Can a triangle and a Circle have the same perimeter?
Mensuration is a part of mathematics that deals with the study of geometric figures and calculation of its parameters like area, length, volume, perimeter, surface area, etc. The study deals with two-dimensional and three-dimensional shapes in which 2D shapes involve the length and breadth of the shape. It does not deal with the thickness of the shape. And, 3D shapes have dimensions of length, width, and height.
Formula chart for mensuration formulas
The determination of parameters in the mensuration of different shapes takes place with the help of standard derived formulas. These formulas make the calculation more convenient for deriving the solution.
2D shapes | Formulas |
---|---|
Triangle | 1/2 base × height |
Rectangle |
perimeter = 2(length + breadth) Area = length × breadth |
Square |
Area = (side)^{2} Perimeter = 4(side) |
Circle |
Diameter = 2 × radius Area = πr^{2} |
3D shapes | Formulas |
---|---|
Sphere |
Volume = 4/3πr^{3} Total surface area = πr(l + radius) |
Cube |
Volume = (side)^{3} Lateral surface area = 4(side)^{2} ^{ }Total surface area = 6(side)^{2} |
Cuboid |
Volume = length × width × height Lateral surface area = 2h(l + b) Total surface area = 2(lb + lh + bh) |
Cone |
Volume = 1/3 πr^{2}h Total surface area = πr(l + radius) |
Can a triangle and a circle have the same perimeter?
Answer:
An equilateral triangle and a circle do not have the same perimeter. Let P_{t }be the perimeter of an equilateral triangle, and, P_{c }is the perimeter of a circle. Now, evaluate the perimeter of an equilateral triangle and a circle.
The perimeter of an equilateral triangle by standard mensuration formula,
P_{t }= a + b + c
As the given triangle is equilateral it’s all sides are equal.
P_{t }= x + x + x
P_{t }= 3x
x = P_{t}/3
Circumference or perimeter of the circle,
P_{c }= 2πr
x = P_{c}/2π
The perimeter of the triangle is greater than a circle.
The perimeter of the triangle > Perimeter of the circle
P_{t }> P_{c}
Hence, the perimeter of an equilateral triangle and a square are the same.
Sample Problems
Question 1: If a circle has a radius of 14cm. Find its area.
Solution:
Given,
Radius of circle(r) =14cm
Area(A) = ?
Now,
Area = πr^{2}
A = 22/7 × 14 × 14
A = 22 × 2 × 14
A = 616cm^{2}
Hence, the area of given circle is 616 cm^{2}.
Question 2: A rhombus has diagonals with lengths 5cm and 8cm respectively. Calculate its area.
Solution:
Given,
Diagonal 1(d_{1}) = 5cm
Diagonal 2(d_{2}) = 8cm
Now,
Area of rhombus(A) = 1/2 × d_{1} × d_{2}
A = 1/2 × 5 × 8
A = 40cm^{2}
Hence, the area of rhombus is 40cm^{2}.
Question 3: If a square has its one side 20cm. What will be its perimeter and area?
Solution:
Given,
Side of square = 20cm
Now,
Perimeter of square(P) = 4(side)
P = 4 × 20
P = 80cm
Then,
Area of square(A) = (side)^{2}
A = 20 × 20
A = 400cm^{2}
Hence, the perimeter of a square is 80cm and its area is 400cm^{2}.
Question: Find the volume of a cuboid having length 10cm, breadth 4cm, and height 5cm.
Solution:
Length(l) = 10cm
Breadth(b) = 4cm
Height (h) = 5cm
Now,
Volume of cuboid (V) = length × breadth × height
V = 10 × 4 × 5
V = 200cm^{3}
Hence, the volume of cuboid is 200cm^{3}.