Geometry Formulas with Questions and Answers

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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 bya² + b² = c²

For example, if the two legs (a and b) are 3 and 4:

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*r²

Arclength and Area of a Circular Sector

Arclength: s = r*t

Area = (1/2) *r² * 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 * r³
Surface Area = 4 * Pi * r²

Volume and Surface Area of a Right Circular Cylinder

Volume = Pi * r² * h
Surface Area = 2 * Pi * r * h

Volume and Surface Area of a Right Circular Cone

Volume = (1/3)* Pi * r² * h
Surface Area = Pi * r * sqrt (r² + h²)

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 meters²

2. The perimeter of a rectangle is 50 feet and its area is 150 feet². Find the length L and the width W of the rectangle, such that L > W.