Aptitude Volume and Surface Area Concepts and formulas

1) Cuboid:

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

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

2) Cube:

Let each edge of a cube = "a" units. Then:

1. Volume of the cube = a³ cubic units.
2. Whole surface area of cube = (6a²) sq. units.
3. 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:

1. Volume of the cylinder = (πr² h) cubic units.
2. Curved surface area of the cylinder = (2πrh) sq. units.
3. Total surface area of the cylinder =(2πrh+2πr²) sq. units.

4) Sphere:

Let r be the radius of the sphere. Then:

1. Volume of the sphere = cubic units.
2. Surface area of the sphere sq. units.
3. Volume of hemisphere cubic units.
4. Curved surface area of the hemisphere = (2 πr²) sq. units.
5. Whole surface area of the hemisphere = (3 πr²) 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:

1. Slant height l
2. Volume of the cone cubic units.
3. Curved surface area of the cone = (πrl) sq. units sq. units.
4. Total surface area of the cone = (πrl+ πr² )= 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.

1. Slant height,
2. Curved surface area = π (r + R) l sq. units.
3. Total surface area = π { (r + R) l + r² + R² } sq. units.
4. Volume cubic units.

Some Quicker methods:

1) For a closed wooden box:

1. Capacity = (external length - 2 x thickness) x (external breadth - 2 x thickness) x (external height - 2 x thickness)
2. Volume of material = External volume - capacity
3. 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.