Shapes and Solids
Square
A square is a four sided regular polygon. The circumference of a square with sides of length s is:
The area of a square is:
Rectangle
The rectangle is a 4 sided polygon, quadrilateral, with right angle corners. The circumference of a rectangle with side lengths of x and y is:
The area of a rectangle is:
Circle
The circle is a shape where all points along the shape are equal distance from a specific point. This point is the center of the circle and the distance to the center of the circle is the radius. The circumference of a circle of radius r is:
The area of a circle is:
Triangle
The triangle is a 3 sided polygon. Triangles can be classified by their sides:
Equilateral triangles: All sides are equal in length.
Isosceles triangles: Two sides are equal in length.
Scalene triangles: All sides have different lengths.
Triangles can also be classified by their angles:
Right triangle: One angle is 90 degrees.
Oblique triangle: Has no angle equal to 90 degrees.
Obtuse triangle: One angle is greater than 90 degrees.
Acute triangle: All angles are less than 90 degrees.
The circumference of a triangle is the sum of all sides of the triangle.
The area of a triangle is determined by its base and height.
Parallelogram
A parallelogram is a 4 sided polygon or quadrilateral with two sets of parallel sides. The opposite sides are equal in length. The circumference of a parallelogram is:
The area of a parallelogram is:
Circular Sector
The circular sector is section of a circle enclosed by two radii. The length of the arc of a circular sector, where r is the radii and theta is the angle, is:
The area of the circular sector is:
Circular Ring
The circular ring or donut shape is composed of a circle with a centered small circle removed from the area. The area of a circular ring, where R is the outer diameter and r is the inner diameter, is:
Trapezoid
A trapezoid is a 4 sided polygon or quadrilateral with one set of parallel sides. The circumference of the trapezoid is:
The area of a trapezoid is:
Sphere
The sphere is a 3 dimensional object, whose surface is continuous and all points of the surface are an equal distance from a fixed point, the center. The surface area of a sphere, where r is the radius, is:
The volume of a sphere is:
Cube
A cube is a three dimensional object bounded by 6 equal square sides. The surface area of a cube , where the length of a side is l, is:
The volume of a cube is:
Rectangular Solid
A rectangular solid is a three dimensional object bounded by rectangular or square sides. The surface area of a rectangular solid, where the lengths of the sides are a, b, c, is:
The volume of a rectangular solid is:
Cylinder
The cylinder, also know as the right circular cylinder, is formed by rotating a line, of length h, around a fixed axis parallel to that line. A cylinder has two ends that are equally sized circles parallel to one another and the circular side is at a right angle to these circular ends. The surface area of cylinder is:
The volume of a cylinder is:
Right Circular Cone
The right circular cone is a three dimensional object created by rotating a right triangle about the vertical side. The surface area of a right circular cone is:
Where the slant height, s is:
The volume of a right circular cone is:
Frustum of a Cone
The frustum of a cone is a portion of cone with the top removed. The top and bottom surfaces are parallel to one another. The volume of a frustum of a cone, where the top radius is r and the bottom radius is R, is:
Pythagorean Theorem
The Pythagorean theorem is based upon the right triangle. And If the three sides of the right triangle are a,b, and c, where c is the hypotenuse, the formula is: