*82*

# Aptitude Volume and Surface Area Concepts and formulas

1) **Cuboid:**

Let length = l, breadth = b, and height = h units

- Volume of cuboid = (l x b x h) cubic units
- Whole surface area of cuboid = 2 (lb + bh +hl) sq. units.
- Diagonal of cuboid = units.

2) **Cube:**

Let each edge of a cube = “a” units. Then:

- Volume of the cube = a
^{3}cubic units. - Whole surface area of cube = (6a
^{2}) sq. units. - Diagonal of the cube

3) **Cylinder:**

Let the radius of the base of a cylinder be r units and height of the cylinder be h units. Then:

- Volume of the cylinder = (πr
^{2}h) cubic units. - Curved surface area of the cylinder = (2πrh) sq. units.
- Total surface area of the cylinder =(2πrh+2πr
^{2}) sq. units.

4) **Sphere:**

Let r be the radius of the sphere. Then:

- Volume of the sphere = cubic units.
- Surface area of the sphere sq. units.
- Volume of hemisphere cubic units.
- Curved surface area of the hemisphere = (2 πr
^{2}) sq. units. - Whole surface area of the hemisphere = (3 πr
^{2}) sq. units.

5) **Right circular cone:**

Let r be the radius of the base, h is the height, and l is the slant height of the cone. Then:

- Slant height l
- Volume of the cone cubic units.
- Curved surface area of the cone = (πrl) sq. units sq. units.
- Total surface area of the cone = (πrl+ πr
^{2})= πr(l+r) sq.units.

6) **Frustum of a right circular cone:**

Let the radius of the base of the frustum = R, the radius of top = r, height = h and slant height = l units.

- Slant height,
- Curved surface area = π (r + R) l sq. units.
- Total surface area = π { (r + R) l + r
^{2}+ R^{2}} sq. units. - Volume cubic units.

## Some Quicker methods:

**1) For a closed wooden box:**

- Capacity = (external length – 2 x thickness) x (external breadth – 2 x thickness) x (external height – 2 x thickness)
- Volume of material = External volume – capacity
- Weight of wood = Volume of wood x density of wood.

**2) Problems involving ratios:**

**I. Two Spheres:**

(i) (Ratio of radii)^{2} = ratio of surface areas.

(ii)Ratio of volumes = (ratio of radii)^{3}

(iii) (Ratio of surface areas)^{3} = (ratio of volumes)^{2}

**II. Two cylinders:**

**a.** When the radii are equal:

(i)Ratio of volumes = ratio of heights.

(ii)Ratio of curved surface areas = ratio of heights

(iii)Ratio of volumes = (ratio of curved surface areas)

**b.** When heights are equal

(i) Ratio of volumes = (ratio of radii)^{2}

(ii)Ratio of curved surface areas = ratio of radii

(iii) Ratio of volumes = (ratio of curved surface areas)^{2}

**c.** When volumes are equal

(i) Ratio of radii

(ii) Ratio of curved surface areas

**d.** When curved surface areas are equal

(i) Ratio of volumes = ratio of radii

(ii)Ratio of volumes = inverse ratio of heights.

(iii) Ratio of radii = inverse ratio of heights.

Aptitude Volume and Surface Area Test Paper 1

Aptitude Volume and Surface Area Test Paper 2

Aptitude Volume and Surface Area Test Paper 3

Aptitude Volume and Surface Area Test Paper 4

Aptitude Volume and Surface Area Test Paper 5