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Comprehensive lesson notes on solid and plane geometry for the technical competency enhancement program 1 offered by the department of civil engineering. The notes cover topics such as polygons, triangles, quadrilaterals, circles, and polyhedrons, including their properties, formulas, and sample problems. The document also includes rules for two chords, two secants, and secant-tangent, as well as spherical components and sample problems.
Typology: Exercises
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After careful study of this lesson, students should be able to:
Polygon – a plane figure with three or more angles.
𝑆𝑢𝑚 𝑜𝑓 𝐼𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒𝑠: 𝑆
𝑖
= 180° (𝑛 − 2 )
𝑆𝑢𝑚 𝑜𝑓 𝐸𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝐴𝑛𝑔𝑙𝑒𝑠: 𝑆
𝑒
= 360°
𝑛
2
2
4 tan (
Regular Polygon – angles are equal and sides are equal
Similar Polygon – two polygons are similar if angles are equal and sides are
proportional
Convex Polygon – having each interior angle less than 180
o
Concave Polygon – having an interior angle greater than 180
o
than the third side.
2
2
2
Quadrilateral Area
Rectangle 𝐿 × 𝑊
Square 𝑠
2
Parallelogram 0. 5 (𝑑
1
)(𝑑
2
)(𝑠𝑖𝑛𝜃)
Rhombus 0. 5 (𝑑
1
)(𝑑
2
)
Trapezoid
ℎ
2
(𝑏
1
2
)
Circle 𝜋𝑟
2
Cyclic Quadrilateral – quadrilateral whose vertices lie on the circumference of a circle.
′
1
2
General Quadrilateral
1. Circle Circumscribed about a Circle
√(𝑎𝑏+𝑐𝑑)(𝑎𝑐+𝑏𝑑)(𝑎𝑑+𝑏𝑐)
4 𝐴
𝑞𝑢𝑎𝑑
𝐴
𝑞𝑢𝑎𝑑
= √
𝑎+𝑏+𝑐+𝑑
2
Polyhedrons – solids whose faces are polygons (3D)
Regular Polyhedrons – are those solids with all its faces identical regular polygons.
There are only five known regular polyhedrons: tetrahedron, hexahedron, octahedron,
dodecahedron, and icosahedron.
Polyhedron Volume Surface Area
Prism 𝐴
𝐵
ℎ 2 𝐴
𝐵
Cylinder 𝐴
𝐵
ℎ 2 𝜋𝑟
2
Pyramid (
1
3
)𝐴
𝐵
ℎ
𝐴
𝐵
Cone (
1
3
)𝐴
𝐵
ℎ
𝜋𝑟
2
Sphere (
4
3
) 𝐴
𝐵
ℎ
4 𝜋𝑟
2
Frustum
(
ℎ
3
) (𝐴
1
2
√
𝐴
1
𝐴
2
) 𝐴 1
𝐴 2
𝐿𝑆𝐴
Truncated Prism 𝐴
𝐵
ℎ
𝑎𝑣𝑒
𝐴 1
𝐴 2
𝐿𝑆𝐴
Spherical Components
2
2
2
2
3
2
