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Definition of Mensuration
Mensuration is a branch of mathematics that completes measurement related activities. In measurement, it is especially related to the formulas of area, volume, and perimeter or perimeter of geometric shapes and their use.
Under mensuration, we talk about two-dimensional and three-dimensional shapes. Where we learn to calculate volume, area, perimeter or perimeter etc.
Definition of Triangle
A flat area bounded by three sides is called a triangle. The symbol ‘∆’ is used for a triangle. Any triangle has three sides, three vertices and three angles. The sum of the three angles of a triangle is 180°.
Triangle Formulas
- Area of triangle = ½ × base × height
Types of Triangle
There are two types of triangles on the basis of sides and angles.
- Triangle Based on Sides
- Triangle Based on Angle
1. Triangle Based on Sides
There are 3 types of triangles on the basis of sides. The perimeter of a triangle is always the sum of its three sides.
equilateral triangle
isosceles triangle
scalene triangle
(a). Equilateral Triangle
The triangle whose all sides are equal is called an equilateral triangle. The value of each angle of an equilateral triangle is 60°.
Equilateral Triangle Formula
- Area of equilateral triangle = √3/4 × side²
- Perimeter of equilateral triangle = 3 × side
- Length of perpendicular from vertex = √3/4 × side
- Radius of semicircle of equilateral triangle R = a/2 × √3
- Radius of circle R = a/√3
(b). Isosceles Triangle
A triangle in which any two of its three sides are equal is called an isosceles triangle.
Isosceles triangle formula
Area of isosceles triangle A = ½ × base × height
Area A = ½ × b × h
Second area A = a/4 √(4b² – a²)
Third area A = ½ × side2 × sinθ
Perimeter of isosceles triangle P = 2a + b
(c). Scalene Triangle
A triangle whose three sides are of different lengths is called a scalene triangle. The three sides of this triangle are of different sizes.
Formula of scalene triangle
- Area of scalene triangle A = ½ × b × h
- A = ½ × base × height
- Area A = ½ × a × b × sinθ
- Semiperimeter of triangle P = ½ ( a + b + c )
- Area A = √s(s – a)(s – b)(s – c)
2. Triangle based on angle There are
3 types of triangles on the basis of angle.
- Acute Triangle
- Right Angle Triangle
- Obtuse Triangle
(a). Acute Triangle
The triangle whose value of each angle is less than 90 degrees is called an acute angled triangle.
The sum of the three sides of an acute triangle is equal to 180 degrees.
Acute Triangle Formula
- Area of acute triangle A = ½ × b × h
- Perimeter of acute triangle = a + b + c
- Area A = √s(s – a)(s – b)(s – c)
- A = ½ × a × b × sinθ
- Area of acute triangle A = ½ × b × h
(b). Right Angle Triangle
The triangle in which one angle is a right angle i.e. 90° is called a right angled triangle.
Right Triangle Formula
Square of length of hypotenuse = square of length of perpendicular + square of length of base
AC² = AB² + BC²
- (hypotenuse)² = (perpendicular)² + (base)²
- Hypotenuse of right triangle = √perpendicular² +base²
- Perpendicular to right triangle = √hypoten² – base²
- Base of right angled triangle = √hypoten² – perpendicular²
- Area of right triangle A = ½ × base × height
- Area = A = ½ × b × h
- Hypotenuse C = √a² + b²
- Perimeter = a + b + c
- Height = (a × b)/c
(c). Obtuse Triangle
A triangle whose one angle is more than 90° is called an obtuse triangle. The sum of each interior angle of this triangle is always equal to 180 degrees.
Obtuse Triangle Formula
- Area A = ½ × b × h
- Perimeter = a + b + c
- Semiperimeter P = ½ ( a + b + c )
- Area A = √s(s – a)(s – b)(s – c)
Definition of Quadrilateral
The flat area bounded by four sides is called a quadrilateral. Any quadrilateral has four sides and four angles. The contribution of the four angles of a quadrilateral is four right angles i.e. 360°.
Line segments AC and BD are called diagonals. Those two sides of a quadrilateral which do not have any common point are called opposite sides. AB, CD and AD, BC are opposite sides.
AB are the opposite sides of CD and AD are the opposite sides of BC.
∠A + ∠B + ∠C + ∠D = 360°
Quadrilateral Formulas
Area of quadrilateral = ½ × product of diagonals
Area of quadrilateral = ½ × d(h₁ + h₂)
Types of Quadrilateral
1. Square
A shape surrounded by four sides, whose four sides are equal and each angle is a right angle i.e. 90°, is called a square.
AC and BD are called diagonals and they are equal to each other i.e. AC = BD
Square Formula
- Area of square = (a side)² = a²
- Area of square = ½ × (product of diagonals) = ½ × AC × BD
- Perimeter of square = 4 × a
- Diagonal of square = a side × √2 = a × √2
- Diagonal of square = √2 × area of square
2. Rectangle
A shape surrounded by four sides, in which opposite sides are parallel and equal and every angle is a right angle, is called a rectangle.
Rectangle Formulas
- Perimeter of rectangle = 2(length + breadth)
- Area of rectangle = length × breadth
- Diagonal of rectangle =√(length² + breadth²)
3. Rhombus
A quadrilateral whose four sides are equal is called a rhombus.
Rhombus Formulas
- Area of rhombus = ½ (product of diagonals)
- Perimeter of rhombus = 4 x side
4. Parallelogram
The quadrilateral whose opposite sides are equal and parallel is called parallelogram.
Parallelogram Formulas
- Area of parallelogram = length x breadth
- Perimeter of parallelogram = 2 (length + breadth)
5. Trapezoid
A quadrilateral whose one pair of sides is parallel is called a trapezium.
Trapezoid Formulas
- Area of trapezium = ½ × height × sum of parallel sides
- Area of trapezium = ½ × h × (AD + BC)
6. Kite-shaped Quadrilateral
In a kite shape, two pairs of adjacent sides are of equal length. That is, a diagonal divides the quadrilateral into two congruent triangles.
Therefore the angles between two pairs of equal sides are equal. And both the diagonals are perpendicular to each other.
Three Dimensional Shapes
Three dimensional shapes include shapes like cube, cuboid, cylinder, cone, sphere, frustum of a cone etc.
1. Cube
The length, width and height of the cube are equal. A cube has six faces, twelve edges and eight corners. It has six faces of equal size. Each face is a square and due to having six faces, it is also called a type of hexahedral.
Cube Formulas
- Volume of cube = a × a × a
- Perimeter of cube = 4 × a × a
- The entire surface area of the cube = 6 a² square centimetres.
- Diagonal of cube = √3a centimeter.
Cuboid
A figure bounded by six faces, in which each face is a rectangle and the opposite faces are equal, is called a cuboid.
Like:- book, brick, match box, trunk etc.
Cuboid Formulas
- Volume of cuboid = length × width × height
- Volume of cuboid = l × b × h
- Perimeter of cuboid = 2(l + b) × h
- Area of all the surfaces of the cuboid = 2(length × width + width × height + height × length)
- Total surface area of cuboid = 2(lb + bh + hl)
- Diagonals of cuboid = √(length)² + (width)² + (height)²
- Diagonal of cuboid = √l² + b² + h²
Cylinders
A cylinder is a three-dimensional solid shape in geometry. Its lateral surface is curved, the ends are circular of equal radius, the cylinder is in its simplest form a roller or a glass of equal diameter.
The body which is formed by the rotation of a straight line always perpendicular to itself on the circumference of a circle is called a cylinder.
Cylinder Formulas
- Volume of cylinder = area of base × height
= πr²h - Curved surface of cylinder = perimeter of base × height = 2πrh
- Total surface area of the cylinder = Area of the curved surface + 2 × Area of the base = 2πrh + 2πr²
= 2πr(r + h) - Volume of hollow cylinder = πh(r₁² – r₂²)
- Curved surface of hollow cylinder = 2πh(r₁² + r₂²)
- Total surface area of hollow cylinder = 2πh(r₁ + r₂) + 2π(r₁² – 2r₂²)
Cone
A cone is a three-dimensional structure formed by lines joining an apex and a base. If the base of a cone is a circle, then it is called a right circular cone.
Cone Formulas
- L = √(r² + h²)
- Volume of the cone = (πr²h)/3 cubic centimeters.
- Curved surface area of the cone = πrl square centimeter.
- Total surface area of the cone = (πrl + πr²) square centimeters.
Circle
Circle Formulas
- Volume of sphere = (4πr³)/3 cubic centimeters
- Curved surface of sphere = 4πr² square centimeter
- Volume of hemisphere = (2πr³)/3 cubic centimeters
- Total surface area of hemisphere = 3πr² square centimeter
- If the radius of the hemisphere is r, then
- Volume of hemisphere = 2/3 πr³
- Work surface of hemisphere = 2πr²
- Total surface area of hemisphere = 3πr²
Mensuration Formulas
- Area of triangle = ½ × base × height
- Area of parallelogram = base × height
- Area of circle = πr² (where r is the radius of the circle.)
- Circumference of circle = 2πr
- Radius of circle = circumference/2r
- Area of equilateral triangle = (√3/4) × side × side
- Perimeter of equilateral triangle = 3 × side
- Length of perpendicular from vertex = (√3/4) × side
- Area of isosceles triangle = (a × √4b²- a²)/4
- Perimeter of isosceles triangle = a + 2b
- Length of perpendicular drawn from vertex A = (√4b² – a²)/2
- Area of scalene triangle = ½ × base × height
- Perimeter of scalene triangle = sum of all three sides = (a + b + c)/2
- Area of scalene triangle = √s(s – a)(s – b)(s – c)
- Area of equilateral triangle = (√3/4) × side × side
- Perimeter of equilateral triangle = 3 × side
- Length of perpendicular from vertex = (√3/4) × side
- Area of isosceles triangle = (a × √4b²- a²)/4
- Perimeter of isosceles triangle = a + 2b
- Length of perpendicular drawn from vertex A = (√4b² – a²)/2
- Area of scalene triangle = ½ × base × height
- Perimeter of scalene triangle = sum of all three sides = (a + b + c)/2
- Area of scalene triangle = √s(s – a)(s – b)(s – c)
- Volume of cube = a × a × a
- Perimeter of cube = 4 × a × a
- The entire surface area of the cube = 6 a² square centimetres.
- Diagonal of cube = √3a centimeter.
- Volume of cuboid = length × width × height
- Volume of cuboid = l × b × h
- Perimeter of cuboid = 2(l + b) × h
- Area of all the surfaces of the cuboid = 2(length × width + width × height + height × length)
- Total surface area of cuboid = 2(lb + bh + hl)
- Diagonals of cuboid = √(length)² + (width)² + (height)²
- Diagonal of cuboid = √l² + b² + h²
- Volume of cylinder = πr²h
- Curved surface of cylinder = perimeter of base × height = 2πrh
- Total surface area of cylinder = 2πr(r + h)
- Volume of hollow cylinder = πh(r₁² – r₂²)
- Curved surface of hollow cylinder = 2πh(r₁² + r₂²)
- Total surface area of hollow cylinder = 2πh(r₁ + r₂) + 2π(r₁² – 2r₂²)
- Volume of cone = ⅓ × area of base × height
- Curved plane of cone = ½ × circumference of base × slant height
- Total surface of the cone = curved surface + area of the base = πr (l + r)
- Slant height of the cone L = √r² + h²
- Curved surface area of a sphere = 4πr² square centimeter
- Volume of sphere = 4/3 πr³ cubic centimeter
- Volume of spherical shell = ⁴⁄₃ π(R³ – r³)
- Entire surface area of spherical shell = ⁴⁄₃ π(R²– r²)
- Volume of largest sphere filled by cube = ¹⁄₆ a³
- Radius of the largest sphere in each cube = a/2
- Surface area of the largest sphere in the cube = πa²
- One side of the largest cube in the sphere = 2R / √3
- Volume of the largest cube in the sphere = 8√3/a × R³
- Surface area of the largest cube in the sphere = 8 r²
- Volume of frustum of a cone = ⅓ (πh) (R² + r² + Rr)
- Curved surface area of frustum = πL(R + r)
- Area of oblique section = π (R + r)³, l² = h² + (R – r)²
- Area of entire surface of frustum = π[R² + r² + l(R + r)]
Mensuration Questions
Q.1 The sides of a triangle are 3 cm, 4 cm and 5 cm respectively, what will be its area?
A.6
B.8
C.10
D.12
Solution:- According to the question,
a = 3 centimeters
b = 4 centimeters
c = 5 centimeters
Sum of the three sides of a triangle = (a + b + c)/2
s = (3 + 4 + 5)/2
s = 12/2
s = 6
Area of triangle = √s(s – a)(s – b)(s – c)
∆ = √6(6 – 3)(6 – 4)(6 – 5)
∆ = √6 × 3 × 2 × 1
∆ = √36
∆ = 6
Ans. 6 square centimeters.
Q.2 A right angled triangle whose base is 6 cm. And hypotenuse 10 cm. , then what is the area?
A. 24 cm.²
B. 30 cm.²
C. 40 cm.²
D. 48 cm.²
Solution:- Height of right angle ∆ = √(10² – 6²)
= √(100 – 36)
= √64
= 8 cm.
Area of ∆ = ½ × base × height
Area of ∆ = ½ × 6 × 8
Area of ∆ = 24 cm.²
Ans. 24 cm.²
Q3. The length of the base of a triangle is 15 meters and height is 12 meters, the area of another triangle is twice the area of this triangle and the length of the base of this triangle is 20 meters, what will be the height of this triangle?
A. 18 meters
B.8 meters
C.28 meters
D.38 meters
Solution:- Area of the first triangle = ½ × base × height
= ½ (15 × 12)
= 90 square meters
Area of second triangle = 2 × 90
Area = 180 square meters
base = 20 meters
Height of second triangle = (area × 2)/base
= (180 × 2)/20
Ans. 18 meters.
Q.4 A right angled triangle whose base is 6 cm. And hypotenuse 10 cm. , then what is the area?
A. 24 cm.²
B. 30 cm.²
C. 40 cm.²
D. 48 cm.²
Solution:- Height of right angle ∆ = √(10² – 6²)
= √(100 – 36)
= √64
= 8 cm.
Area of ∆ = ½ × base × height
Area of ∆ = ½ × 6 × 8
Area of ∆ = 24 cm.²
Ans. 24 cm.²
Q.5 The sides of a triangle are 3 cm respectively. 4 cm. And 5 cm. is the area of the triangle?
A. 6 square cm.
B. √23 square cm.
C. √12 square cm.
D. √32 cm.
Solution:- According to the question,
The sides of the triangle are 3 cm respectively. 4 cm. And 5 cm. Are.
a = 3, b = 4, c = 5
S = (a + b + c)/2
S = (3 + 4 + 5)/2
S = 12/2
S = 6 centimeters
Area of ∆ = √s(s – a)(s – b)(s – c)
Area of ∆ = √6(6 – 3)(6 – 4)(6 – 5)
Area of ∆ = √6 × 3 × 2 × 1
Area of ∆ = √6 × 6
Area of ∆ = 6 square cm.
Ans. 6 square cm.
FAQ
Ans. Mensuration is a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc.
Ans. Mensuration Formulas : Area of rectangle (A) is equal to length(l) × Breath(b) Perimeter of a rectangle (P) is equal to 2 × Length (l) + Breath (b)
Ans. a division of mathematics that studies geometric figure calculation and its parameters such as area, length, volume, lateral surface area, surface area, etc.
Ans. figuring out lengths, volumes, shapes, surface areas, and other parameters on 2D and 3D figures.
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