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Deriving Formulas Worksheet
Directions: For each of the following problems, you are to derive the specified area formula using the
given area formulas. Be sure to include diagrams for each problem.
1. Derive the formula for the area of an equilateral triangle using the formula for a triangle.
Given: Atriangle =
where b = base of ∆ and h = height/altitude of ∆ and
equilateral ∆
Prove: Aequilateraltriangle =
Since the ∆ is equilateral, all sides are congruent and all angles are congruent. Therefore all sides are s
and all angles are 60.
2. Prove that the area of a trapezoid is ½ h(b1 + b2) by drawing altitudes and use the rectangle and the
triangles formed.
Given: Atriangle =
where b = base of ∆ and h = height/altitude of ∆
Arectangle = bh where b = base of rectangle and h = height/altitude of
rectangle.
Prove: Atrapezoid =
The trapezoid can be split into 3 shapes when the altitudes are drawn in: 2 congruent triangles
(assuming isosceles trapezoid) and 1 rectangle.
3. Prove that the area of a trapezoid is ½ h(b1 + b2) by drawing a diagonal and use the two triangles
formed.
Given: Atriangle =
where b = base of ∆ and h = height/altitude of ∆
Diagonal of Trapezoid splits trapezoid into 2 ∆
Prove: Atrapezoid =
The trapezoid can be split into 2 triangles when 1 diagonal is used.
4. Derive the formula for the area of a rhombus using the formula for the area of a triangle.
Given: Atriangle =
where b = base of ∆ and h = height/altitude of ∆
Prove: Arhombus =
A rhombus is made up of 2 congruent triangles. Diagonals of rhombus are perpendicular bisectors
of each other.
Half of 1 diagonal serves as the height of a triangle while the other diagonal serves as the base.
Deriving Formulas WS
Rev A
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Period:
Date:
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5. Derive a formula to find the area of any triangle, given the measures of two sides of the triangle and
their included angle.
Given: Atriangle =
where b = base of ∆ and h = height/altitude of ∆
Find: Area of any triangle
c is the base of the triangle and h is the height of the triangle. Need to find h. Hint use trig ratios.
Deriving Formulas WS
Rev A