# Mensuration And mensuration formula

**Mensuration** is a branch of mathematics that deals with the measurement of geometric shapes and figures, The mensuration formula encompasses concepts such as area, volume, perimeter, and surface area. It plays a vital role in solving real-world problems involving the calculation of sizes, quantities, and dimensions of various objects and spaces.

**Question 1: Perimeter of a Regular Octagon**

**Question: Calculate the perimeter of a regular octagon with a side length of 9 inches.**

Answer: The perimeter of a regular octagon is found by multiplying the side length by 8 (the number of sides). Perimeter = 9 inches * 8 = 72 inches.

**Question 2: Circumference of a Circle**

**Question: Find the circumference of a circle with a diameter of 14 centimeters.**

Answer: The circumference (C) of a circle is calculated as C = Ï€d, where “d” is the diameter. Substituting the value: C = Ï€ * 14 cm â‰ˆ 43.98 cm.

**Question 3: Volume of a Rectangular Prism**

**Question: Determine the volume of a rectangular prism with dimensions 6 meters, 8 meters, and 10 meters.**

Answer: The volume (V) of a rectangular prism is given by V = length * width * height. Substituting the values: V = 6 meters * 8 meters * 10 meters = 480 cubic meters.

**Question 4: Area of a Circle**

**Question: Calculate the area of a circle with a radius of 5 inches.**

Answer: The area (A) of a circle is found as A = Ï€rÂ², where “r” is the radius. Substituting the value: A = Ï€ * (5 inches)Â² â‰ˆ 78.54 square inches.

**Question 5: Surface Area of a Cylinder**

**Question: Find the total surface area of a cylinder with a radius of 2 meters and a height of 12 meters.**

Answer: The total surface area (SA) of a cylinder is calculated as SA = 2Ï€rÂ² + 2Ï€rh, where “r” is the radius and “h” is the height. Substituting the values: SA = 2Ï€ * (2 meters)Â² + 2Ï€ * 2 meters * 12 meters â‰ˆ 100.53 square meters.

**Question 6: Volume of a Pyramid**

**Question: Determine the volume of a square pyramid with a base side length of 7 centimeters and a height of 9 centimeters.**

Answer: The volume (V) of a pyramid is given by V = (1/3) * base area * height. Substituting the values: V = (1/3) * (7 cm)Â² * 9 cm = 147 cubic centimeters.

**Question 7: Area of a Trapezoid**

**Question: Calculate the area of a trapezoid with bases of lengths 12 meters and 8 meters and a height of 6 meters.**

Answer: The area (A) of a trapezoid is found as A = (1/2) * (sum of bases) * height. Substituting the values: A = (1/2) * (12 meters + 8 meters) * 6 meters = 60 square meters.

**Question 8: Similar Triangles**

**Question: Two triangles are similar. If one triangle has a height of 15 inches and the other has a height of 9 inches, and the corresponding bases are in the ratio of 3:5, find the height of the second triangle.**

Answer: The heights of similar triangles are proportional to the corresponding sides. So, (15 inches / x) = (9 inches / 5). Solve for “x”: x = (15 inches * 5) / 9 inches â‰ˆ 8.33 inches.

**Question 9: Area of a Rhombus**

**Question: Find the area of a rhombus with diagonals of lengths 10 centimeters and 12 centimeters.**

Answer: The area (A) of a rhombus is given by A = (1/2) * product of diagonals. Substituting the values: A = (1/2) * 10 cm * 12 cm = 60 square centimeters.

**Question 10: Volume of a Cone**

**Question: Calculate the volume of a cone with a radius of 8 inches and a height of 15 inches.**

Answer: The volume (V) of a cone is found as V = (1/3)Ï€rÂ²h, where “r” is the radius and “h” is the height. Substituting the values: V = (1/3)Ï€ * (8 inches)Â² * 15 inches â‰ˆ 301.59 cubic inches.

**Question 11: Surface Area of a Sphere**

**Question: Determine the total surface area of a sphere with a radius of 10 centimeters.**

Answer: The total surface area (SA) of a sphere is calculated as SA = 4Ï€rÂ², where “r” is the radius. Substituting the value: SA = 4Ï€ * (10 cm)Â² â‰ˆ 1256.64 square centimeters.

**Question 12: Volume of a Cylinder**

**Question: Find the volume of a cylinder with a radius of 5 meters and a height of 7 meters.**

Answer: The volume (V) of a cylinder is calculated as V = Ï€rÂ²h, where “r” is the radius and “h” is the height. Substituting the values: V = Ï€ * (5 meters)Â² * 7 meters â‰ˆ 549.78 cubic meters.

**Question 13: Interior Angle Sum of a Polygon**

**Question: Calculate the sum of the interior angles in a hexagon.**

Answer: The sum of the interior angles in a polygon can be found using the formula (n-2) * 180 degrees, where “n” is the number of sides. For a hexagon: (6-2) * 180 degrees = 4 * 180 degrees = 720 degrees.

**Question 14: Exterior Angle of a Pentagon**

**Question: Determine the measure of the exterior angle of a regular pentagon.**

Answer: In a regular polygon, all exterior angles have the same measure. To find it, use the formula: 360 degrees/number of sides. For a pentagon: 360 degrees / 5 = 72 degrees.

**Question 15: Diagonals of a Square**

**Question: Calculate the number of diagonals in a square.**

Answer: A square has two diagonals.

**Question 16: Surface Area of a Rectangular Prism**

**Question: Find the total surface area of a rectangular prism with dimensions 4 meters, 6 meters, and 8 meters.**

Answer: The total surface area (SA) of a rectangular prism is given by SA = 2lw + 2lh + 2wh, where “l,” “w,” and “h” are the length, width, and height, respectively. Substituting the values: SA = 2(4 m * 6 m) + 2(4 m * 8 m) + 2(6 m * 8 m) = 208 square meters.

**Question 17: Volume of a Triangular Prism**

**Question: Determine the volume of a triangular prism with a triangular base of an area of 24 square centimeters and a height of 10 centimeters.**

Answer: The volume (V) of a triangular prism is calculated as V = (1/2) * base area * height. Substituting the values: V = (1/2) * 24 square centimeters * 10 centimeters = 120 cubic centimeters.

**Question 18: Area of a Sector**

**Question: Find the area of a sector with a central angle of 60 degrees in a circle with a radius of 8 inches.**

Answer: The area (A) of a sector is calculated as (Î¸/360) * Ï€rÂ², where “Î¸” is the central angle in degrees, and “r” is the radius. Substituting the values: A = (60/360) * Ï€ * (8 inches)Â² â‰ˆ 33.51 square inches.

**Question 19: Altitude of a Triangle**

**Question: Given a triangle with a base of 10 meters and an area of 30 square meters, find the length of the altitude to the base.**

Answer: The area (A) of a triangle is calculated as A = (1/2) * base * altitude. To find the altitude (h), rearrange the formula: h = (2A) / base = (2 * 30 square meters) / 10 meters = 6 meters.

**Question 20: Volume of a Sphere**

**Question: Determine the volume of a hemisphere with a radius of 6 centimeters.**

Answer: The volume (V) of a hemisphere is half the volume of a sphere, so V = (1/2) * (4/3)Ï€rÂ³. Substituting the value: V = (1/2) * (4/3)Ï€ * (6 cm)Â³ â‰ˆ 452.39 cubic centimeters.

**Problem 21: Area of a Rectangle**

**Problem: Calculate the area of a rectangle with a length of 10 meters and a width of 6 meters.**

Solution: The area (A) of a rectangle is found by multiplying its length and width. Substituting the values: A = 10 meters * 6 meters = 60 square meters.

**Problem 22: Volume of a Cuboid**

**Problem: Determine the volume of a cuboid with dimensions of 4 meters, 5 meters, and 6 meters.**

Solution: The volume (V) of a cuboid is calculated as V = length * width * height. Substituting the values: V = 4 meters * 5 meters * 6 meters = 120 cubic meters.

**Problem 23: Perimeter of a Square**

**Problem: Find the perimeter of a square with a side length of 8 centimeters.**

Solution: The perimeter (P) of a square is given by the formula P = 4 * side length. Substituting the value: P = 4 * 8 centimeters = 32 centimeters.

**Problem 24: Surface Area of a Cylinder**

**Problem: Calculate the total surface area of a cylinder with a radius of 3 inches and a height of 10 inches.**

Solution: The total surface area (SA) of a cylinder is found as SA = 2Ï€rÂ² + 2Ï€rh, where “r” is the radius and “h” is the height. Substituting the values: SA = 2Ï€ * (3 inches)Â² + 2Ï€ * 3 inches * 10 inches â‰ˆ 188.50 square inches.

**Problem 25: Volume of a Sphere**

**Problem: Determine the volume of a sphere with a radius of 5 centimeters.**

Solution: The volume (V) of a sphere is calculated as V = (4/3)Ï€rÂ³, where “r” is the radius. Substituting the value: V = (4/3)Ï€ * (5 cm)Â³ â‰ˆ 523.60 cubic centimeters.

**Problem 26: Area of a Triangle**

**Problem: Find the area of a triangle with a base of 12 meters and a height of 8 meters.**

Solution: The area (A) of a triangle is given by A = (1/2) * base * height. Substituting the values: A = (1/2) * 12 meters * 8 meters = 48 square meters.

**Problem 27: Volume of a Cone**

**Problem: Calculate the volume of a cone with a radius of 6 inches and a height of 9 inches.**

Solution: The volume (V) of a cone is found as V = (1/3)Ï€rÂ²h, where “r” is the radius and “h” is the height. Substituting the values: V = (1/3)Ï€ * (6 inches)Â² * 9 inches â‰ˆ 339.29 cubic inches.

**Problem 28: Perimeter of a Parallelogram**

**Problem: Find the perimeter of a parallelogram with adjacent sides of lengths 10 meters and 15 meters.**

Solution: The perimeter (P) of a parallelogram is calculated as P = 2 * (sum of adjacent sides). Substituting the values: P = 2 * (10 meters + 15 meters) = 50 meters.

**Problem 29: Surface Area of a Pyramid**

**Problem: Determine the surface area of a square pyramid with a base side length of 8 inches and a slant height of 10 inches.**

Solution: The surface area (SA) of a square pyramid is found as SA = base area + (1/2) * perimeter of base * slant height. Substituting the values: SA = (8 inches)Â² + (1/2) * 4 * 8 inches * 10 inches â‰ˆ 264 square inches.

**Problem 30: Area of a Trapezium**

**Problem: Calculate the area of a trapezium with bases of lengths 6 meters and 8 meters and a height of 5 meters.**

Solution: The area (A) of a trapezium is given by A = (1/2) * (sum of bases) * height. Substituting the values: A = (1/2) * (6 meters + 8 meters) * 5 meters = 35 square meters.

# MCQ IMPORTANT QUESTIONS On Mensuration And mensuration formula

**Question 1: What is the formula for calculating the area of a triangle?**

A) A = Ï€rÂ²

B) A = 2Ï€rh

C) A = 1/2 * base * height

D) A = Ï€d

**Answer: C) A = 1/2 * base * height**

**Question 2: What is the perimeter of a rectangle with a length of 12 meters and a width of 8 meters?**

A) 16 meters

B) 32 meters

C) 40 meters

D) 96 meters

**Answer: B) 32 meters**

**Question 3: In a right triangle, which side is opposite to the right angle?**

A) Hypotenuse

B) Base

C) Altitude

D) None of the above

**Answer: A) Hypotenuse**

**Question 4: What is the volume of a cube with an edge length of 5 centimeters?**

A) 15 cubic centimeters

B) 25 cubic centimeters

C) 50 cubic centimeters

D) 125 cubic centimeters

**Answer: D) 125 cubic centimeters**

**Question 5: How many degrees are there in the interior angles of a hexagon?**

A) 90 degrees

B) 120 degrees

C) 180 degrees

D) 720 degrees

**Answer: D) 720 degrees**

**Question 6: What is the formula for calculating the circumference of a circle?**

A) C = 2Ï€r

B) C = Ï€rÂ²

C) C = 2Ï€rh

D) C = 1/2 * Ï€d

**Answer: A) C = 2Ï€r**

**Question 7: If two triangles have the same shape but different sizes, they are called:**

A) Congruent triangles

B) Isosceles triangles

C) Similar triangles

D) Equilateral triangles

**Answer: C) Similar triangles**

**Question 8: What is the sum of the interior angles in a quadrilateral?**

A) 90 degrees

B) 180 degrees

C) 360 degrees

D) 720 degrees

**Answer: B) 180 degrees**

**Question 9: In a parallelogram, opposite sides are:**

A) Congruent and parallel

B) Congruent and perpendicular

C) Congruent and intersecting

D) None of the above

**Answer: A) Congruent and parallel**

**Question 10: What is the formula for calculating the volume of a cylinder?**

A) V = Ï€rÂ²

B) V = 2Ï€r

C) V = Ï€rÂ²h

D) V = 1/2 * Ï€d

**Answer: C) V = Ï€rÂ²h**