The meaning of 3-D solids is 3 dimensional solids. In Geometry, a three dimensional shape is defined as a shape or solid having three dimensions- length, breadth (or width) and height (or depth). They have volume. Students can also download the HD printout of Surface Area Formulas and Volume Formulas of 3-D Solids and can stick on their pinboard. This will help them to revise and learn all the formulas for doing sums on 3 d shapes.
Surface Area Formulas and Volume Formulas of 3-D Solids
Few example of 3-D shapes are cuboid, cube, cone, cylinder, sphere, hemisphere. Below are the formulas for surface area and volume of 3-d solids.
Surface Area and Volume Formulas of Cuboid
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Cuboid is a solid bounded by 6 rectangular plane surfaces.
Cuboid is a solid bounded by 6 rectangular plane surfaces.
If l be the length of the cuboid, b its breadth, h be the height and d be the diagonal, then
- Diagonal = √(l2 +b2 + h2)
- Lateral Surface Area/ Area of 4 walls = 2h (l +b)
- Total Surface Area = (lb + bh + hl)
- Volume = l x b x h
Surface Area and Volume Formulas of Cube
Cube is a solid bounded by 6 square plane surfaces.
If a be the edge of the cube, and d be the diagonal, then
- Diagonal = (√3)a
- Lateral Surface Area/ Area of 4 walls = 4a2
- Total Surface Area = 6a2
- Volume = a3
Surface Area and Volume Formulas of Cylinder
A solid obtained by revolving a rectangular lamina about one of its sides is called a right circular cylinder. The cross sections of a cylinder are two congruent circles. Below are the formulas for surface area and volume of a solid cylinder and a hollow cylinder.
Surface area and volume of Solid Cylinder
Let r be the radius and h be the height of a solid cylinder, then
- Circumference of base = 2πr
- Area of base = πr2
- Curved Surface area = 2πrh
- Total Surface Area = 2πr2 + 2πrh
- Volume = πr2h
Surface area and volume of Hollow Cylinder
Let R and r be the external and internal radii of a hollow cylinder and h be the height, then
- Thickness of cylinder = R – r
- Area of circular ring (cross-section) = π (R2 _ r2)
- External Curved Surface area = 2πRh
- Internal Curved Surface area = 2πrh
- Total Surface Area = 2πRh + 2πrh + 2π (R2 _ r2)
- Volume of the material = πh (R2 _ r2)
Surface Area and Volume Formulas of Cone
A solid obtained by revolving a right angled triangular lamina about any side (other than the hypotenuse) is called a right circular cone. Below are the formulas for surface area and volume of a cone.
Let r be the radius, h be the height and l be the slant height of the cone, then
- Slant height (l) = √(h2 + r2)
- Area of Base = πr2
- Curved (lateral) surface area = πrl
- Total Surface Area = πr2 + πrl
Surface Area and Volume Formulas of Sphere
A sphere is a three dimensional figure, which is made up of all points in space that lie at a constant distance from a fixed point. The constant distance is called radius and the fixed point is called the centre of the sphere.
Surface Area and Volume Formulas of a Solid Sphere
Let r be the radius of the solid sphere, then
- Surface Area = 4πr2
- Volume = 4/3 πr3
Surface Area and Volume Formulas of a Spherical Shell
Let R and r be the external and internal radii of a spherical shell, then
- Thickness of shell = R – r
- Surface Area = 4π (R2 + r2)
- Volume = 4/3 π (R3 – r3)
Surface Area and Volume Formulas of a Hemisphere
When a sphere is cut by a plane passing through its centre, then the sphere is divided into two equal parts. Each part is called a hemisphere.
Let r be the radius of the hemisphere, then
- Curved Surface Area = 2πr2
- Total Surface Area = 3πr2
- Volume = 2/3 πr3
Surface Area and Volume Formulas of a Hemipherical Shell
Let R and r be the external and internal radii of a Hemispherical shell, then
- Thickness of shell = R – r
- Area of base =
- External Curved Surface Area = 2πR2
- Internal Curved Surface Area = 2πr2
- Total Surface Area = 2πR2 + 2πr2+ (R2 – r2)
- Volume = 2/3 π (R3 – r3)