Mathematics · Quantitative Aptitude

Mensuration of Solids

209 Questions

Mensuration of solids focuses on calculating the volume and surface area of three dimensional shapes like cylinders and cones. These geometry problems require applying standard mathematical formulas. They frequently appear in state public service and engineering exams.

Cylinder volume and areaCone slant heightSurface area ratiosHollow pipe calculationsDimensional optimization

Mensuration of Solids Questions

Multiple choice

Find the surface area of a cylinder with radius 5 cm and height 10 cm.

  1. 314 cm²

  2. 157 cm²

  3. 78.5 cm²

  4. 39.25 cm²
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a cylinder is given by SA = 2πr(r + h), where r is the radius and h is the height. In this case, r = 5 cm and h = 10 cm, so SA = 2π(5)(5 + 10) = 314 cm².

Multiple choice

Find the volume of a cone with radius 3 cm and height 4 cm.

  1. 12π cm³

  2. 24π cm³

  3. 36π cm³

  4. 48π cm³
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a cone is given by V = ½πr²h, where r is the radius and h is the height. In this case, r = 3 cm and h = 4 cm, so V = ½π(3²)(4) = 12π cm³.

Multiple choice

Find the surface area of a cone with radius 6 cm and height 8 cm.

  1. 150.79 cm²

  2. 75.39 cm²

  3. 37.69 cm²

  4. 18.84 cm²
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a cone is given by SA = πr(r + l), where r is the radius, l is the slant height, and h is the height. In this case, r = 6 cm and h = 8 cm. To find l, we use the Pythagorean theorem: l² = r² + h². Substituting the values, we get l² = 6² + 8² = 100. Therefore, l = 10 cm. Now we can calculate the surface area: SA = π(6)(6 + 10) = 150.79 cm².

Multiple choice

A cylinder has a radius of 6 cm and a height of 8 cm. What is the volume of the cylinder?

  1. 288π cm³

  2. 144π cm³

  3. 72π cm³

  4. 36π cm³
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a cylinder is given by V = πr²h, where r is the radius and h is the height. In this case, r = 6 cm and h = 8 cm, so V = π(6²)(8) = 288π cm³.

Multiple choice

A cone has a radius of 5 cm and a height of 12 cm. What is the volume of the cone?

  1. 314 cm³

  2. 157 cm³

  3. 78.5 cm³

  4. 39.25 cm³
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a cone is given by V = ½πr²h, where r is the radius and h is the height. In this case, r = 5 cm and h = 12 cm, so V = ½π(5²)(12) = 314 cm³.

Multiple choice

A company wants to design a cylindrical can with a volume of 500 cubic centimeters. What dimensions will minimize the surface area of the can?

  1. Radius: 5 cm, Height: 10 cm

  2. Radius: 6.32 cm, Height: 7.96 cm

  3. Radius: 7.07 cm, Height: 7.07 cm

  4. Radius: 8 cm, Height: 6.25 cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using calculus, we can find the dimensions that minimize the surface area while satisfying the volume constraint.

Multiple choice

What is the Euler characteristic of a sphere?

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The Euler characteristic of a space is a topological invariant that is defined as the alternating sum of the number of vertices, edges, and faces in a triangulation of the space. The Euler characteristic of a sphere is 2 because it can be triangulated into two triangles.

Multiple choice

What is the formula for the volume of a cylinder?

  1. $V = \pi r^2 h$
  2. $V = \frac{1}{3}\pi r^2 h$
  3. $V = 2\pi r^2 h$
  4. $V = 4\pi r^2 h$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the volume of a cylinder is $V = \pi r^2 h$, where r is the radius of the base of the cylinder and h is the height of the cylinder.

Multiple choice

What is the formula for the surface area of a cylinder?

  1. $S = 2\pi r h + 2\pi r^2$
  2. $S = \pi r h + 2\pi r^2$
  3. $S = \pi r h - 2\pi r^2$
  4. $S = 2\pi r h - 2\pi r^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the surface area of a cylinder is $S = 2\pi r h + 2\pi r^2$, where r is the radius of the base of the cylinder and h is the height of the cylinder.

Multiple choice

What is the formula for the volume of a cone?

  1. $V = \frac{1}{3}\pi r^2 h$
  2. $V = \frac{1}{2}\pi r^2 h$
  3. $V = \pi r^2 h$
  4. $V = 2\pi r^2 h$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$, where r is the radius of the base of the cone and h is the height of the cone.

Multiple choice

What is the formula for the surface area of a cone?

  1. $S = \pi r h + \pi r^2$
  2. $S = 2\pi r h + \pi r^2$
  3. $S = \pi r h - \pi r^2$
  4. $S = 2\pi r h - \pi r^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the surface area of a cone is $S = \pi r h + \pi r^2$, where r is the radius of the base of the cone and h is the height of the cone.

Multiple choice

What is the formula for the surface area of a pyramid?

  1. $S = Bh + \frac{1}{2}Pl$
  2. $S = Bh + Pl$
  3. $S = Bh - \frac{1}{2}Pl$
  4. $S = Bh - Pl$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the surface area of a pyramid is $S = Bh + \frac{1}{2}Pl$, where B is the area of the base of the pyramid, h is the height of the pyramid, P is the perimeter of the base of the pyramid, and l is the slant height of the pyramid.

Multiple choice

What is the formula for the surface area of a prism?

  1. $S = 2Bh + Pl$
  2. $S = Bh + Pl$
  3. $S = Bh - Pl$
  4. $S = 2Bh - Pl$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the surface area of a prism is $S = 2Bh + Pl$, where B is the area of the base of the prism, h is the height of the prism, P is the perimeter of the base of the prism, and l is the slant height of the prism.

Multiple choice

What is the formula for the volume of a cylinder with radius (r) and height (h)?

  1. \(\pi r^2h\)
  2. \(2\pi r^2h\)
  3. \(\pi r^3\)
  4. \(2\pi r^3\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The volume of a cylinder is given by the formula (\pi r^2h), where (r) is the radius and (h) is the height of the cylinder.

Multiple choice

What is the formula for the surface area of a cylinder with radius (r) and height (h)?

  1. \(2\pi r^2 + 2\pi rh\)
  2. \(\pi r^2 + 2\pi rh\)
  3. \(2\pi r^2\)
  4. \(\pi r^2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The surface area of a cylinder is given by the formula (2\pi r^2 + 2\pi rh), where (r) is the radius and (h) is the height of the cylinder.