This document provides information about plane and solid geometry. It defines key shapes and formulas for calculating areas and volumes. For plane geometry, it covers triangles, rectangles, squares, quadrilaterals, regular polygons, circles, parabolic and elliptic segments. For solid geometry, it defines polyhedrons, prisms, cylinders, cones, pyramids, spheres, ellipsoids and paraboloids. It provides formulas to calculate properties like areas, volumes, surface areas, circumferences and more for these various geometric shapes.

1. Topic: PLANE AND SOLID GEOMETRY
PLANE GEOMETRY
Plane geometry is all about shapes, like lines, circles and triangles that are drawn on same flat surface called plane.
TRIANGLE
When two sides, a and b and an included angle θ is given:
1
AT = ab (sin θ)
2
When three sides, a, b and c is given:
Hero’s Formula:
A T = s(s − a)(s − b)(s − c)
When base, b and height, h is given:
a+b+c
s=
1 2
AT = bh
2
When angles A, B and C and one side, a is given:
a2(sin B)(sin C)
AT =
2 sin A
RECTANGLE SQUARE
Area: A = ab
Perimeter: P = 2(a + b)
Area: A = a2
2 2
Diagonal: d= a +b Perimeter: P = 4a)
Diagonal: d=a 2
GENERAL QUADRILATERAL When four sides, a, b, c and d and Cyclic Quadrilateral
two opposite angles A and C are
given:
1 1
A = bc sin C + ad sin A
2 2
Parallelogram
When diagonal, d1 and d2 and included
angle, θ are given:
Radius of circumscribed circle:
1
A = d1 d2 sin θ 1
2 A = d1 d2 sin θ
2 (ab + cd)(ac + bd)(ad + bc)
r =
When four sides, a, b, c and d and 4A
included angle, θ are given: Rhombus, θ = 90o
A = (s − a)(s − b)(s − c)(s − d) − abcd cos2 θ
a+b+c+d
s=
2
Where:
1 1
θ= (sum of two opposite angles) A = d1d2
2 2
REGULAR POLYGONS Equilateral polygons are polygons with 360 o
equal sides Exterior angle: θ=
n
Equiangular polygons are polygons
Area:
with equal interior angles
1 2 1
Regular polygons are polygons that A= R (sin θ) n = xr n
are both equilateral and equiangular. 2 2
Perimeter: P = (x)(n)
n−2
Interior angles = * 180o
n
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2. Topic: PLANE AND SOLID GEOMETRY
CIRCLE
Sector of a circle Segment of a circle
A = A Sector + A Triangle
A = A Sector − A Triangle
1 2
A= r θrad 1 2 1
D2 2 A= r α rad + r 2 sin θ
A = πr 2 = π 1 1 2 2
4 A = r 2 θ rad − r 2 sin θ
Arc: S = rθ rad 2 2
Circumference = 2πr = πD
Note: 180o = 2π rad
PARABOLIC SEGMENT ELLIPSE
A = πab
2
A= bh
3
Circle Escribed (Excircle) about a triangle
Circle circumscribed about a triangle
AT
Radius of escribed circle: ra =
s−a
abc
Radius of circumscribed circle: r = Circle Circumscribed about a Quadrilateral
4A T
Circle inscribed in a triangle
A quad
Radius of circumscribed circle: r=
AT s
Radius of inscribed triangle: r =
s A quad = abcd
a+b+c
s=
2 a+b+c+d
s=
2
SOLID GEOMETRY
Solid geometry treats shapes and figures that do not lie in the same plane.
Polyhedrons Prism
Polyhedrons are three dimensional solid with flat polygon Prism is a polyhedron whose base and ends have the same size and
faces and straight edges. shape extended in parallel planes whose sides are a parallelogram.
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3. Topic: PLANE AND SOLID GEOMETRY
Rectangular Prism
Right Circular Cylinder Spherical Segment of One Base
Volume: V = abc
Lateral area: A L = 2(ab + ac)
V = πr 2h
Surface area: A S = 2(ab + bc + ac) πh2
V= (3r − h)
Right Circular Cone 3
Face diagonal: d1 = a2 + b 2
Spherical Segment of Two Bases
Space diagonal: d2 = a2 + b 2 + c 2
Truncated Prism
1 2
V= πr h
3
Lateral area: AL = πrL
πh
V= (3a2 + 3b2 + h2 )
Frustum of a Right Circular Cone 6
∑h
V = Ar
n
Ar = area of right section
Ellipsoid
n = number of sides
Pyramid
πh 2
V= (R + r 2 + Rr)
3
4
V = πabc
3
Lateral area: AL = π(R + r)L
1
V = A bh Paraboloid Revolution
3 Sphere
Ab = area of the base
Frustum of a pyramid
1 2
V = πr h
2
V=
h
3
(A1 + A 2 + A1 A 2 ) V=
4 3
πr
Lateral area:
⎡ 3
3⎤
3 4πr ⎢⎛ r 2 ⎞2 ⎛ r ⎞ ⎥
Ab = area of the base AL = ⎜ 2⎟
+h ⎟ −⎜ ⎟ ⎥
⎢ ⎜2⎟
3h2 ⎢⎜ 4
⎝ ⎠ ⎝ ⎠ ⎥
⎣ ⎦
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