Plane & Solid Figure Geometry Formulas
1. Plane Figures (2Dimensional)
1.1 Basic Definitions
A plane figure lies in a flat surface. The main measures are perimeter (total length around the figure) and area (amount of space inside).
1.2 Common Shapes
| Shape | Perimeter Formula | Area Formula |
|---|---|---|
| Square | P = 4a | A = a |
| Rectangle | P = 2(l + w) | A = lw |
| Right Triangle | P = a + b + (a + b) | A = ab |
| General Triangle | P = a + b + c | A = bh = [s(sa)(sb)(sc)] |
| Parallelogram | P = 2(b + h) | A = bh |
| Rhombus | P = 4a | A = (dd) |
| Circle | C = 2r = d | A = r |
| Regular Polygon (n sides) | P = ns | A = Pa = nsa |
1.3 Special Cases
- Equilateral Triangle: All sides equal a. Height h = (3/2)a. Area = A = (3/4)a.
- Regular Hexagon: Six equal sides a. Area = A = (33/2)a.
- Sector of a Circle: Central angle (radians). Arc length = L = r. Area = A = r.
2. Solid Figures (3Dimensional)
2.1 Basic Definitions
A solid figure occupies space. The key quantities are surface area (total area covering the solid) and volume (space contained).
2.2 Common Solids
| Solid | Surface Area Formula | Volume Formula |
|---|---|---|
| Cube | SA = 6a | V = a |
| Rectangular Prism | SA = 2(lw + lh + wh) | V = lwh |
| Cylinder | SA = 2r(r + h) | V = rh |
| Sphere | SA = 4r | V = (4/3)r |
| Cone (right circular) | SA = r(r + l) | V = (1/3)rh |
| Pyramid (regular base) | SA = B + Pl | V = (1/3)Bh |
| Ellipsoid | SA 4[(a^p b^p + a^p c^p + b^p c^p)/3]^{1/p} (p1.6075) | V = (4/3)abc |
2.3 Derivation Highlights
Cylinder: The lateral surface is a rectangle rolled into a tube; its area equals the rectangles width (circumference) times its height: 2rh. Adding the top and bottom circles gives the total surface area.
Cone: The slant height l = (r + h). Lateral area is the area of a sector of a circle with radius l and arc length equal to the base circumference, giving r l. Adding the base yields the full surface area.
Pyramid: For a regular pyramid, the base area B is known, and each triangular face has area sl where s is the side length of the base and l the slant height. Summing over all faces gives Pl where P is the perimeter of the base.
3. Conversion Between Plane & Solid Formulas
Many solid formulas are extensions of plane ones. For example, the volume of a prism is the area of its base multiplied by the height (V = A_baseh). Understanding the base geometry therefore directly leads to the volumetric result.
3.1 Example: Volume of a Hexagonal Prism
Base: regular hexagon with side a. Area of base A = (33/2)a. Height h. Volume:
V = Ah = (33/2)ah
3.2 Example: Surface Area of a Spherical Segment
When a sphere is sliced by two parallel planes, the resulting band (spherical segment) has surface area:
SA = 2rh_segment
where h_segment is the distance between the two cutting planes. This mirrors the lateral area of a cylinder, showing the geometric link between circles and spheres.4. Quick Reference Cheat Sheet
- Circle circumference: C = 2r
- Circle area: A = r
- Cylinder volume: V = rh
- Cylinder surface area: SA = 2r(r + h)
- Sphere volume: V = (4/3)r
- Sphere surface area: SA = 4r
- Right cone volume: V = (1/3)rh
- Right cone surface area: SA = r(r + (r+h))
- Rectangular prism volume: V = lwh
- Rectangular prism surface area: SA = 2(lw + lh + wh)
Geometry Lesson Plan: Plane Figures And Tangrams and Reference File Download Link
High School Plane Geometry Through Transformations and Reference File Download Link