# Surface Area Formulas

Surface area refers to the space occupied by the geometrical three-dimensional shape. It is denoted by the sum of the individual surfaces of the sides of a three-dimensional figure. The surface area of geometrical 3D figures can be categorized into two broad categories of total surface area and lateral surface area.

The total surface area includes the area of all the surfaces of the figure/ object(base + sides) whereas the lateral surface area of geometrical figures includes the only surface of the sides. The lateral surface area doesn’t include the area of base in its calculation. Lateral surface area or curved surface area is simply the sum of the area of sides of the geometrical figure.

Formulas below

Lateral Surface Area = Area of side faces of a geometrical figureTotal Surface Area = LSA + Area of top and base

**Formulas of Surface Area of Different Geometrical Figures**

The formulas for Lateral Surface Area (LSA) and Total Surface Area(TSA) of different 3-D Geometrical Figures are given below

**Surface area of Cube**

- A cube is a symmetrical figure consisting of six square side faces.
- Here each face has ‘a’ units of length as sides.

LSA of cube = 4×a^{2}TSA of cube = 6×a^{2}

**Surface area of Cuboid**

- A cuboid is a geometrical figure consisting of six rectangular side faces.
- Here faces have ‘l’ units of length, ‘b’ units of breadth, and ‘h’ units of height.

LSA of cube = 2×(hl + bh)TSA of cube = 2×(hl + bh + bh)

**3. Surface area of Sphere**

- The sphere is a 3D circular figure.
- Here, the radius has ‘r’ units of length.

LSA of sphere = 4πr^{2}TSA of sphere = 4πr^{2}Here, TSA of sphere = LSA of sphere

**Surface area of the Hemisphere**

- When a sphere gets sliced into two equal parts, each part is called a hemisphere.
- Here, the radius has ‘r’ units of length.

LSA of hemisphere = 2πr^{2}TSA of hemisphere = 3πr^{2}

**Surface Area of Cylinder**

- A cylinder is a 3D figure with two circular bases.
- Here circular base has ‘r’ units of radius and ‘h’ units of height.

LSA of cylinder = 2πrhTSA of cylinder = 2πr^{2}+ 2πrh

**Surface area of Pyramid**

- A pyramid is a 3D figure having triangular sides on a single given base.
- Here, the base can be a triangular, square, pentagon, or any 2-Dimensional shape.

LSA of Pyramid = 1/2 * (Perimeter of base) * HeightTSA of Pyramid = [ 1/2 * (Perimeter of base) * Height ] + Area of base

### Sample Questions

**Question 1: Find the Lateral surface of a Sphere with a radius of 4 cm.**

**Answer:**

The formula of Lateral Surface Area of Sphere = 4πr

^{2}.Given r is 4cm.

= 4×3.14 × r × r

= 4 × 3.14 × 4 × 4

= 200.96 cm

^{2}

**Question 2: Find the Total surface of a Hemi- Sphere with a radius of 6 cm.**

**Answer:**

Formula of Lateral Surface Area of Hemi- Sphere = 2πr

^{2}.Given that r is 6cm

= 2 × 3.14× r ×r

= 2 * 3.14 * 6 *6

= 226.08 cm

^{2}

**Question 3: Find the Total surface of a Cube with a side of 10 m.**

Answer:The formula of Total Surface Area of Cube = 6a

^{2}Given a is 10cm

= 6× a ×a

= 6× 10 × 10

= 600 m

^{2}

**Question 4: Find the Lateral surface of a Cuboid with a length of 10 cm, breadth of 8 cm, and height of 6 cm.**

**Answer:**

Formula of Lateral Surface Area of Cuboid = 2×(hl + bh)

Given l is 10cm, b is 8cm, h is 6cm

= 2× (h×l + b×h)

= 2× (6×10 + 8×6)

= 2× (60 + 48)

= 2×(108)

= 216 cm

^{2}

**Question 5: Find the Total surface of a Cylinder with a radius of 4 cm and height of 6 cm.**

**Answer:**

Formula of Total Surface Area of Cylinder = 2πr

^{2}+ 2πrhGiven r is 4cm and h is 6cm

= 2×3.14×r×r + 2×3.14×r×h

= 2×3.14×4×4 + 2×3.14×4×6

= 100.48 + 150.72

= 251.2 cm

^{2}

**Question 6: Find the Lateral surface of a Pyramid with the base as an equilateral triangle of side 5 cm and height of 8 cm.**

**Answer:**

Formula of Lateral Surface Area of Pyramid = 1/2 × (Perimeter of base) × Height

Given h is 8cm and the base is an equilateral triangle with a side of 5cm

= 1/2 × (Perimeter of base) ×Height

= 1/2 × (5 + 5 + 5) ×Height

= 1/2 × (5 + 5 + 5) × 8

= 1/2 × (15) × 8

= 60 cm

^{2}