Area of plane surfaces - Important Formulas, shortcuts and Tricks:
A rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle.
(A) Area of a rectangle = (Length × Breadth)
∴Length = (Area/Breadth) and Breadth = (Area/Length).
(B) Perimeter of a rectangle = 2(Length + Breadth)
(C) Diagonal of a rectangle = √(Length² + Breadth²) = √(l² + b²)
(D) Area of 4 walls of a room = 2 (Length + Breadth) × Height
A square is any quadrilateral with four right angles and all of its sides are of equal length.
(A) Diagonal of a square = √2 × a
(B) Area of a square = (side)² = ½ (diagonal)²
(C) Perimeter of a square = 4a, where a is each side
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted by ΔABC
(A) Area of a triangle = ½ × Base × Height
Where a, b, c are the sides of triangle and s = ½ (a + b + c)
(C) Area of an equilateral triangle = √3/4 × (side)²
(D) Height of a equilateral triangle = √3/2 × a
(E) Radius of incircle of an equilateral triangle of side a = a/2√3
(F) Radius of circumcircle of an equilateral triangle of side a = a/√3
(G) Radius of in-circle of a triangle of area Δ and semi-perimeter s = Δ/s
A parallelogram is a (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
(A) Area of Parallelogram = (Base × Height)
(B) Perimeter of Parallelogram = 2 × (sum of adjacent sides)
If all the sides of a Parallelogram are equal, it is called a rhombus. The diagonals of rhombus bisect each other at right angles.
If d₁ and d₂ are the diagonals of a rhombus, then
(A) Side of rhombus = ½ × √(d₁² + d₂²)
(B) Area of rhombus = ½ × (product of diagonals) = ½ × d₁d₂
(C) Perimeter of rhombus = 2 × √(d₁² + d₂²)
A trapezium is a quadrilateral that has only one pair of parallel sides.
(A) Area of a trapezium = ½ × (sum of parallel sides) × distance between them.
A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same. The fixed point is called the centre and the constant distance is known as the radius of the circle. If r = radius, then,
(A) Area of a circle = πr², where, r is the radius.
(B) Circumference of a circle = 2πr
(C) Length of an arc = 2πrθ/360, where θ is the central angle.
(D) Area of a sector = ½(arc × r) = πr²θ/360
(E) Circumference of a semicircle = πr
(F) Area of a semicircle = πr²/2
(G) Area of a quadrant circle = πr²/4
(H) If two circles touch internally/externally, then the distance between their centres is equal to difference/sum
(I) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
(J) The number of revolutions completed by rotating a wheel in one minute = Distance moved in one minute/ Circumference
(I) Results on Triangles:
- Sum of the angles of a triangle is 180 degrees.
- Sum of any two sides of a triangle is greater than the third side.
- Pythagoras theorem: In a right angle triangle, (Hypotenuse)² = (base)² + (Height)²
- The line joining the midpoint of a side of a triangle to the opposite vertex is called the Median.
- The point where the three medians of a triangle meet is called Centroid. Centroid divides each of the medians in the ratio 2:1.
- In an isosceles triangle, the altitude from the vertex bi-sects the base
- The median of a triangle divides it into two triangles of the same area.
- Area of a triangle formed by joining the midpoints of the sides of a given triangle is one-fourth of the area of the given triangle.
II. Results on Quadrilaterals:
- The diagonals of a parallelogram bisect each other.
- Each diagonal of a parallelogram divides it into two triangles of the same area
- The diagonals of a rectangle are equal and bisect each other.
- The diagonals of a square are equal and bisect each other at right angles.
- The diagonals of a rhombus are unequal and bisect each other at right angles.
- A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
- Of all the parallelograms of a given sides, the parallelogram which is a rectangle has the greatest area.
How many bricks of 20 cm × 10 cm will be required to pave the floor of a hall which is 24 m long and 16 m wide?
Let, No. of bricks required be n
∴n = (2400 cm × 1600 cm)/ (20 cm × 10 cm) = 19200
∴ No. of bricks required is 19200.
A rectangular lawn is 80 meters long and 60 meters wide. The time taken by a man to along its diagonal at the speed of 18 km/hour is.
Diagonal, d = √(80² + 60²) = √10000 = 100
Speed, v = 18 km/hr = 18 × (5/18) = 5 m/sec.
∴ Time taken by the man is, t = d/v = 100/5 = 20 sec.