# Areas

## Area of a Circle

$A=πr^2$

π is a constant that can, for the purposes of the SAT, be written as 3.14 (or 3.14159)

*r*is the radius of the circle (any line drawn from the center point straight to the edge of the circle)

## Circumference of a Circle

#### $C=2πr (or C=πd)$

*d*is the diameter of the circle. It is a line that bisects the circle through the midpoint and touches two ends of the circle on opposite sides. It is twice the radius.

## Area of a Rectangle

* *$A=lw$

*l*is the length of the rectangle*w*is the width of the rectangle

## Area of a Triangle

$A=12bh$

*b*is the length of the base of triangle (the edge of one side)*h*is the height of the triangleIn a right triangle, the height is the same as a side of the 90-degree angle. For non-right triangles, the height will drop down through the interior of the triangle, as shown above.

## The Pythagorean Theorem

$a^2+b^2=c^2$

In a right triangle, the two smaller sides (

*a*and*b*) are each squared. Their sum is the equal to the square of the hypotenuse (c, longest side of the triangle).

## Properties of Special Right Triangle: Isosceles Triangle

An isosceles triangle has two sides that are equal in length and two equal angles opposite those sides.

An isosceles right triangle always has a 90-degree angle and two 45 degree angles.

The side lengths are determined by the formula: x, x, x√2, with the hypotenuse (side opposite 90 degrees) having a length of one of the smaller sides *√2.

E.g., An isosceles right triangle may have side lengths of 12, 12, and 12√2.

## Properties of Special Right Triangle: 30, 60, 90 Degree Triangle

A 30, 60, 90 triangle describes the degree measures of the triangle's three angles.

The side lengths are determined by the formula: x, x√3, and 2x

The side opposite 30 degrees is the smallest, with a measurement of x.

The side opposite 60 degrees is the middle length, with a measurement of x√3.

The side opposite 90 degree is the hypotenuse (longest side), with a length of 2x.

For example, a 30-60-90 triangle may have side lengths of 5, 5√3, and 10.

## Volume of a Rectangular Solid

$V=lwh$

*l*is the length of one of the sides.*h*is the height of the figure.*w*is the width of one of the sides.

#### Volume of a Cylinder

$V=πr^2h$

r is the radius of the circular side of the cylinder.

h is the height of the cylinder.

## Volume of a Sphere

$V=(43)πr^3$

r is the radius of the sphere.

## Volume of a Cone

$V=(13)πr^2h$

r is the radius of the circular side of the cone.

h is the height of the pointed part of the cone (as measured from the center of the circular part of the cone).

## Volume of a Pyramid

$V=(13)lwh$

l is the length of one of the edges of the rectangular part of the pyramid.

h is the height of the figure at its peak (as measured from the center of the rectangular part of the pyramid).

w is the width of one of the edges of the rectangular part of the pyramid.

**Law: the number of degrees in a circle is 360**

**Law: the number of degrees in a circle is 360**

**Law: the number of radians in a circle is 2π**

**Law: the number of radians in a circle is 2π**

**Law: the number of degrees in a triangle is 180**

**Law: the number of degrees in a triangle is 180**

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