#### 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__

__Examples__

- 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

- A rectangle has a perimeter of 74 m. Find
- the length of the rectangle if its breadth is 17 m.
- Perimeter of a rectangle = 2(L + B)
- Perimeter = 74 m,
- Breadth = 17 m,
- ∴ 74 = 2(L + 17)
^{74}/_{2}= L + 17- 37 = L + 17
- L = 37 – 17
- ∴ Length of the rectangle = 20 m.

- 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.

- the length of the rectangle if its breadth is 17 m.
- 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.

- 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.

- Calculate the circumference of a circle of radius 3
^{1}/_{2}m. (Let π = 3^{1}/_{7}).- Circumference of a circle = 2πr,
- R = 3
^{1}/_{2 }, - ∴ Circumference of a circle = 2 × 3
^{1}/_{7 }× 3^{1}/_{2} - = 2 ×
^{22}/_{7 }×^{7}/_{2} - = 2 × 11
- ∴ Circumference of a circle = 22 m

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- Find the area of a circle of radius 3
^{1}/_{2}meters. (Let π =^{22}/_{7})- Area of a circle = πr²
- =
^{22}/_{7 }× (3^{1}/_{2})² - =
^{22}/_{7 }×^{7}/_{2}×^{7}/_{2} - =
^{11}/_{2}× 7 - =
^{77}/_{2} - ∴ The area of the circle = 38
^{1}/_{2}m²

- 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