In arithmetic, the traditional name for the geometry of three-dimensional Euclidean space is solid geometry. Solid geometry deals with the measurement of different strong figures or Polyhedrons (three-dimensional figures) including cuboids, cubes, cylinders, spheres, pyramids & cones and also with the estimation of volumes of them.

A figure that has depth in addition to width and height is a space figure or three dimensional figures. A tennis ball, a box, a bicycle and a redwood tree are all everyday examples of space figures.

Having all flat faces in a three-dimensional figure is known as a polyhedron. Cuboids and cube belong to polyhedrons whereas sphere, cylinder and cone areÂ not polyhedrons.

A figure that has depth in addition to width and height is a space figure or three dimensional figures. A tennis ball, a box, a bicycle and a redwood tree are all everyday examples of space figures.

Having all flat faces in a three-dimensional figure is known as a polyhedron. Cuboids and cube belong to polyhedrons whereas sphere, cylinder and cone areÂ not polyhedrons.

A cross-section is the shape of a particular two-dimensional “slice” of a space figure.

Eg: By the cross-section of the cylinder, a circle is formed.

**Volume**:

- Measure of how much three-dimensional figure takes-up is known as volume.
- Just as area is used to measure a plane figure, volume is used to measure a space figure.

**Surface area**:

The total area of all faces of the figure is the surface area of a three-dimensional figure.

**Polyhedrons:-**

A rectangular solid has six rectangular surfaces called faces as shown in the figure below:

l – Length

w – width

h – height

- Adjacent faces are perpendicular to each other.
- Each line segment that is the intersection of two faces is called an edge.
- Vertex is each point at which the edges intersect.
- 12 edges and 8 vertices are there.

**Cuboids**: Prisms with a rectangular base and edges that are perpendicular to its base are known as cuboids.

Eg: A cardboard box looks a lot like a cuboid.

Volume of cuboids: Volume of a rectangular solid takes the formula for area of a rectangle and adds another dimension.

Volume = l x w x h

Surface area of cuboid: The sum of the areas of the congruent pairs is the surface area of cuboid. i.e.

Surface area = 2lw + 2lh + 2wh

Longest diagonal of cuboids: Every rectangular solid has four diagonals, each with the same length, that connect each pair of opposite vertices. The diagonal of a rectangular solid ‘d’ is the line segment whose endpoints are opposite corners of the solid.

d = \(\sqrt{(l^2 + w^2 + h^2)}\)

**Cubes**: A three-dimensional figure having six matching square sides is known as a cube. It is as shown in the figure below:

Volume of a cube = \(S^3\)

Surface area of cube: The formula for finding the surface area is the same as the formula for finding the surface of a rectangular solid, except with S substituted in for l, w and h since a cube is just a rectangular solid:

Surface area of a cube = 6\(S^2\)

Diagonal length of a cube: The formula for the diagonal of a cube is also adapted from the formula for the diagonal length of a rectangular solid, with a substituted for l, w and h.

Diagonal length of a cube = S\(\sqrt{3}\)

**Other figures:-**

**Cylinder**: A three-dimensional figure having two congruent circular bases that are parallel. It is as shown in the figure below:

Since, a cylinder has a circular base, the volume of a cylinder is equal to the area of the circle that is the base times the height:

Volume of a cylinder = Ï€\(r^2\)h

Surface area of a cylinder: Surface area of the entire cylinder is the sum of the areas of the curved surface and two cross section areas.

Surface area of a cylinder = 2Ï€rh + 2Ï€\(r^2\)

**Sphere**: A three dimensional figure having all of its points the same distance from its centre is a sphere.

The distance from the centre to the surface of the sphere is called its radius.

Volume of the sphere = Ï€\(\frac{4}{3}r^3\)cubic units

Surface area of the sphere = 4Ï€\(r^2\)

Surface area of the sphere = 4Ï€\(r^2\)

**Cone**: A cone is a three-dimensional figure having a circular base and a single vertex.

Volume of the cone = Ï€\(\frac{1}{3}r^2\)h

Curved surface area of the cone (S) = Ï€r\(\sqrt{r^2 + h^2}\)

Total surface area (S) = Ï€rl + Ï€\(r^2\) = Ï€r(l + r)

Slant height, l = \(\sqrt{r^2 + h^2}\)

Curved surface area of the cone (S) = Ï€r\(\sqrt{r^2 + h^2}\)

Total surface area (S) = Ï€rl + Ï€\(r^2\) = Ï€r(l + r)

Slant height, l = \(\sqrt{r^2 + h^2}\)

1. Volume of cuboid = l x w x h

2. Surface area of cuboid = 2lw + 2lh + 2wh

3. Diagonal length of a cuboid = \(\sqrt{(l^2 + w^2 + h^2)}\)

4. Volume of a cube = \(S^3\)

5. Surface area of a cube = 6\(S^2\)

6. Diagonal length of a cube = S\(\sqrt{3}\)

7. Volume of a cylinder = Ï€\(r^2\)h

8. Surface area of a cylinder = 2Ï€rh + 2Ï€\(r^2\)

9. Volume of the sphere = Ï€\(\frac{4}{3}r^3\)cubic units

10. Surface area of the sphere = 4Ï€\(r^2\)

2. Surface area of cuboid = 2lw + 2lh + 2wh

3. Diagonal length of a cuboid = \(\sqrt{(l^2 + w^2 + h^2)}\)

4. Volume of a cube = \(S^3\)

5. Surface area of a cube = 6\(S^2\)

6. Diagonal length of a cube = S\(\sqrt{3}\)

7. Volume of a cylinder = Ï€\(r^2\)h

8. Surface area of a cylinder = 2Ï€rh + 2Ï€\(r^2\)

9. Volume of the sphere = Ï€\(\frac{4}{3}r^3\)cubic units

10. Surface area of the sphere = 4Ï€\(r^2\)

Problems

Given that,

All the dimensions to calculate the volume of the rectangular solid, which is 4, 8 and 16.

This is also the volume of the cube.

Therefore, the length of an edge of the cube is the cubic root of (16 x 8 x 4) = 8.

All the dimensions to calculate the volume of the rectangular solid, which is 4, 8 and 16.

This is also the volume of the cube.

Therefore, the length of an edge of the cube is the cubic root of (16 x 8 x 4) = 8.

**Model 2:**A cylinder building contains wheat. Radius and depth of the building are 20 feet and 50 feet respectively. Find the volume of this cylindrical building?

**Solution:**

Given that,

Radius = 20 feet

Depth = 50 feet

Volume of the cylindrical building (V) = Ï€\(r^2\)h

V = Ï€\((20)^2\) x 50

V = 20000Ï€

Therefore, the volume of the building is 20000Ï€ cubic feet.

Radius = 20 feet

Depth = 50 feet

Volume of the cylindrical building (V) = Ï€\(r^2\)h

V = Ï€\((20)^2\) x 50

V = 20000Ï€

Therefore, the volume of the building is 20000Ï€ cubic feet.

**Model 3:**Radius is 5 cm and height is 10 cm of a cone.

Quantity A | Quantity B |

The volume of the cone | 300\({cm}^3\) |

A. The quantity on the left is greater.

B. The quantity on the right is greater.

C. Both are equal.

D. The relationship cannot be determined without further information.

**Solution:**

Given that,

Radius = 5 cm

height = 10 cm

Now, volume of the cone (V) = Ï€\(\frac{1}{3}r^2\)h

V = Ï€\(\frac{1}{3}5^2\) x 10

V = (250Ï€)\(\frac{1}{3}\)

V = 261.67\({cm}^3\)

Hence, option (B) is correct answer.

Radius = 5 cm

height = 10 cm

Now, volume of the cone (V) = Ï€\(\frac{1}{3}r^2\)h

V = Ï€\(\frac{1}{3}5^2\) x 10

V = (250Ï€)\(\frac{1}{3}\)

V = 261.67\({cm}^3\)

Hence, option (B) is correct answer.

**Model 4:**Radius of a sphere is 6cm.

Quantity A | Quantity B |

The surface area of the sphere | 452.16 sq.cm |

A. The quantity on the left is greater.

B. The quantity on the right is greater.

C. Both are equal.

D. The relationship cannot be determined without further information.

**Solution:**

Given that,

Radius of a sphere is 6cm.

Now, Surface area of sphere (S) = 4Ï€\(r^2\)

S = 4 x 3.14 \(6^2\)

S = 12.56 x 36

S = 452.16 sq.cm

Thus, the two values are equal.

Therefore, correct option is (C).

Radius of a sphere is 6cm.

Now, Surface area of sphere (S) = 4Ï€\(r^2\)

S = 4 x 3.14 \(6^2\)

S = 12.56 x 36

S = 452.16 sq.cm

Thus, the two values are equal.

Therefore, correct option is (C).

**Model 5:**

Quantity A | Quantity B |

The volume of a cylinder if the radius is doubled | The volume of a cylinder if the height is doubled |

A. The quantity on the left is greater.

B. The quantity on the right is greater.

C. Both are equal.

D. The relationship cannot be determined without further information.

**Solution:**

The volume of the cylinder multiplies the square of the radius by the height.

Doubling the radius of a cylinder will change the volume more fundamentally because of squaring involved.

Hence, option (A) is right answer.

Doubling the radius of a cylinder will change the volume more fundamentally because of squaring involved.

Hence, option (A) is right answer.