Question 11

We can understand a 3D object by looking at it from different directions, creating 2D pictures called views.

Step 1 — Understanding Views

Let us imagine a 3D object placed in front of us. We can look at it from different angles. Each angle gives us a flat, 2D picture of the object.

The front view is what we see when we look directly at the object from the front. The top view is what we see when we look directly down on the object from above. The side view is what we see when we look directly at one side of the object. For this problem, we will assume a standard orientation for each solid, usually with its main axis vertical.

Step 2 — Cube Views

Let us consider a cube with each side of length $l$. We place the cube so its faces are perfectly aligned with our viewing directions.

When we look at the cube from the front, we see one of its square faces. This square has dimensions $l$ by $l$.

When we look down on the cube from the top, we see its top square face. This square also has dimensions $l$ by $l$.

When we look at the cube from the side, we see one of its side square faces. This square also has dimensions $l$ by $l$.

$\boxed{\text{Front view: square } l \times l \text{, Top view: square } l \times l \text{, Side view: square } l \times l}$

Step 3 — Cuboid Views

Let us consider a cuboid with length $l$, breadth $b$, and height $h$. We place the cuboid with its length along the front-back direction, breadth along the left-right direction, and height vertically.

When we look at the cuboid from the front, we see a rectangular face. This rectangle has length $l$ and height $h$.

When we look down on the cuboid from the top, we see its top rectangular face. This rectangle has length $l$ and breadth $b$.

When we look at the cuboid from the side (e.g., the right side), we see a rectangular face. This rectangle has breadth $b$ and height $h$.

$\boxed{\text{Front view: } l \times h \text{ rectangle, Top view: } l \times b \text{ rectangle, Side view: } b \times h \text{ rectangle}}$

Step 4 — Parallelepiped Views

A parallelepiped is a 3D shape where all six faces are parallelograms. Let us imagine it resting on one of its parallelogram bases.

When we look at the parallelepiped from the front, we see one of its parallelogram faces. This is a parallelogram.

When we look down on the parallelepiped from the top, we see its top face. This is also a parallelogram.

When we look at the parallelepiped from the side, we see one of its side faces. This is also a parallelogram.

$\boxed{\text{Front view: parallelogram, Top view: parallelogram, Side view: parallelogram}}$

Step 5 — Cylinder Views

Let us consider a cylinder with its axis placed vertically. Imagine it standing upright, like a soda can.

When we look at the cylinder from the front, we see its curved surface. This appears as a rectangle. The width of this rectangle is the diameter of the cylinder's base, and its height is the cylinder's height.

When we look down on the cylinder from the top, we see its circular top base. This is a circle.

When we look at the cylinder from the side, we again see its curved surface. This also appears as a rectangle, identical to the front view.

$\boxed{\text{Front view: rectangle, Top view: circle, Side view: rectangle}}$

Step 6 — Cone Views

Let us consider a cone with its axis placed vertically. Imagine it standing upright, with its circular base on the ground and its tip pointing up.

When we look at the cone from the front, we see its triangular shape. This appears as an isosceles triangle. The base of this triangle is the diameter of the cone's base.

When we look down on the cone from the top, we see its circular base. We also see the very tip (apex) of the cone in the center of the circle.

When we look at the cone from the side, we again see its triangular shape. This also appears as an isosceles triangle, identical to the front view.

$\boxed{\text{Front view: isosceles triangle, Top view: circle (with center dot), Side view: isosceles triangle}}$

Step 7 — Prism (square base, axis vertical) Views

Let us consider a prism with a square base, and its axis placed vertically. Imagine it standing upright, with its square base on the ground.

When we look at this prism from the front, we see one of its rectangular side faces. This is a rectangle.

When we look down on the prism from the top, we see its square top base. This is a square.

When we look at the prism from the side, we see another rectangular side face. This is also a rectangle, identical to the front view because the base is square.

$\boxed{\text{Front view: rectangle, Top view: square, Side view: rectangle}}$

Step 8 — Pyramid (square base, axis vertical) Views

Let us consider a pyramid with a square base, and its axis placed vertically. Imagine it standing upright, with its square base on the ground and its apex pointing up.

When we look at this pyramid from the front, we see one of its triangular faces. This appears as an isosceles triangle.

When we look down on the pyramid from the top, we see its square base. We also see the edges connecting the corners of the base to the apex, which appear as diagonals within the square.

When we look at the pyramid from the side, we again see one of its triangular faces. This also appears as an isosceles triangle, identical to the front view.

$\boxed{\text{Front view: isosceles triangle, Top view: square (with diagonals), Side view: isosceles triangle}}$

Answer

(a) Cube: Front view, Top view, and Side view are all squares of dimension $l \times l$. (b) Cuboid: Front view is $l \times h$ rectangle, Top view is $l \times b$ rectangle, Side view is $b \times h$ rectangle. (c) Parallelepiped: Front view, Top view, and Side view are all parallelograms. (d) Cylinder (axis vertical): Front view is a rectangle, Top view is a circle, Side view is a rectangle. (e) Cone (axis vertical): Front view is an isosceles triangle, Top view is a circle (with center dot), Side view is an isosceles triangle. (f) Prism (square base, axis vertical): Front view is a rectangle, Top view is a square, Side view is a rectangle. (g) Pyramid (square base, axis vertical): Front view is an isosceles triangle, Top view is a square (with diagonals), Side view is an isosceles triangle.