8.G

Geometry

8.G.1Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:8.G.10Use the formulas or informal reasoning to find the arc length, areas of sectors, surface areas and volumes of pyramids, cones, and spheres. For example, given a circle with a 60 central angle, students identify the arc length as 1/6 of the total circumference (1/6 = 60/360).8.G.11Investigate the relationship between the formulas of three dimensional geometric shapes;8.G.11aGeneralize the volume formula for pyramids and cones (V = (1/3)Bh).8.G.11bGeneralize surface area formula of pyramids and cones (SA = B + (1/2)Pl).8.G.12Solve real-world and mathematical problems involving arc length, area of two-dimensional shapes including sectors, volume and surface area of three-dimensional objects including pyramids, cones and spheres.8.G.1aAn angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one -degree angle,” and can be used to measure angles.8.G.1bAn angle that turns through n one-degree angles is said to have an angle measure of n degrees.8.G.2Measure angles in whole-number degrees using a protractor. Draw angles of specified measure using a protractor and straight edge.8.G.3Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g. by using an equation with a symbol for the unknown angle measure.8.G.4Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and use them to solve simple equations for an unknown angle in a figure.8.G.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.8.G.6Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on drawing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.8.G.7Explain a proof of the Pythagorean Theorem and its converse.8.G.8Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. For example: Finding the slant height of pyramids and cones.8.G.9Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
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