Comprehensive Guide to Geometry Formulas: Area, Perimeter, and Volume Calculations
This guide covers essential geometry formulas for calculating the area, perimeter, circumference, and volume of various shapes, including triangles, rectangles, circles, sectors, spheres, cones, and cylinders. You’ll also find useful solutions and examples at the end. Whether you’re studying for a test or solving geometry problems, this guide will help you understand and apply these formulas effectively.
Right Triangle and Pythagora’s theorem
Pythagora’s theorem: The two sides a and b of a right triangle and the hypotenuse c are related bya 2 + b 2 = c 2
- 3² + 4² = 9 + 16 = 25
- The hypotenuse c = √25 = 5
This formula helps to calculate one side when the other two sides are known.
Area and Perimeter of Triangle
Perimeter = a + b + c
There are several formulas for the area.
If the base b and the corresponding height h are known, we use the formula
Area = (1 / 2) * b * h.
If two sides and the angle between them are known, we use one of the formulas, depending on which side and which angle are known
Area = (1 / 2)* b * c sin A
Area = (1 / 2)* a * c sin B
Area = (1 / 2)* a * b sin C .
If all three sides are known, we may use Heron’s formula for the area.
Area = sqrt [ s(s – a)(s – b)(s – c) ] , where s = (a + b + c)/2.
Area and Perimeter of Rectangle
- Perimeter of a rectangle:
Perimeter = 2L + 2W (where L = length and W = width) - Area of a rectangle:
Area = L * W
Area of Parallelogram
- Area = base * height
The Area of Trapezoid
A trapezoid (or trapezium in some regions) is a four-sided shape with one pair of parallel sides. These parallel sides are called bases (denoted as a and b), and the distance between them is the height (denoted as h).
- Area = (1 / 2)(a + b) * h
Circumference of a Circle and Area of a Circular Region
- Circumference = 2*Pi*r
- Area = Pi*r 2
Arclength and Area of a Circular Sector
Arclength: s = r*t
Area = (1/2) *r 2 * t
where t is the central angle in RADIANS.
Volume and Surface Area of a Rectangular Solid
- Volume = L*W*H
- Surface Area = 2(L*W + H*W + H*L)
Volume and Surface Area of a Sphere
- Volume = (4/3)* Pi * r 3
- Surface Area = 4 * Pi * r 2
Volume and Surface Area of a Right Circular Cylinder
- Volume = Pi * r 2 * h
- Surface Area = 2 * Pi * r * h
Volume and Surface Area of a Right Circular Cone
- Volume = (1/3)* Pi * r 2 * h
- Surface Area = Pi * r * sqrt (r 2 + h 2)
Questions and Answers For Geometry Formulas
1. A rectangle has a perimeter of 320 meters and its length L is 3 times its width W. Find the dimensions W and L, and the area of the rectangle.
Solution and answer:
- Use the formula of the perimeter to write.
2 L + 2 W = 320 - We now rewrite the statement “its length L is 3 times its width W” into a mathematical equation as follows:
L = 3 W - We substitute L in the equation 2 L + 2 W = 320 by 3 W.
2(3 W) + 2 W = 320 - Expand and group like terms.
8 W = 320 - Solve for W.
W = 40 meters - Use the equation L = 3 W to find L.
L = 3 W = 120 meters - Use the formula of the area.
Area = L W = 120 * 40 = 4800 meters 2
2. The perimeter of a rectangle is 50 feet and its area is 150 feet 2. Find the length L and the width W of the rectangle, such that L > W.
Solution and answer:
- Use the formula of the perimeter to write
2 L + 2 W = 50 - and the formula of the area to write
L W = 150 - Divide all terms in the equation 2 L + 2 W = 50 by 2 to obtain
L + W = 25 - Solve the above for W
W = 25 – L - Substitute W by 25 – L in the equation L W = 150
L(25 – L) = 150 - Expand the above equation and rewrite with right term equal to zero.
-L 2 + 25 L – 150 = 0
Sources: PinterPandai, Wikipedia, Pioneer Mathematics
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