The study of forms, sizes, and the characteristics of space is the focus of the intriguing mathematical field of geometry. Within geometry, flat figures and solid forms are two basic types. Gaining an understanding of these notions is necessary for both practical applications in a variety of professions, including computer graphics, engineering, and architecture, as well as for understanding more advanced mathematical theories. Now let's explore these fascinating geometric forms' characteristics.

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## Plane Figures

Plane figures, which are essentially two-dimensional (2D) forms, consist solely of length and breadth and lie flat on a plane. They lack substance. The attributes of a few typical plane figures are as follows:

### 1. Triangles

A triangle is a three-sided polygon that has three angles. Triangles are categorized according to their angles and sides:

Equilateral Triangle**: A **Triangle with an equilateral triangle has three equal sides and angles.

Isosceles Triangle**: **Two sides and two angles of an isosceles triangle are equal.

Scalene Triangle**:** Each angle and side is unique.

Right Triangle**: **One 90-degree angle makes up a right triangle.

Properties:

. Every inner angle adds up to 180 degrees.

. The formula to compute the area is as follows: Area = 1/2 × base × height

### 2. Quadrilaterals

Polygons having four sides and four angles are called quadrilaterals. Quadrilateral types include:

Square**: **Every angle is 90 degrees, and all sides and angles are equal.

Rectangle**: **Each angle is 90 degrees, and the opposite sides are equal.

Parallelogram**: **Equal and parallel sides on either side.

Rhombus**: **Rhombuses have equal angles on both sides and opposing angles.

Trapezoid**:** There is just one parallel pair of opposing sides.

Properties:

. Every internal angle adds up to 360 degrees.

. Different formulae are used to calculate area; for instance, area = length × width may be used to get the area of a rectangle.

### 3. Circles

A circle is made up of all the points on a plane that are equally spaced out from the center, which is a fixed point.

Properties:

- The radius is the length of the circle from the center to any point on the circle.
- Two times the radius is the diameter.
- The formula for the circumference (perimeter) is C = 2πr, where r is the radius.
- The area is 2 A = πr \ 2.

## Solid Shapes

Three-dimensional (3D) solid forms have three dimensions: height, breadth, and depth. These forms have volume and take up space. The characteristics of a few popular solid forms are as follows:

### 1. Cubes

Six equal square faces make up a cube, which is a solid object.

Properties:

. Every edge has the same length.

. Every angle is ninety degrees.

. The formula for volume is V = a * 3 (where an is an edge's length).

. SA = 6a 2 is the surface area.

### 2. Rectangular Prisms

Six rectangular faces make up a rectangular prism sometimes referred to as a cuboid.

Properties:

. Equal faces are opposite ones.

. The formula for volume is V=l×w×h, where l, w, and h stand for length, breadth, and height, respectively.

. The formula for the surface area is SA=2lw+2lh+2wh.

### 3. Spheres

Every point on the surface of a sphere is equally spaced from the center, making it a completely round three-dimensional form.

Properties:

. The radius is the length of the sphere measured from the center to any point on it.

. The volume is V= 3 4 πr 3 and is equal to V = 4 3 πr 3.

. SA=4πr 2 is the surface area, and SA = 4 πr 2.

### 4. Cylinders

A curved surface connects the two parallel circular bases of a cylinder.

Properties:

. The distance between the bases is the height (h).

. The formula for volume is V = πr − 2 h.

. The formula for the surface area is SA=2πr(h+r) = 2 π (ℎ + r).

### 5. Cones

A cone has one vertex and a circular base.

Properties:

. The height (h) is the base-to-vertex distance measured perpendicularly.

. The distance between the vertex and the base's edge is known as the slant height (l).

. The volume is V = 3 1 πr 2 h, where V = 1 3πr 2 ℎ.

. SA=πr(l+r) is the surface area, and SA = πr ( l+ r).

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### FAQs (Frequently Asked Questions)

**Q1. What are plane figures?**

Ans: Plane figures are two-dimensional (2D) shapes that lie flat on a plane. They have length and width but no depth. Examples include triangles, quadrilaterals, and circles.

**Q2. What are the properties of triangles?**

Ans: Triangles are classified based on their sides and angles:

- Equilateral Triangle: All sides and angles are equal.
- Isosceles Triangle: Two sides and two angles are equal.
- Scalene Triangle: All sides and angles are different.
- Right Triangle: Has one 90-degree angle. The sum of the interior angles in any triangle is always 180 degrees.

**Q3. What are quadrilaterals, and what are their properties?**

Ans: Quadrilaterals are polygons with four sides and four angles. Types include squares, rectangles, parallelograms, rhombuses, and trapezoids. The sum of the interior angles in any quadrilateral is always 360 degrees.

**Q4. What are solid shapes?**

Ans: Solid shapes are three-dimensional (3D) objects that have length, width, and height (or depth). They occupy space and have volume. Examples include cubes, rectangular prisms, spheres, cylinders, and cones.

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