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#### Perimeter and Area of Plane Shapes Examples of Plane Shapes

##### PROPERTIES OF SOME PLANE SHAPES
• Rectangle: • Every interior angle of a rectangle is equal to 90°, i.e. ∠A = ∠B = ∠C = ∠D = 90°
• The sum of the interior angles is equal to 360°.
• The opposite sides are equal and parallel to one another, i.e. AB = CD & AC = BD.
• The diagonals are congruent.
• The diagonals bisect each other, i.e AO = OD & BO = OC.
• The diagonals bisect each other at different angles. One of them is an acute angle, while the other is an obtuse angle.
• Any diagonal of a rectangle is a diameter of its circumcircle.
• Square: • A square has 4 vertices and 4 sides.
• Each of the interior angles is equal 90°, i.e. ∠A = ∠B = ∠C = ∠D = 90°.
• The sum of the interior angles is equal to 360°.
• Each side of a square is equal, i.e. AB = BD = CD = AC
• The opposite sides of a square are parallel to each other.
• The diagonals of a square are equal.
• The diagonals of the square bisect each other at 90°.
• The diagonal of a square divides it into two similar isosceles triangles.
• Circle: • Circles are said to be congruent if they have equal radii.
• Equal chords and equal circles have equal circumference.
• The diameter of a circle is the longest chord of a circle.
• The radius drawn perpendicular to the chord bisects the chord.
• Circles having different radii are similar.
• A circle can circumscribe a rectangle, trapezium, triangle, square, kite.
• A circle can be inscribed inside a square, triangle, and kite.
• The chords that are equidistant from the center are equal in length.
• The distance from the center of the circle to the longest chord (diameter) is zero.
• The perpendicular distance from the center of the circle decreases when the length of the chord increases.
• If the tangents are drawn at the end of the diameter, they are parallel to each other.
• An isosceles triangle is formed when the radii joining the ends of a chord to the center of a circle.
• Kite: • Two disjoint pairs of consecutive sides are congruent by definition, i.e. AB = AD and BC = CD.
• The diagonals are perpendicular; they meet at right angles.
• One diagonal (segment AC) is the perpendicular bisector of the other diagonal (segment BD)
• The diagonal AC bisects a pair of opposite angles (angle A and angle C).
• The opposite angles at the endpoints of the cross diagonal are congruent (angle D and angle B)
##### PERIMETER OF PLANE SHAPES
• The perimeter of a shape is a measure of the distance round the boundary or edge of the shape.
• With the usual lettering, the perimeter of a rectangle is 2(L + B), square is 4L, circle is 2πr or πd.
• The perimeter of a circle is called the circumference.
##### AREA OF PLANE SHAPES
• The area of a shape is a measure of its surface.
• Area is measured in units such as cm² or m², which are derived from the SI units of length.
• The areas of common plane shapes are as follows:
• Rectangle = L × B
• Square = L²
• Parallelogram = Base × Height
• Triangle = ½ × Base × Height
• Trapezium = ½ h(a + b)
• Circle = πr²
###### Examples
1. Calculate the perimeter of a football field which measures 80 m by 50 m.
• Perimeter of field = 2(L + B)
•                               = 2(80 + 50)
•                               = 2 × 130
• ∴ The perimeter of the field = 260 m
2. A rectangle has a perimeter of 74 m. Find
1. the length of the rectangle if its breadth is 17 m.
• Perimeter of a rectangle = 2(L + B)
• Perimeter = 74 m,
• ∴ 74 = 2(L + 17)
• 74/2 = L + 17
• 37 = L + 17
• L = 37 – 17
• ∴ Length of the rectangle = 20 m.
2. the breadth of the rectangle if its length is 25 m.
• Perimeter of a rectangle = 2(L + B)
• Perimeter = 74 m,
• Length = 25 m,
• ∴ 74 = 2(25 + B)
• 74/2 = 25 + B
• 37 = 25 + B
• B = 37 – 25
• ∴ Breadth of the rectangle = 12 m.
3. Calculate the perimeter of a square of side 12.3 cm.
• Perimeter of a square = 4L,
• L = 12.3 cm,
• ∴ Perimeter = 4 × 12.3
• ∴ Perimeter of the square = 49.2 cm.
4. A square lawn has a perimeter of 56 m. Find the length of the side of the lawn.
• Perimeter of a square = 4L,
• Perimeter = 56 m,
• ∴ Length of the side = 56/4
• ∴ Length of the side of the lawn = 14 m.
5. Calculate the circumference of a circle of radius 31/2 m. (Let π = 31/7).
• Circumference of a circle = 2πr,
• R = 31/2 ,
• ∴ Circumference of a circle = 2 × 31/7 × 31/2
•                                                = 2 × 22/7 × 7/2
•                                                = 2 × 11
• ∴ Circumference of a circle = 22 m
6. A bicycle wheel has a diameter of 65 cm. During a journey, the wheel makes 1000 complete revolutions. How many meters does the bicycle travel? (Let π = 3.14)
• Circumference of a circle = πd,
• Distance traveled in one revolution = 3.14 × 65 cm
• Distance traveled in 1000 revolutions = (3.14 × 65 × 1000 m)/100
• ∴ Distance traveled in 1000 revolutions = 31.4 × 65 m
• ∴ Distance traveled in 1000 revolutions = 2041 m
7. Calculate the area of a rectangle 6 cm by 3.5 cm.
• Area of a rectangle = L × B
• L = 6 cm
• B = 3.5 cm
• ∴ The area of the rectangle = 6 × 3.5
• ∴ The area of the rectangle = 21 cm²
8. The area of a rectangle is 224 cm². If its length is 16 cm, calculate the breadth.
• Area of a rectangle = L × B
• Area = 224 cm²
• L = 16 cm
• ∴ 224 = 16 × B
• B = 224/16
• ∴ Breadth of the rectangle = 14 cm²
9. The area of a square plot is 144 m². Calculate the length of a side of the plot.
• Area of a square = L²
• Area = 144 m²
• ∴ 144 = L²
• ∴ L = √144
• ∴ L = 12 m
10. An assembly area is in the shape of a 30 m by 30 m square. Part of the area is a concrete rectangle 25 m by 5m, the rest is grass. Calculate the area of the grass. • Area of assembly area = L²
•                                          = 30²
• ∴ Area of assembly area = 900 m²
• Area of concrete = L × B
•                                = 25 × 5
• ∴ Area of concrete = 125 m²
• ∴ Area of grass = 900 – 125
• ∴ Area of grass = 775 m²
11. Calculate the area of a parallelogram if its base is 9.2 cm and its height is 6 cm.
• Area of parallelogram = base × height
•                                            = 9.2 × 6
• ∴ Area of the Parallelogram = 55.2 cm²
12. In the figure below, the base of the parallelogram is 6 cm and its height is 4 cm. Calculate the area of the parallelogram. If the length of the other side of the parallelogram is 8 cm, calculate the corresponding height, h. • Area of parallelogram = 6 × 4
• ∴ The area of parallelogram = 24 cm²
• Also, the area of the parallelogram = 8 × h
• ∴ 24 = 8 × h
• ∴ h = 24/8
• ∴ The height = 3 cm
13. Calculate the area of the triangle shown in the figure below. • Area of a triangle = ½ × B × H
•                                      = ½ × 9 × 12
• ∴ The area of the triangle = 54 cm²
14. In the diagram below, the diagonal AC divides the trapezium into two triangles. The height of each triangle is 8 cm. Calculate the area of the trapezium. • Area of Δ ACB = ½ × 13 × 8 = 52 cm²
• Area of Δ ACB = ½ × 6 × 8 = 24 cm²
• Area of trapezium = Area of Δ ACB + Area of Δ ACB
•                                      = 52 + 24
• ∴ The area of the trapezium = 76 cm²
15. Find the area of a circle of radius 31/2 meters. (Let π = 22/7)
• Area of a circle = πr²
•                              = 22/7 × (31/2
•                              = 22/7 × 7/2 × 7/2
•                              = 11/2 × 7
•                              = 77/2
• ∴ The area of the circle = 381/2
16. Calculate the area of the shape in the figure below. (Let π = 22/7). The radius of the semi-circle is 7 m. • Area of shape = Area of Big Rectangle + Area of Small Rectangle + Area of Semi Circle
• Area of big rectangle = 16 × 10 = 160 m²
• Area of small rectangle = 14 × 1 = 14 m²
• Area of semi circle = ½ × 22/7 × 7 × 7 = 77 m²
• ∴ The area of the shape = 160 + 14 + 77
• ∴ The area of the shape = 251 m²