Area, Circumference, and Volume Formulas Here are the basic formulas for finding area, circumference, and volume. They will be discussed in detail in the following sections. Triangles The sum of the measures of the three angles in a triangle always equals 180 degrees. a b c a + b + c = 180° Circle Rectangle Triangle r l w h b A = lw A = 1 _ 2 bh C = 2πr A = πr 2 Cylinder Rectangular Solid h l V = πr 2 h w r h V = lwh C = Circumference A = Area r = Radius l = Length w = Width h = Height v = Volume b = Base – THEA MATH REVIEW– 155 Exterior Angles An exterior angle can be formed by extending a side from any of the three vertices of a triangle. Here are some rules for working with exterior angles: ■ An exterior angle and an interior angle that share the same vertex are supplementary. In other words, exterior angles and interior angles form straight lines with each other. ■ An exterior angle is equal to the sum of the non-adjacent interior angles. ■ The sum of the exterior angles of a triangle equals 360 degrees. Example m∠1 + m∠2 = 180° m∠1 = m∠3 + m∠5 m∠3 + m∠4 = 180° m∠4 = m∠2 + m∠5 m∠5 + m∠6 = 180° m∠6 = m∠3 + m∠2 m∠1 + m∠4 + m∠6 = 360° C LASSIFYING TRIANGLES It is possible to classify triangles into three categories based on the number of congruent (indicated by the sym- bol: ≅) sides. Sides are congruent when they have equal lengths. Scalene Triangle Isosceles Triangle Equilateral Triangle no sides congruent more than 2 congruent sides all sides congruent It is also possible to classify triangles into three categories based on the measure of the greatest angle: Acute Triangle Right Triangle Obtuse Triangle greatest angle is acute greatest angle is 90° greatest angle is obtuse 1 2 356 4 – THEA MATH REVIEW– 156 ANGLE-SIDE RELATIONSHIPS Knowing the angle-side relationships in isosceles, equilateral, and right triangles is helpful. ■ In isosceles triangles, congruent angles are opposite congruent sides. ■ In equilateral triangles, all sides are congruent and all angles are congruent. The measure of each angle in an equilateral triangle is always 60°. ■ In a right triangle, the side opposite the right angle is called the hypotenuse and the other sides are called legs. The box in the angle of the 90-degree angle symbolizes that the triangle is, in fact, a right triangle. Hypotenuse Leg Leg 60° 60° 60° x x x 66 48° 48° – THEA MATH REVIEW– 157 Pythagorean Theorem The Pythagorean theorem is an important tool for working with right triangles. It states: a 2 + b 2 = c 2 ,where a and b represent the legs and c represents the hypotenuse. This theorem makes it easy to find the length of any side as long as the measure of two sides is known. So, if leg a = 1 and leg b = 2 in the triangle below, it is possible to find the measure of the hypotenuse, c. a 2 + b 2 = c 2 1 2 + 2 2 = c 2 1 + 4 = c 2 5=c 2 ͙5 ෆ = c P YTHAGOREAN TRIPLES Sometimes, the measures of all three sides of a right triangle are integers. If three integers are the lengths of a right triangle, we call them Pythagorean triples. Some popular Pythagorean triples are: 3, 4, 5 5, 12, 13 8, 15, 17 9, 40, 41 The smaller two numbers in each triple represent the length of the legs, and the largest number represents the length of the hypotenuse. MULTIPLES OF PYTHAGOREAN TRIPLES Whole-number multiples of each triple are also triples. For example, if we multiply each of the lengths of the triple 3, 4, 5 by 2, we get 6, 8, 10. This is also a triple. Example If given a right triangle with sides measuring 6, x, and a hypotenuse 10, what is the value of x? 3, 4, 5 is a Pythagorean triple, and a multiple of that is 6, 8, 10. Therefore, the missing side length is 8. 1 2 c – THEA MATH REVIEW– 158 COMPARING TRIANGLES Triangles are said to be congruent (indicated by the symbol: ≅) when they have exactly the same size and shape. Two triangles are congruent if their corresponding parts (their angles and sides) are congruent. Sometimes, it is easy to tell if two triangles are congruent by looking at them. However, in geometry, it must be able to be proven that the triangles are congruent. There are a number of ways to prove that two triangles are congruent: Side-Side-Side (SSS) If the three sides of one triangle are congruent to the three corresponding sides of another triangle, the triangles are congruent. Side-Angle-Side (SAS) If two sides and the included angle of one triangle are congruent to the cor- responding two sides and included angle of another triangle, the triangles are congruent. Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to the cor- responding two angles and included side of another triangle, the triangles are congruent. Used less often but also valid: Angle-Angle-Side (AAS) If two angles and the non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, the triangles are congruent. Hypotenuse-Leg (Hy-Leg) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, the triangles are congruent. 30˚ 30˚ SAS ≅ SAS 5' 5' 7' 7' 30˚ 30˚ ASA ≅ ASA 5' 5' SSS ≅ SSS 5' 5' 7' 7' 30˚ 30˚ AAS ≅ AAS 7' 7' Hy-Leg ≅ Hy-Leg 6' 6' 10' 10' 50˚ 50˚ 9' 9' 50˚ 50˚ ≅ ≅ ≅ ≅ ≅ – THEA MATH REVIEW– 159 Example Determine if these two triangles are congruent. Although the triangles are not aligned the same, there are two congruent corresponding sides, and the angle between them (150°) is congruent. Therefore, the triangles are congruent by the SAS pos- tulate. Example Determine if these two triangles are congruent. Although the triangles have two congruent corresponding sides, and a corresponding congruent angle, the 150° angle is not included between them. This would be “SSA,” but SSA is not a way to prove that two triangles are congruent. Area of a Triangle Area is the amount of space inside a two-dimensional object. Area is measured in square units, often written as unit 2 . So, if the length of a triangle is measured in feet, the area of the triangle is measured in feet 2 . A triangle has three sides, each of which can be considered a base of the triangle. A perpendicular line seg- ment drawn from a vertex to the opposite base of the triangle is called the altitude, or the height. It measures how tall the triangle stands. It is important to note that the height of a triangle is not necessarily one of the sides of the triangle. The cor- rect height for the following triangle is 8, not 10. The height will always be associated with a line segment (called an altitude) that comes from one vertex (angle) to the opposite side and forms a right angle (signified by the box). In other words, the height must always be perpendicular to (form a right angle with) the base. Note that in an obtuse triangle, the height is outside the triangle, and in a right triangle the height is one of the sides. Obtuse Triangle b h Right Triangle b h b h Acute Triangle 8" 150˚ 11" 8" 150˚ 11" 8" 150˚ 6" 8" 150˚ 6" – THEA MATH REVIEW– 160 . 360 degrees. Example m∠1 + m∠2 = 180° m∠1 = m∠3 + m∠5 m∠3 + m 4 = 180° m 4 = m∠2 + m∠5 m∠5 + m∠6 = 180° m∠6 = m∠3 + m∠2 m∠1 + m 4 + m∠6 = 360° C LASSIFYING TRIANGLES It is possible to classify. right triangle, we call them Pythagorean triples. Some popular Pythagorean triples are: 3, 4, 5 5, 12, 13 8, 15, 17 9, 40 , 41 The smaller two numbers in each triple represent the length of the legs, and. symbolizes that the triangle is, in fact, a right triangle. Hypotenuse Leg Leg 60° 60° 60° x x x 66 48 ° 48 ° – THEA MATH REVIEW– 157 Pythagorean Theorem The Pythagorean theorem is an important tool for