# Square

on . Posted in Plane Geometry

• Square (a two-dimensional figure) is a quadrilateral with four equal side lengths and 90° interior angles.
• Abbreviated as SQ
• Circumcircle is a circle that passes through all the vertices of a two-dimensional figure.
• Diagonal is a line from one vertices to another that is non adjacent.
• Inscribed circle is the largest circle possible that can fit on the inside of a two-dimensional figure.
• Polygon (a two-dimensional figure) is a closed plane figure for which all edges are line segments and not necessarly congruent.
• Quadrilateral (a two-dimensional figure) is a polygon with four sides.
• a ∥ c
• b ∥ d
• a = b = c = d
• ∠A = ∠B = ∠C = ∠D = 360°
• 4 interior angles are 90°
• 2 diagonals
• 4 edges
• 4 vertex

### Area of a Square formulas

$$A_{area} \;=\; a^2$$

$$A_{area} \;=\; D'^2\;/\;2$$

$$A_{area} \;=\; 4 \; r^2$$

$$A_{area} \;=\; 2 \; R^2$$

Symbol English Metric
$$A_{area}$$ = area $$in^2$$ $$mm^2$$
$$D'$$ = diagonal $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$
$$r$$ = inside radius $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$

### Circumcircle Radius of a Square formulas

$$R \;=\; a\;/\; \sqrt {2}$$

$$R \;=\; D' \;/\;2$$

Symbol English Metric
$$R$$ = outside radius $$in$$ $$mm$$
$$D'$$ = diagonal $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Diagonal of a Square formulas

$$D' \;=\; a \; \sqrt {2}$$

$$D' \;=\; \sqrt {2 \; A_{area} }$$

$$D' \;=\; 2 \; R$$

$$D' \;=\; 2 \; r \; \sqrt{2}$$

$$D' \;=\; 2\;R$$

$$D' \;=\; P\;/\; 2\; \sqrt{2}$$

Symbol English Metric
$$D'$$ = diagonal $$in$$ $$\large{ mm }$$
$$A_{area}$$ = area $$in^2$$ $$mm^2$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$
$$r$$ = inside radius $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$
$$P$$ = perimeter $$in$$ $$mm$$

### Distance from Centroid of a Square formulas

$$C_x \;=\; a \;/\; 2$$

$$C_y \;=\; a \;/\; 2$$

Symbol English Metric
$$C$$ = distance from centroid $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Elastic Section Modulus of a Square formulas

$$S \;=\; a^3 \;/\; 6$$
Symbol English Metric
$$S$$ = elastic section modulus $$in^3$$ $$mm^3$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Inscribed Circle Radius of a Square formulas

$$r \;=\; a\;/\;2$$

$$r \;=\; P\;/\;8$$

$$r \;=\; D\;/\;2\; \sqrt{2}$$

$$r \;=\; \sqrt{A_{area} } \;/\;2$$

$$r \;=\; R\;/\; \sqrt{2}$$

Symbol English Metric
$$r$$ = inside radius $$in$$ $$mm$$
$$A_{area}$$ = area $$in^2$$ $$mm^2$$
$$D'$$ = diagonal $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$
$$P$$ = perimeter $$in$$ $$mm$$

### Perimeter of a Square formulas

$$P = 4\;a$$

$$P = 4\; \sqrt{A_{area} }$$

$$P = 2\;D'\; \sqrt{2}$$

$$P = 4\;R\; \sqrt{2}$$

$$P = 8\;r$$

Symbol English Metric
$$P$$ = perimeter $$in$$ $$mm$$
$$A_{area}$$ = area $$in^2$$ $$mm^2$$
$$D'$$ = diagonal $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$
$$r$$ = inside radius $$in$$ $$mm$$
$$R$$ = outside radius $$in$$ $$mm$$

### Side of a Square formulas

$$a = \sqrt{ A_{area} }$$
Symbol English Metric
$$a, b, c, d$$ = edge $$in$$ $$mm$$
$$A_{area}$$ = area $$in^2$$ $$mm^2$$

### Plastic Section Modulus of a Square formulas

$$Z = a^3 \;/\; 4$$
Symbol English Metric
$$Z$$ = plastic section modulus $$in^3$$ $$mm^3$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Polar Moment of Inertia of a Square formulas

$$J_{z} = a^4\;/\;6$$

$$J_{z1} = 2\;a^4\;/\;3$$

Symbol English Metric
$$J$$ = torsional constant $$in^4$$ $$mm^4$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Radius of Gyration of a Square formulas

$$k_{x} = a \;/\; 2\; \sqrt{3}$$

$$k_{y} = a \;/\; 2\; \sqrt{3}$$

$$k_{z} = a \;/\; \sqrt{6}$$

$$k_{x1} = a \;/\; \sqrt{3}$$

$$k_{y1} = a \;/\; \sqrt{3}$$

$$k_{z1} = \sqrt{ \frac {2}{3} \;a }$$

Symbol English Metric
$$k$$ = radius of gyration $$in$$ $$mm$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Second Moment of Area of a Square formulas

$$I_{x} = a^4\;/\;12$$

$$I_{y} = a^4\;/\;12$$

$$I_{x1} = a^4\;/\;3$$

$$I_{y1} = a^4\;/\;3$$

Symbol English Metric
$$I$$ = moment of inertia $$in^4$$ $$mm^4$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$

### Torsional Constant formulas

$$J = 2.25 \; ( a \;/\; 2 )^4$$
Symbol English Metric
$$J$$ = torsional constant $$in^4$$ $$mm^4$$
$$a, b, c, d$$ = edge $$in$$ $$mm$$