Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
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๐ = included in formula booklet โข ๐ซ = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
๐ = included in formula booklet โข ๐ซ = not in formula booklet
2D & 3D Geometry
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
๐ = included in formula booklet โข ๐ซ = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
๐ = included in formula booklet โข ๐ซ = not in formula booklet
The area of a triangle is given by
where b is the base and h is the height.
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In a right angled triangle with sides a, b and hypotenuse (longest side) c, Pythagoras' Theorem states
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In a right angled triangle with an angle ฮธ<90ยฐ, the trigonometric ratios sin, cos and tan are defined by
where opposite and adjacent refer to the side lengths of the sides opposite and adjacent to ฮธ, while hypotenuse is the length of the longest side.
In any triangle ABC, the angles add up to 180ยฐ:
Example
A triangle has angles 35ยฐ and 55ยฐ. Find the missing angle.
so x=180โ35โ55=90ยฐ.
If we know the value of sinฮธ, cosฮธ or tanฮธ in a right angled triangle, we can find ฮธ using an inverse trigonometric function on a calculator. These functions are sinโ1, cosโ1 and tanโ1 and satisfy
whenever ฮธ<90ยฐ, which is always true in a right angled triangle.
The trigonometric ratios sin, cos and tan are actually functions that relate an angle ฮธ to a ratio of sides. The values of sin, cos and tan for specific angles can be found on the calculator. For example
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When we know two side lengths and the angle between the two sides, we can find the area even if we don't directly know the height by using the fact that
Thus
The previously found formula for area
applies to any pair of sides and the angle between them. Since the area is the same no matter which sides we use:
Multiplying everything by 2 and dividing by abc:
Flipping the numerator and denominator gives the form that appears in the formula booklet:
The sine rule is primarily used when we know two angles and a side. When we know two sides and an angle, the version with angles in the numerator is easier to work with.
The cosine rule is a generalization of Pythagoras' theorem for non-right-angled triangles. It states that
The cosine rule is primarily used when we
know two sides and the angle between them, and want to find the third side,
know all three sides and want to find an angle.
The ambiguous case of the sine rule is a consequence of the symmetry of the sin function:
Therefore, if a given triangle has a specific sin(A) and it is not specified whether A is acute / obtuse, then there are two possible lengths for the side opposite A:
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When we type sinโ1(x) into the calculator, it will always return an acute angle. To find the corresponding obtuse angle, we take
Angles of elevation and depression describe how far upward or downward you look relative to a horizontal line when observing an object.
The angle of elevation is the angle formed by looking upward from the horizontal line to an object above your line of sight.
The angle of depression is the angle formed by looking downward from the horizontal line to an object below your line of sight.
These angles are always measured relative to a horizontal line, never vertical. Because the lines of sight form alternate interior angles with horizontal lines, the angle of elevation from one viewpoint equals the angle of depression from the other viewpoint.
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A true bearing describes a direction measured clockwise from the north direction (0ยฐ) around a full circle up to 360ยฐ. Bearings are always given using three digits (e.g. 045ยฐ,120ยฐ,270ยฐ) to avoid confusion.
A bearing of 000โ points directly north.
090โ points east.
180โ points south.
270โ points west.
When working with true bearings, clearly draw a compass rose to visualize directions and measure angles clockwise from the north line.
One very useful fact about bearings is that returning in the direction something came from means adding or subtracting 180ยฐ from the original bearing - since it is doing a U-turn.
Whether you add or subtract depends on whether the original bearing is smaller or bigger than 180ยฐ, since the resulting bearing must be between 0 and 360ยฐ.
The ratio of a circle's perimeter to its diameter is constant in all circles. This constant is called ฯ (pi). Since the diameter is twice the radius, the circumference of a circle is
The area of a circle is
where r is the radius of the circle.
An arc is defined by the radius r of the circle and the angle ฮธ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle ฮธ, the arc length is calculated as
Since the perimeter of a full circle is 2ฯr, the angle ฮธ corresponding to a full circle (360ยฐ) is
So
Some key angles in radians and degrees:
An arc is part of the circumference of a circle. It is defined by the radius r of the circle and the angle ฮธ that the arc "sweeps out" over the circle's perimeter.
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Since the arc length is a fraction of the overall circumference determined by the value of the angle ฮธ, the arc length is calculated as
A sector is an area enclosed between a circular arc and the two radii of the circle touching each end of the arc:
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The area of a circle is ฯr2, and there are 2ฯ radians in a circle. Therefore, a sector with central angle ฮธ is 2ฯฮธโ of a full circle, and has area
so
One radian is the interior angle of an arc which has a length equivalent to the radius r of the circle. Since the circumference of a circle is given by 2ฯr, then, there are 2ฯ total radians in a circle (the equivalent of 360ยฐ).
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A sphere is a perfectly round, three-dimensional geometric shape where every point on its surface is exactly the same distance (the radius) from a single central point. It's the three-dimensional analog of a circle. For example, a ball or globe is spherical in shape. The surface area A and volume V of a sphere are given by:
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A cylinder is a three-dimensional geometric shape formed by two identical circular bases connected by a curved lateral surface. The segment connecting the centers of the circular bases is called the axis, which is perpendicular to each base in a right cylinder (the type usually studied).
The volume V of a cylinder with radius r is given by:
The curved surface of a cylinder (excluding the circular ends) is given by:
If we include the circular ends, each with area ฯr2, we get
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A prism is a three-dimensional solid shape consisting of two parallel, congruent faces called bases, connected by rectangular lateral faces. Prisms are named according to the shape of their basesโfor example, triangular prism, rectangular prism, or hexagonal prism.
The volume V of a prism is calculated by multiplying the area A of its base by its height h:
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A right circular cone is a three-dimensional geometric shape whose apex (vertex) lies directly above the center of its circular base.
Key Parts:
Circular Base: Flat circle with radius r.
Apex (Vertex): The point directly above the center of the base.
Height (h): Perpendicular distance from apex to base center.
Slant Height (l): Distance along the cone's surface from apex to edge of base.
Formulas:
Volume:
Surface Area of curved surface:
Slant Height Relationship:
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A right pyramid is a three-dimensional shape with a polygonal base and triangular lateral faces, in which the apex (vertex) is located directly above the center (centroid) of the base.
Key Parts:
Polygonal base: a flat polygon (triangle, square, pentagon, etc.)
Apex (vertex): the point positioned vertically above the base's centroid
Height (h): perpendicular distance from apex to base centroid
Slant height (l): distance along a lateral face from the apex perpendicular to an edge of the base
The volume of a right pyramid is given by
where A is the area of the base.
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