the largest of the 3 angles is a right angle.
- Acute Triangle: the largest of the 3 angles is an acute angle (less than 90 degrees).
- Obtuse Triangle: the largest of the 3 angles is an obtuse angle (more than 90 degrees).
- According to their sides:
- Equilateral Triangle: all 3 sides are the same length.
- Isosceles Triangle: it has 2 (or more) sides that are of equal length. (An equilateral triangle is also isosceles.)
- Scalene Triangle: no 2 sides are of equal measure.
The Rectangle
The rectangle is a shape that has 4 sides. A rectangle has two equal sides of one length and two equal sides of a different length. A rectangle is like a stretched square. The distinguishing characteristic of a rectangle is that all 4 angles measure 90 degrees.
The Square
A square has four sides, but not just any four sides. A square’s four sides are all the same length. A square with one-inch sides is smaller than a square with three-inch sides because one is less than three. A square also has four corners. The square is a type of rectangle, but also a type of rhombus. It has characteristics of both of these. That is to say, all 4 angles are right angles, and all 4 sides are equal in length.
The Trapezium
The trapezium also has 4 sides. It has two sides that are parallel but the other 2 are not.
The Rhombus
The rhombus is a shape formed by 4 straight lines. Its 4 sides measure the same length but, unlike the rectangle, any of all 4 angles measure 90 degrees.
The Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle. A Parallelogram is a flat shape with opposite sides parallel and equal in length.
Definition of a Cube: A cube is a special type of cuboid whose length, breadth and height are equal. Each face of the cube is a square and all angles are right angles. Cube is thus a special case of a cuboid. It is also called a square prism.
Properties of a Cube:
- It has six faces
- All the faces are congruent
- Each face is parallel to the one opposite to it
- Has 12 edges and 8 vertices
Cuboid: A cuboid is a rectangular prism with six rectangular faces and all angles are right angles. It can also be called a square prism if at least two of the lengths are equal.
Properties of a cuboid:
- It has 6 faces
- Each face is congruent and parallel to the one opposite to it.
- It has 12 edges and 8 vertices
PERIMETERS OF REGULAR AND IRREGULAR SHAPES
Perimeter simply means addition of all the sides of a given plane shape. Plane shapes can be classified into two types:
- Regular shapes.
- Irregular shapes.
PERIMETERS OF REGULAR PLANE SHAPES
The regular plane shapes are as follows:
- Square
- Rectangle
- Triangle
- Circle
- Kite
- Trapezium
- Rhombus
- Parallelogram
Example 1
Find the perimeter of the rectangle below:
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Solution
Perimeter = sum of all sides OR 2L + 2B
The shape has four sides. 2(7) + 2(5)
Perimeter = 7 + 7 + 5 + 5 14 + 10
Perimeter = 24cm 24cm
Example 2
Find the perimeter of the square below:
|
Solution
A square has four sides that are all equal
Perimeter = sum of all sides OR 4 Lenght
Perimeter = 5 + 5 + 5 + 5 4 5
Perimeters = 20cm 20cm
Example 3
Find the perimeter of the shape below:
Solution
The shape (a trapezium) has four sides.
Perimeter = sum of all sides
Perimeter = 80 + 50 + 100 + 110
Perimeter = 340m
PERIMETER OF IRREGULAR PLANE SHAPES
Example 1. Find the perimeter of the shape below:
Solution
The shape has 5 sides.
Perimeter = sum of all sides
Perimeter = 5 + 4 + 4 + 4 + 3
Perimeter = 20in
Example 2. Find the perimeter of the shape below:
Solution
The shape has six sides. First find the missing sides
There are 3 missing sides, the first one is 5m, the second one is 7m and the third one is ( 5 + 11 = 16m )
Perimeters = (7 +5) + 5 + 5 + 11 + 7 + 16
Perimeters = 56m
AREAS OF PLANE SHAPES
Area is measured in square units. The area of regular plane shapes is given in the table below;
Examples: find the area of the rectangle below:
Solution
Area of rectangle = length bredth
Area of rectangle = 9 4
Area of rectangle = 36m2
Example 2: find the area of the shape below:
Solution
The shape above is a trapezium.
Area of a trapezium =
Area of trapezium = (6 + 8) 4
Area of trapezium = 14 4
Area of trapezium =
Area of trapezium =
Area of trapezium = 28
Example 3: find the area of the shape below:
Solution
The shape is a square.
Area of a square = length length
Area of square = 8 8
Area of square = 64cm2
Examples 4: find the area of the shape below:
Solution
The shape is a circle. = 3.14 or 22/7
Area of a circle =
Area of circle = 3.14 2 2
Area of circle = 12.56cm2
Example 5: find the area of the shape below:
Solution
The shape is a triangle.
Area of a triangle = base of triangle height of triangle
Area of triangle = 11 10
Area of triangle =
Area of triangle =
Area of triangle = 55cm2
AREA OF IRREGULAR PLANE SHAPES
When finding the areas of irregular plane shapes, split the irregular plane shape in regular plane shapes. Find their areas and sum them all together.
Example: find the area of the shape below:
Solution
The shape can be split into two regular plane shapes A and B
A = a rectangle with length 8m and breadth 4m
B = a square with length 10m
Area of A = 8 4
Area of A = 32m2
Area of B = 10 10
Area of B = 100m2
Area of the irregular plane shape = 32 + 100
Area of the irregular plane shape = 132m2
AREA OF SHADED PORTIONS
Example: find the area of the shaded portion in the figure below:
Solution
The shaded portion is between a rectangle (the bigger shape) and a square (the smaller shape)
Area of the rectangle = length breadth
Area of rectangle = 20
Area of rectangle = 300cm2
Area of the square = length length
Area of square = 8 8
Area of square = 64cm2
Area of shaded portion = area of rectangle – area of square
Area of shaded portion = 300 – 64
Area of shaded portion = 236cm2
Area of Complex figures
A complex figure is made up of two or more shapes. To find the area of a complex figure, break the figure into smaller parts.
Example 6: Find the area of the swimming pool at Dew’s hotel.
Solution:
Step 1: Break up the figure into smaller parts. Look for rectangles and squares.
Step 2: Find the area of each part.
Square
Area = side x side
A = 10m x 10m
A = 100 square meters.
Rectangles
A = length x width
A = 12m x 6m
A = 72 square meters
Step 3: Add the areas.
The area is 72 + 100 = 172 square meters.
Quiz
- Find the perimeter of the following shapes:
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