Surface Area of Simple Solids

Measurement is a branch of mathematics that deals with the measurement of length, area, and volume of various geometric shapes. In this lesson, we are going to focus on understanding the concept of surface area of simple solids. Simple solids include shapes such as cubes, cuboids, spheres, cylinders, and cones. The surfaces of these solids can be flat or curved. The surface area of these solids is formed by adding all the areas of their surfaces together.

Understanding surface area

The surface area of a solid object is defined as the total area occupied by the surface of that object. It is measured in square units like square centimetre (cm 2), square metre (m 2) etc. It is important to understand that surface area is different from volume. While volume measures the space inside a solid, surface area measures the space on the surface of a solid.

Surface area of a cube

A cube is a solid figure having six equal square faces. Let us consider each face of the cube. If each side of the cube is s, then the area of one face is s × s = s 2 Since the cube has 6 faces, the total surface area (TSA) is:

TSA = 6 × s 2

TSA = 6 × 4 2 = 6 × 16 = 96 cm 2

Surface area of cuboid

A cuboid is a box-shaped solid object with six rectangular faces. Unlike a cube, its faces can have different areas. If the length, width, and height of a cuboid are l, b, and h respectively, the surface area of the cuboid is given by:

TSA = 2(lb + bh + hl)

TSA = 2(10 × 5 + 5 × 3 + 3 × 10) = 2(50 + 15 + 30) = 2 × 95 = 190 m 2

Surface area of a cylinder

A cylinder is a solid geometric figure having straight parallel sides and a circular or oval cross section. The surface of a cylinder has two circular bases and one curved side. If r is the radius of the base and h is the height of the cylinder, then the curved surface area (CSA) and total surface area (TSA) are calculated as:

CSA = 2πrh TSA = 2πr(r + h)

CSA = 2π × 3 × 12 = 72π ≈ 226.2 cm 2

TSA = 2π × 3 × (3 + 12) = 2π × 3 × 15 = 90π ≈ 282.6 cm 2

Surface area of a cone

A cone is a three-dimensional geometric figure that tapers smoothly from a flat base to a point called the apex. The surface area of a cone is composed of a circular base and a curved surface. If r is the radius of the base, and l is the slant height of the cone, the surface area is given by:

CSA = πrl TSA = πr(r + l)

CSA = π × 5 × 13 = 65π ≈ 204.2 cm 2

TSA = π × 5 × (5 + 13) = π × 5 × 18 = 90π ≈ 282.6 cm 2

Surface area of a sphere

A sphere is a perfectly round geometric object in three-dimensional space, shaped like a round ball. The surface area of a sphere depends on its radius. If r is the radius of the sphere, the formula for surface area is:

SA = 4πr 2

r = diameter/2 = 24/2 = 12 cm

SA = 4π × (12) 2 = 4π × 144 = 576π ≈ 1808.6 cm 2

Summary and conclusion

Understanding the surface areas of simple solids is important because it helps solve real-world problems, from wrapping gifts correctly to designing various containers and structures. Remember these key points:

• A cube has six equal square faces. Its total surface area is 6s 2.
• A cuboid has rectangular faces, and its total surface area is 2(lb + bh + hl).
• The curved surface area of a cylinder is 2πrh, and its total surface area is 2πr(r + h).
• The curved surface area of a cone is πrl, and its total surface area is πr(r + l).
• The surface area of a sphere is 4πr 2.

By understanding and using these formulas, you can quickly determine the surface area of these simple solids for practical purposes.

Comments