GEOMETRY
Math Formulas for Geometric Shapes
Special Right Triangle: 30º-60º-90º
Congruent 30º-60º-90º triangles are formed when an altitude is drawn in an equilateral triangle. Remember that the altitude in an equilateral triangle will bisect the angle and is the perpendicular bisector of the side. If the side of the equilateral triangle is set to a length of 2 units, the Pythagorean Theorem will find the length of the altitude to be units
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Pattern Formulas:Short Leg is half of the Hypotenuse: \(SL = \frac{1}{2}H\)
Long Leg is half the Hypotenuse: \(LL = \frac{1}{2}* H * \sqrt{3}\)
Combining the two formulas above: \(LL = SL* \sqrt{3}\)
Special Right Triangle: 45º-45º-90º
Our first observation is that a 45º-45º-90º triangle is an "isosceles right triangle". This tells us that if we know the length of one of the legs, we will know the length of the other leg. This will reduce our work when trying to find the sides of the triangle. Remember that an isosceles triangle has two congruent sides and congruent base angles (in this case 45º and 45º).
Congruent 45º-45º-90º triangles are formed when a diagonal is drawn in a square. Remember that a square contains 4 right angles and its diagonal bisects the angles. If the side of the square is set to a length of 1 unit, the Pythagorean Theorem will find the length of the diagonal to be units.
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Surface Area and Volume of a Cylinder
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Surface Area and Volume of a Cone
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Surface Area and Volume of a Rectangular Prism
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Surface Area and Volume of a Pyramid
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Surface Area and Volume of a Prism
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Area of a Circle Sector
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Area and Perimeter of a Parallelogram
The parallelogram has two sets of opposite sides that run parallel to one another. The shape is a quadrangle, so it has four sides: two sides of one length (a) and two sides of another length (b).
To find out the perimeter of any parallelogram, use this simple formula:
Perimeter = 2a + 2bWhen you need to find the area of a parallelogram, you will need the height (h). This is the distance between two parallel sides. The base (b) is also required and this is the length of one of the sides.
Area = b x hKeep in mind that the b in the area formula is not the same as the b in the perimeter formula. You can use any of the sides—which were paired as a and b when calculating perimeter—though most often we use a side that is perpendicular to the height.
Area and Perimeter of a Rectangle
The rectangle is also a quadrangle. Unlike the parallelogram, the interior angles are always equal to 90 degrees. Also, the sides opposite one another will always measure the same length.
To use the formulas for perimeter and area, you will need to measure the rectangle's length (l) and its width (w).
Perimeter = 2h + 2wArea = h x wArea and Perimeter of a Square
The square is even easier than the rectangle because it is a rectangle with four equal sides. That means you only need to know the length of one side (s) in order to find its perimeter and area.
Perimeter = 4sArea = s2Area and Perimeter of a Trapezoid
The trapezoid is a quadrangle that can look like a challenge, but it's actually quite easy. For this shape, only two sides are parallel to one another, though all four sides can be of different lengths. This means that you will need to know the length of each side (a, b1, b2, c) to find a trapezoid's perimeter.
Perimeter = a + b1 + b2 + cTo find the area of a trapezoid, you will also need the height (h). This is the distance between the two parallel sides.
Area and Perimeter of a Hexagon
A six-sided polygon with equal sides is a regular hexagon. The length of each side is equal to the radius (r). While it may seem like a complicated shape, calculating the perimeter is a simple matter of multiplying the radius by the six sides.
Perimeter = 6rFiguring out the area of a hexagon is a little more difficult and you will have to memorize this formula:
Area = (3√3/2 )r2Area = 1/2 (b1 + b2) x hArea and Perimeter of an Octagon
A regular octagon is similar to a hexagon, though this polygon has eight equal sides. To find the perimeter and area of this shape, you will need the length of one side (a).
Perimeter = 8aArea = ( 2 + 2√2 )a2