# Solid Geometry Problems

#### Chapter 51

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# Solid Geometry Problems

### Description

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.

### Concepts

Measurement terms:-
Cross-Section:
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:

From the figure, there are three dimensions in it. They are
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: Cube’s length, width and height are all equal, so the formula for the volume of a cube is
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:

Volume of a cylinder: The product of the area of its base and its height is the volume of a cylinder.
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€$$r^2$$

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

If r is the radius of the circular base, h is height of the cone and l is the slant height then
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 = $$\sqrt{r^2 + h^2}$$

### Formula

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€$$r^2$$

### Model Problems

Model 1:A cube and a rectangular solid are equal in volume. Find the length of an edge of the cube, if the length of the edges of the rectangular solid are 4, 8 and 16?
Solution:
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.

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,
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,
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).

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.